Provided by: liblapack-doc_3.11.0-2build1_all bug

NAME

       complexOTHERauxiliary - complex

SYNOPSIS

   Functions
       subroutine clabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
           CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
       subroutine clacgv (N, X, INCX)
           CLACGV conjugates a complex vector.
       subroutine clacn2 (N, V, X, EST, KASE, ISAVE)
           CLACN2 estimates the 1-norm of a square matrix, using reverse communication for
           evaluating matrix-vector products.
       subroutine clacon (N, V, X, EST, KASE)
           CLACON estimates the 1-norm of a square matrix, using reverse communication for
           evaluating matrix-vector products.
       subroutine clacp2 (UPLO, M, N, A, LDA, B, LDB)
           CLACP2 copies all or part of a real two-dimensional array to a complex array.
       subroutine clacpy (UPLO, M, N, A, LDA, B, LDB)
           CLACPY copies all or part of one two-dimensional array to another.
       subroutine clacrm (M, N, A, LDA, B, LDB, C, LDC, RWORK)
           CLACRM multiplies a complex matrix by a square real matrix.
       subroutine clacrt (N, CX, INCX, CY, INCY, C, S)
           CLACRT performs a linear transformation of a pair of complex vectors.
       complex function cladiv (X, Y)
           CLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
       subroutine claein (RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK, EPS3, SMLNUM, INFO)
           CLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by
           inverse iteration.
       subroutine claev2 (A, B, C, RT1, RT2, CS1, SN1)
           CLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian
           matrix.
       subroutine clags2 (UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
           CLAGS2
       subroutine clagtm (TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB)
           CLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a
           tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which
           may be 0, 1, or -1.
       subroutine clahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)
           CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,
           using the double-shift/single-shift QR algorithm.
       subroutine clahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
           CLAHR2 reduces the specified number of first columns of a general rectangular matrix A
           so that elements below the specified subdiagonal are zero, and returns auxiliary
           matrices which are needed to apply the transformation to the unreduced part of A.
       subroutine claic1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
           CLAIC1 applies one step of incremental condition estimation.
       real function clangt (NORM, N, DL, D, DU)
           CLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
           absolute value of any element of a general tridiagonal matrix.
       real function clanhb (NORM, UPLO, N, K, AB, LDAB, WORK)
           CLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
           or the element of largest absolute value of a Hermitian band matrix.
       real function clanhp (NORM, UPLO, N, AP, WORK)
           CLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
           or the element of largest absolute value of a complex Hermitian matrix supplied in
           packed form.
       real function clanhs (NORM, N, A, LDA, WORK)
           CLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
           absolute value of any element of an upper Hessenberg matrix.
       real function clanht (NORM, N, D, E)
           CLANHT returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
           or the element of largest absolute value of a complex Hermitian tridiagonal matrix.
       real function clansb (NORM, UPLO, N, K, AB, LDAB, WORK)
           CLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
           or the element of largest absolute value of a symmetric band matrix.
       real function clansp (NORM, UPLO, N, AP, WORK)
           CLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
           or the element of largest absolute value of a symmetric matrix supplied in packed
           form.
       real function clantb (NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
           CLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
           or the element of largest absolute value of a triangular band matrix.
       real function clantp (NORM, UPLO, DIAG, N, AP, WORK)
           CLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
           or the element of largest absolute value of a triangular matrix supplied in packed
           form.
       real function clantr (NORM, UPLO, DIAG, M, N, A, LDA, WORK)
           CLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
           or the element of largest absolute value of a trapezoidal or triangular matrix.
       subroutine clapll (N, X, INCX, Y, INCY, SSMIN)
           CLAPLL measures the linear dependence of two vectors.
       subroutine clapmr (FORWRD, M, N, X, LDX, K)
           CLAPMR rearranges rows of a matrix as specified by a permutation vector.
       subroutine clapmt (FORWRD, M, N, X, LDX, K)
           CLAPMT performs a forward or backward permutation of the columns of a matrix.
       subroutine claqhb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
           CLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ.
       subroutine claqhp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
           CLAQHP scales a Hermitian matrix stored in packed form.
       subroutine claqp2 (M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)
           CLAQP2 computes a QR factorization with column pivoting of the matrix block.
       subroutine claqps (M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
           CLAQPS computes a step of QR factorization with column pivoting of a real m-by-n
           matrix A by using BLAS level 3.
       subroutine claqr0 (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK,
           INFO)
           CLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
           from the Schur decomposition.
       subroutine claqr1 (N, H, LDH, S1, S2, V)
           CLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3
           matrix H and specified shifts.
       subroutine claqr2 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND,
           SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
           CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect
           and deflate fully converged eigenvalues from a trailing principal submatrix
           (aggressive early deflation).
       subroutine claqr3 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND,
           SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
           CLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect
           and deflate fully converged eigenvalues from a trailing principal submatrix
           (aggressive early deflation).
       subroutine claqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK,
           INFO)
           CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
           from the Schur decomposition.
       subroutine claqr5 (WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, S, H, LDH, ILOZ, IHIZ, Z,
           LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)
           CLAQR5 performs a single small-bulge multi-shift QR sweep.
       subroutine claqsb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
           CLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by
           spbequ.
       subroutine claqsp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
           CLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors
           computed by sppequ.
       subroutine clar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT,
           ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
           CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1
           through bn of the tridiagonal matrix LDLT - λI.
       subroutine clar2v (N, X, Y, Z, INCX, C, S, INCC)
           CLAR2V applies a vector of plane rotations with real cosines and complex sines from
           both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
       subroutine clarcm (M, N, A, LDA, B, LDB, C, LDC, RWORK)
           CLARCM copies all or part of a real two-dimensional array to a complex array.
       subroutine clarf (SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
           CLARF applies an elementary reflector to a general rectangular matrix.
       subroutine clarfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK,
           LDWORK)
           CLARFB applies a block reflector or its conjugate-transpose to a general rectangular
           matrix.
       subroutine clarfb_gett (IDENT, M, N, K, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
           CLARFB_GETT
       subroutine clarfg (N, ALPHA, X, INCX, TAU)
           CLARFG generates an elementary reflector (Householder matrix).
       subroutine clarfgp (N, ALPHA, X, INCX, TAU)
           CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
       subroutine clarft (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
           CLARFT forms the triangular factor T of a block reflector H = I - vtvH
       subroutine clarfx (SIDE, M, N, V, TAU, C, LDC, WORK)
           CLARFX applies an elementary reflector to a general rectangular matrix, with loop
           unrolling when the reflector has order ≤ 10.
       subroutine clarfy (UPLO, N, V, INCV, TAU, C, LDC, WORK)
           CLARFY
       subroutine clargv (N, X, INCX, Y, INCY, C, INCC)
           CLARGV generates a vector of plane rotations with real cosines and complex sines.
       subroutine clarnv (IDIST, ISEED, N, X)
           CLARNV returns a vector of random numbers from a uniform or normal distribution.
       subroutine clarrv (N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MINRGP, RTOL1, RTOL2, W,
           WERR, WGAP, IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO)
           CLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and
           the eigenvalues of L D LT.
       subroutine clartv (N, X, INCX, Y, INCY, C, S, INCC)
           CLARTV applies a vector of plane rotations with real cosines and complex sines to the
           elements of a pair of vectors.
       subroutine clascl (TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
           CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
       subroutine claset (UPLO, M, N, ALPHA, BETA, A, LDA)
           CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to
           given values.
       subroutine clasr (SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA)
           CLASR applies a sequence of plane rotations to a general rectangular matrix.
       subroutine claswp (N, A, LDA, K1, K2, IPIV, INCX)
           CLASWP performs a series of row interchanges on a general rectangular matrix.
       subroutine clatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
           CLATBS solves a triangular banded system of equations.
       subroutine clatdf (IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)
           CLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes
           a contribution to the reciprocal Dif-estimate.
       subroutine clatps (UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
           CLATPS solves a triangular system of equations with the matrix held in packed storage.
       subroutine clatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
           CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real
           tridiagonal form by an unitary similarity transformation.
       subroutine clatrs (UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
           CLATRS solves a triangular system of equations with the scale factor set to prevent
           overflow.
       subroutine clauu2 (UPLO, N, A, LDA, INFO)
           CLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular
           matrices (unblocked algorithm).
       subroutine clauum (UPLO, N, A, LDA, INFO)
           CLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular
           matrices (blocked algorithm).
       subroutine crot (N, CX, INCX, CY, INCY, C, S)
           CROT applies a plane rotation with real cosine and complex sine to a pair of complex
           vectors.
       subroutine cspmv (UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
           CSPMV computes a matrix-vector product for complex vectors using a complex symmetric
           packed matrix
       subroutine cspr (UPLO, N, ALPHA, X, INCX, AP)
           CSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.
       subroutine csrscl (N, SA, SX, INCX)
           CSRSCL multiplies a vector by the reciprocal of a real scalar.
       subroutine ctprfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B,
           LDB, WORK, LDWORK)
           CTPRFB applies a complex 'triangular-pentagonal' block reflector to a complex matrix,
           which is composed of two blocks.
       integer function icmax1 (N, CX, INCX)
           ICMAX1 finds the index of the first vector element of maximum absolute value.
       integer function ilaclc (M, N, A, LDA)
           ILACLC scans a matrix for its last non-zero column.
       integer function ilaclr (M, N, A, LDA)
           ILACLR scans a matrix for its last non-zero row.
       integer function izmax1 (N, ZX, INCX)
           IZMAX1 finds the index of the first vector element of maximum absolute value.
       real function scsum1 (N, CX, INCX)
           SCSUM1 forms the 1-norm of the complex vector using the true absolute value.

Detailed Description

       This is the group of complex other auxiliary routines

Function Documentation

   subroutine clabrd (integer M, integer N, integer NB, complex, dimension( lda, * ) A, integer
       LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAUQ,
       complex, dimension( * ) TAUP, complex, dimension( ldx, * ) X, integer LDX, complex,
       dimension( ldy, * ) Y, integer LDY)
       CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

       Purpose:

            CLABRD reduces the first NB rows and columns of a complex general
            m by n matrix A to upper or lower real bidiagonal form by a unitary
            transformation Q**H * A * P, and returns the matrices X and Y which
            are needed to apply the transformation to the unreduced part of A.

            If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
            bidiagonal form.

            This is an auxiliary routine called by CGEBRD

       Parameters
           M

                     M is INTEGER
                     The number of rows in the matrix A.

           N

                     N is INTEGER
                     The number of columns in the matrix A.

           NB

                     NB is INTEGER
                     The number of leading rows and columns of A to be reduced.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the m by n general matrix to be reduced.
                     On exit, the first NB rows and columns of the matrix are
                     overwritten; the rest of the array is unchanged.
                     If m >= n, elements on and below the diagonal in the first NB
                       columns, with the array TAUQ, represent the unitary
                       matrix Q as a product of elementary reflectors; and
                       elements above the diagonal in the first NB rows, with the
                       array TAUP, represent the unitary matrix P as a product
                       of elementary reflectors.
                     If m < n, elements below the diagonal in the first NB
                       columns, with the array TAUQ, represent the unitary
                       matrix Q as a product of elementary reflectors, and
                       elements on and above the diagonal in the first NB rows,
                       with the array TAUP, represent the unitary matrix P as
                       a product of elementary reflectors.
                     See Further Details.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,M).

           D

                     D is REAL array, dimension (NB)
                     The diagonal elements of the first NB rows and columns of
                     the reduced matrix.  D(i) = A(i,i).

           E

                     E is REAL array, dimension (NB)
                     The off-diagonal elements of the first NB rows and columns of
                     the reduced matrix.

           TAUQ

                     TAUQ is COMPLEX array, dimension (NB)
                     The scalar factors of the elementary reflectors which
                     represent the unitary matrix Q. See Further Details.

           TAUP

                     TAUP is COMPLEX array, dimension (NB)
                     The scalar factors of the elementary reflectors which
                     represent the unitary matrix P. See Further Details.

           X

                     X is COMPLEX array, dimension (LDX,NB)
                     The m-by-nb matrix X required to update the unreduced part
                     of A.

           LDX

                     LDX is INTEGER
                     The leading dimension of the array X. LDX >= max(1,M).

           Y

                     Y is COMPLEX array, dimension (LDY,NB)
                     The n-by-nb matrix Y required to update the unreduced part
                     of A.

           LDY

                     LDY is INTEGER
                     The leading dimension of the array Y. LDY >= max(1,N).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             The matrices Q and P are represented as products of elementary
             reflectors:

                Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

             Each H(i) and G(i) has the form:

                H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

             where tauq and taup are complex scalars, and v and u are complex
             vectors.

             If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
             A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
             A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

             If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
             A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
             A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

             The elements of the vectors v and u together form the m-by-nb matrix
             V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
             the transformation to the unreduced part of the matrix, using a block
             update of the form:  A := A - V*Y**H - X*U**H.

             The contents of A on exit are illustrated by the following examples
             with nb = 2:

             m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

               (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
               (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
               (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
               (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
               (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
               (  v1  v2  a   a   a  )

             where a denotes an element of the original matrix which is unchanged,
             vi denotes an element of the vector defining H(i), and ui an element
             of the vector defining G(i).

   subroutine clacgv (integer N, complex, dimension( * ) X, integer INCX)
       CLACGV conjugates a complex vector.

       Purpose:

            CLACGV conjugates a complex vector of length N.

       Parameters
           N

                     N is INTEGER
                     The length of the vector X.  N >= 0.

           X

                     X is COMPLEX array, dimension
                                    (1+(N-1)*abs(INCX))
                     On entry, the vector of length N to be conjugated.
                     On exit, X is overwritten with conjg(X).

           INCX

                     INCX is INTEGER
                     The spacing between successive elements of X.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clacn2 (integer N, complex, dimension( * ) V, complex, dimension( * ) X, real EST,
       integer KASE, integer, dimension( 3 ) ISAVE)
       CLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating
       matrix-vector products.

       Purpose:

            CLACN2 estimates the 1-norm of a square, complex matrix A.
            Reverse communication is used for evaluating matrix-vector products.

       Parameters
           N

                     N is INTEGER
                    The order of the matrix.  N >= 1.

           V

                     V is COMPLEX array, dimension (N)
                    On the final return, V = A*W,  where  EST = norm(V)/norm(W)
                    (W is not returned).

           X

                     X is COMPLEX array, dimension (N)
                    On an intermediate return, X should be overwritten by
                          A * X,   if KASE=1,
                          A**H * X,  if KASE=2,
                    where A**H is the conjugate transpose of A, and CLACN2 must be
                    re-called with all the other parameters unchanged.

           EST

                     EST is REAL
                    On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
                    unchanged from the previous call to CLACN2.
                    On exit, EST is an estimate (a lower bound) for norm(A).

           KASE

                     KASE is INTEGER
                    On the initial call to CLACN2, KASE should be 0.
                    On an intermediate return, KASE will be 1 or 2, indicating
                    whether X should be overwritten by A * X  or A**H * X.
                    On the final return from CLACN2, KASE will again be 0.

           ISAVE

                     ISAVE is INTEGER array, dimension (3)
                    ISAVE is used to save variables between calls to SLACN2

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             Originally named CONEST, dated March 16, 1988.

             Last modified:  April, 1999

             This is a thread safe version of CLACON, which uses the array ISAVE
             in place of a SAVE statement, as follows:

                CLACON     CLACN2
                 JUMP     ISAVE(1)
                 J        ISAVE(2)
                 ITER     ISAVE(3)

       Contributors:
           Nick Higham, University of Manchester

       References:
           N.J. Higham, 'FORTRAN codes for estimating the one-norm of
             a real or complex matrix, with applications to condition estimation', ACM Trans.
           Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

   subroutine clacon (integer N, complex, dimension( n ) V, complex, dimension( n ) X, real EST,
       integer KASE)
       CLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating
       matrix-vector products.

       Purpose:

            CLACON estimates the 1-norm of a square, complex matrix A.
            Reverse communication is used for evaluating matrix-vector products.

       Parameters
           N

                     N is INTEGER
                    The order of the matrix.  N >= 1.

           V

                     V is COMPLEX array, dimension (N)
                    On the final return, V = A*W,  where  EST = norm(V)/norm(W)
                    (W is not returned).

           X

                     X is COMPLEX array, dimension (N)
                    On an intermediate return, X should be overwritten by
                          A * X,   if KASE=1,
                          A**H * X,  if KASE=2,
                    where A**H is the conjugate transpose of A, and CLACON must be
                    re-called with all the other parameters unchanged.

           EST

                     EST is REAL
                    On entry with KASE = 1 or 2 and JUMP = 3, EST should be
                    unchanged from the previous call to CLACON.
                    On exit, EST is an estimate (a lower bound) for norm(A).

           KASE

                     KASE is INTEGER
                    On the initial call to CLACON, KASE should be 0.
                    On an intermediate return, KASE will be 1 or 2, indicating
                    whether X should be overwritten by A * X  or A**H * X.
                    On the final return from CLACON, KASE will again be 0.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:
           Originally named CONEST, dated March 16, 1988.
            Last modified: April, 1999

       Contributors:
           Nick Higham, University of Manchester

       References:
           N.J. Higham, 'FORTRAN codes for estimating the one-norm of
             a real or complex matrix, with applications to condition estimation', ACM Trans.
           Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

   subroutine clacp2 (character UPLO, integer M, integer N, real, dimension( lda, * ) A, integer
       LDA, complex, dimension( ldb, * ) B, integer LDB)
       CLACP2 copies all or part of a real two-dimensional array to a complex array.

       Purpose:

            CLACP2 copies all or part of a real two-dimensional matrix A to a
            complex matrix B.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies the part of the matrix A to be copied to B.
                     = 'U':      Upper triangular part
                     = 'L':      Lower triangular part
                     Otherwise:  All of the matrix A

           M

                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrix A.  N >= 0.

           A

                     A is REAL array, dimension (LDA,N)
                     The m by n matrix A.  If UPLO = 'U', only the upper trapezium
                     is accessed; if UPLO = 'L', only the lower trapezium is
                     accessed.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,M).

           B

                     B is COMPLEX array, dimension (LDB,N)
                     On exit, B = A in the locations specified by UPLO.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,M).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clacpy (character UPLO, integer M, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( ldb, * ) B, integer LDB)
       CLACPY copies all or part of one two-dimensional array to another.

       Purpose:

            CLACPY copies all or part of a two-dimensional matrix A to another
            matrix B.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies the part of the matrix A to be copied to B.
                     = 'U':      Upper triangular part
                     = 'L':      Lower triangular part
                     Otherwise:  All of the matrix A

           M

                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrix A.  N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The m by n matrix A.  If UPLO = 'U', only the upper trapezium
                     is accessed; if UPLO = 'L', only the lower trapezium is
                     accessed.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,M).

           B

                     B is COMPLEX array, dimension (LDB,N)
                     On exit, B = A in the locations specified by UPLO.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,M).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clacrm (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real,
       dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, real,
       dimension( * ) RWORK)
       CLACRM multiplies a complex matrix by a square real matrix.

       Purpose:

            CLACRM performs a very simple matrix-matrix multiplication:
                     C := A * B,
            where A is M by N and complex; B is N by N and real;
            C is M by N and complex.

       Parameters
           M

                     M is INTEGER
                     The number of rows of the matrix A and of the matrix C.
                     M >= 0.

           N

                     N is INTEGER
                     The number of columns and rows of the matrix B and
                     the number of columns of the matrix C.
                     N >= 0.

           A

                     A is COMPLEX array, dimension (LDA, N)
                     On entry, A contains the M by N matrix A.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A. LDA >=max(1,M).

           B

                     B is REAL array, dimension (LDB, N)
                     On entry, B contains the N by N matrix B.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B. LDB >=max(1,N).

           C

                     C is COMPLEX array, dimension (LDC, N)
                     On exit, C contains the M by N matrix C.

           LDC

                     LDC is INTEGER
                     The leading dimension of the array C. LDC >=max(1,N).

           RWORK

                     RWORK is REAL array, dimension (2*M*N)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clacrt (integer N, complex, dimension( * ) CX, integer INCX, complex, dimension( *
       ) CY, integer INCY, complex C, complex S)
       CLACRT performs a linear transformation of a pair of complex vectors.

       Purpose:

            CLACRT performs the operation

               (  c  s )( x )  ==> ( x )
               ( -s  c )( y )      ( y )

            where c and s are complex and the vectors x and y are complex.

       Parameters
           N

                     N is INTEGER
                     The number of elements in the vectors CX and CY.

           CX

                     CX is COMPLEX array, dimension (N)
                     On input, the vector x.
                     On output, CX is overwritten with c*x + s*y.

           INCX

                     INCX is INTEGER
                     The increment between successive values of CX.  INCX <> 0.

           CY

                     CY is COMPLEX array, dimension (N)
                     On input, the vector y.
                     On output, CY is overwritten with -s*x + c*y.

           INCY

                     INCY is INTEGER
                     The increment between successive values of CY.  INCY <> 0.

           C

                     C is COMPLEX

           S

                     S is COMPLEX
                     C and S define the matrix
                        [  C   S  ].
                        [ -S   C  ]

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   complex function cladiv (complex X, complex Y)
       CLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.

       Purpose:

            CLADIV := X / Y, where X and Y are complex.  The computation of X / Y
            will not overflow on an intermediary step unless the results
            overflows.

       Parameters
           X

                     X is COMPLEX

           Y

                     Y is COMPLEX
                     The complex scalars X and Y.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine claein (logical RIGHTV, logical NOINIT, integer N, complex, dimension( ldh, * ) H,
       integer LDH, complex W, complex, dimension( * ) V, complex, dimension( ldb, * ) B, integer
       LDB, real, dimension( * ) RWORK, real EPS3, real SMLNUM, integer INFO)
       CLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by
       inverse iteration.

       Purpose:

            CLAEIN uses inverse iteration to find a right or left eigenvector
            corresponding to the eigenvalue W of a complex upper Hessenberg
            matrix H.

       Parameters
           RIGHTV

                     RIGHTV is LOGICAL
                     = .TRUE. : compute right eigenvector;
                     = .FALSE.: compute left eigenvector.

           NOINIT

                     NOINIT is LOGICAL
                     = .TRUE. : no initial vector supplied in V
                     = .FALSE.: initial vector supplied in V.

           N

                     N is INTEGER
                     The order of the matrix H.  N >= 0.

           H

                     H is COMPLEX array, dimension (LDH,N)
                     The upper Hessenberg matrix H.

           LDH

                     LDH is INTEGER
                     The leading dimension of the array H.  LDH >= max(1,N).

           W

                     W is COMPLEX
                     The eigenvalue of H whose corresponding right or left
                     eigenvector is to be computed.

           V

                     V is COMPLEX array, dimension (N)
                     On entry, if NOINIT = .FALSE., V must contain a starting
                     vector for inverse iteration; otherwise V need not be set.
                     On exit, V contains the computed eigenvector, normalized so
                     that the component of largest magnitude has magnitude 1; here
                     the magnitude of a complex number (x,y) is taken to be
                     |x| + |y|.

           B

                     B is COMPLEX array, dimension (LDB,N)

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           RWORK

                     RWORK is REAL array, dimension (N)

           EPS3

                     EPS3 is REAL
                     A small machine-dependent value which is used to perturb
                     close eigenvalues, and to replace zero pivots.

           SMLNUM

                     SMLNUM is REAL
                     A machine-dependent value close to the underflow threshold.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     = 1:  inverse iteration did not converge; V is set to the
                           last iterate.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine claev2 (complex A, complex B, complex C, real RT1, real RT2, real CS1, complex SN1)
       CLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.

       Purpose:

            CLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix
               [  A         B  ]
               [  CONJG(B)  C  ].
            On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
            eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
            eigenvector for RT1, giving the decomposition

            [ CS1  CONJG(SN1) ] [    A     B ] [ CS1 -CONJG(SN1) ] = [ RT1  0  ]
            [-SN1     CS1     ] [ CONJG(B) C ] [ SN1     CS1     ]   [  0  RT2 ].

       Parameters
           A

                     A is COMPLEX
                    The (1,1) element of the 2-by-2 matrix.

           B

                     B is COMPLEX
                    The (1,2) element and the conjugate of the (2,1) element of
                    the 2-by-2 matrix.

           C

                     C is COMPLEX
                    The (2,2) element of the 2-by-2 matrix.

           RT1

                     RT1 is REAL
                    The eigenvalue of larger absolute value.

           RT2

                     RT2 is REAL
                    The eigenvalue of smaller absolute value.

           CS1

                     CS1 is REAL

           SN1

                     SN1 is COMPLEX
                    The vector (CS1, SN1) is a unit right eigenvector for RT1.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             RT1 is accurate to a few ulps barring over/underflow.

             RT2 may be inaccurate if there is massive cancellation in the
             determinant A*C-B*B; higher precision or correctly rounded or
             correctly truncated arithmetic would be needed to compute RT2
             accurately in all cases.

             CS1 and SN1 are accurate to a few ulps barring over/underflow.

             Overflow is possible only if RT1 is within a factor of 5 of overflow.
             Underflow is harmless if the input data is 0 or exceeds
                underflow_threshold / macheps.

   subroutine clags2 (logical UPPER, real A1, complex A2, real A3, real B1, complex B2, real B3,
       real CSU, complex SNU, real CSV, complex SNV, real CSQ, complex SNQ)
       CLAGS2

       Purpose:

            CLAGS2 computes 2-by-2 unitary matrices U, V and Q, such
            that if ( UPPER ) then

                      U**H *A*Q = U**H *( A1 A2 )*Q = ( x  0  )
                                        ( 0  A3 )     ( x  x  )
            and
                      V**H*B*Q = V**H *( B1 B2 )*Q = ( x  0  )
                                       ( 0  B3 )     ( x  x  )

            or if ( .NOT.UPPER ) then

                      U**H *A*Q = U**H *( A1 0  )*Q = ( x  x  )
                                        ( A2 A3 )     ( 0  x  )
            and
                      V**H *B*Q = V**H *( B1 0  )*Q = ( x  x  )
                                        ( B2 B3 )     ( 0  x  )
            where

              U = (   CSU    SNU ), V = (  CSV    SNV ),
                  ( -SNU**H  CSU )      ( -SNV**H CSV )

              Q = (   CSQ    SNQ )
                  ( -SNQ**H  CSQ )

            The rows of the transformed A and B are parallel. Moreover, if the
            input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
            of A is not zero. If the input matrices A and B are both not zero,
            then the transformed (2,2) element of B is not zero, except when the
            first rows of input A and B are parallel and the second rows are
            zero.

       Parameters
           UPPER

                     UPPER is LOGICAL
                     = .TRUE.: the input matrices A and B are upper triangular.
                     = .FALSE.: the input matrices A and B are lower triangular.

           A1

                     A1 is REAL

           A2

                     A2 is COMPLEX

           A3

                     A3 is REAL
                     On entry, A1, A2 and A3 are elements of the input 2-by-2
                     upper (lower) triangular matrix A.

           B1

                     B1 is REAL

           B2

                     B2 is COMPLEX

           B3

                     B3 is REAL
                     On entry, B1, B2 and B3 are elements of the input 2-by-2
                     upper (lower) triangular matrix B.

           CSU

                     CSU is REAL

           SNU

                     SNU is COMPLEX
                     The desired unitary matrix U.

           CSV

                     CSV is REAL

           SNV

                     SNV is COMPLEX
                     The desired unitary matrix V.

           CSQ

                     CSQ is REAL

           SNQ

                     SNQ is COMPLEX
                     The desired unitary matrix Q.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clagtm (character TRANS, integer N, integer NRHS, real ALPHA, complex, dimension( *
       ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( ldx, * )
       X, integer LDX, real BETA, complex, dimension( ldb, * ) B, integer LDB)
       CLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal
       matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or
       -1.

       Purpose:

            CLAGTM performs a matrix-vector product of the form

               B := alpha * A * X + beta * B

            where A is a tridiagonal matrix of order N, B and X are N by NRHS
            matrices, and alpha and beta are real scalars, each of which may be
            0., 1., or -1.

       Parameters
           TRANS

                     TRANS is CHARACTER*1
                     Specifies the operation applied to A.
                     = 'N':  No transpose, B := alpha * A * X + beta * B
                     = 'T':  Transpose,    B := alpha * A**T * X + beta * B
                     = 'C':  Conjugate transpose, B := alpha * A**H * X + beta * B

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           NRHS

                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrices X and B.

           ALPHA

                     ALPHA is REAL
                     The scalar alpha.  ALPHA must be 0., 1., or -1.; otherwise,
                     it is assumed to be 0.

           DL

                     DL is COMPLEX array, dimension (N-1)
                     The (n-1) sub-diagonal elements of T.

           D

                     D is COMPLEX array, dimension (N)
                     The diagonal elements of T.

           DU

                     DU is COMPLEX array, dimension (N-1)
                     The (n-1) super-diagonal elements of T.

           X

                     X is COMPLEX array, dimension (LDX,NRHS)
                     The N by NRHS matrix X.

           LDX

                     LDX is INTEGER
                     The leading dimension of the array X.  LDX >= max(N,1).

           BETA

                     BETA is REAL
                     The scalar beta.  BETA must be 0., 1., or -1.; otherwise,
                     it is assumed to be 1.

           B

                     B is COMPLEX array, dimension (LDB,NRHS)
                     On entry, the N by NRHS matrix B.
                     On exit, B is overwritten by the matrix expression
                     B := alpha * A * X + beta * B.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(N,1).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex,
       dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, integer ILOZ, integer IHIZ,
       complex, dimension( ldz, * ) Z, integer LDZ, integer INFO)
       CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,
       using the double-shift/single-shift QR algorithm.

       Purpose:

               CLAHQR is an auxiliary routine called by CHSEQR to update the
               eigenvalues and Schur decomposition already computed by CHSEQR, by
               dealing with the Hessenberg submatrix in rows and columns ILO to
               IHI.

       Parameters
           WANTT

                     WANTT is LOGICAL
                     = .TRUE. : the full Schur form T is required;
                     = .FALSE.: only eigenvalues are required.

           WANTZ

                     WANTZ is LOGICAL
                     = .TRUE. : the matrix of Schur vectors Z is required;
                     = .FALSE.: Schur vectors are not required.

           N

                     N is INTEGER
                     The order of the matrix H.  N >= 0.

           ILO

                     ILO is INTEGER

           IHI

                     IHI is INTEGER
                     It is assumed that H is already upper triangular in rows and
                     columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
                     CLAHQR works primarily with the Hessenberg submatrix in rows
                     and columns ILO to IHI, but applies transformations to all of
                     H if WANTT is .TRUE..
                     1 <= ILO <= max(1,IHI); IHI <= N.

           H

                     H is COMPLEX array, dimension (LDH,N)
                     On entry, the upper Hessenberg matrix H.
                     On exit, if INFO is zero and if WANTT is .TRUE., then H
                     is upper triangular in rows and columns ILO:IHI.  If INFO
                     is zero and if WANTT is .FALSE., then the contents of H
                     are unspecified on exit.  The output state of H in case
                     INF is positive is below under the description of INFO.

           LDH

                     LDH is INTEGER
                     The leading dimension of the array H. LDH >= max(1,N).

           W

                     W is COMPLEX array, dimension (N)
                     The computed eigenvalues ILO to IHI are stored in the
                     corresponding elements of W. If WANTT is .TRUE., the
                     eigenvalues are stored in the same order as on the diagonal
                     of the Schur form returned in H, with W(i) = H(i,i).

           ILOZ

                     ILOZ is INTEGER

           IHIZ

                     IHIZ is INTEGER
                     Specify the rows of Z to which transformations must be
                     applied if WANTZ is .TRUE..
                     1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

           Z

                     Z is COMPLEX array, dimension (LDZ,N)
                     If WANTZ is .TRUE., on entry Z must contain the current
                     matrix Z of transformations accumulated by CHSEQR, and on
                     exit Z has been updated; transformations are applied only to
                     the submatrix Z(ILOZ:IHIZ,ILO:IHI).
                     If WANTZ is .FALSE., Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z. LDZ >= max(1,N).

           INFO

                     INFO is INTEGER
                      = 0:  successful exit
                      > 0:  if INFO = i, CLAHQR failed to compute all the
                             eigenvalues ILO to IHI in a total of 30 iterations
                             per eigenvalue; elements i+1:ihi of W contain
                             those eigenvalues which have been successfully
                             computed.

                             If INFO > 0 and WANTT is .FALSE., then on exit,
                             the remaining unconverged eigenvalues are the
                             eigenvalues of the upper Hessenberg matrix
                             rows and columns ILO through INFO of the final,
                             output value of H.

                             If INFO > 0 and WANTT is .TRUE., then on exit
                     (*)       (initial value of H)*U  = U*(final value of H)
                             where U is an orthogonal matrix.    The final
                             value of H is upper Hessenberg and triangular in
                             rows and columns INFO+1 through IHI.

                             If INFO > 0 and WANTZ is .TRUE., then on exit
                                 (final value of Z)  = (initial value of Z)*U
                             where U is the orthogonal matrix in (*)
                             (regardless of the value of WANTT.)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:

                02-96 Based on modifications by
                David Day, Sandia National Laboratory, USA

                12-04 Further modifications by
                Ralph Byers, University of Kansas, USA
                This is a modified version of CLAHQR from LAPACK version 3.0.
                It is (1) more robust against overflow and underflow and
                (2) adopts the more conservative Ahues & Tisseur stopping
                criterion (LAWN 122, 1997).

   subroutine clahr2 (integer N, integer K, integer NB, complex, dimension( lda, * ) A, integer
       LDA, complex, dimension( nb ) TAU, complex, dimension( ldt, nb ) T, integer LDT, complex,
       dimension( ldy, nb ) Y, integer LDY)
       CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so
       that elements below the specified subdiagonal are zero, and returns auxiliary matrices
       which are needed to apply the transformation to the unreduced part of A.

       Purpose:

            CLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
            matrix A so that elements below the k-th subdiagonal are zero. The
            reduction is performed by an unitary similarity transformation
            Q**H * A * Q. The routine returns the matrices V and T which determine
            Q as a block reflector I - V*T*v**H, and also the matrix Y = A * V * T.

            This is an auxiliary routine called by CGEHRD.

       Parameters
           N

                     N is INTEGER
                     The order of the matrix A.

           K

                     K is INTEGER
                     The offset for the reduction. Elements below the k-th
                     subdiagonal in the first NB columns are reduced to zero.
                     K < N.

           NB

                     NB is INTEGER
                     The number of columns to be reduced.

           A

                     A is COMPLEX array, dimension (LDA,N-K+1)
                     On entry, the n-by-(n-k+1) general matrix A.
                     On exit, the elements on and above the k-th subdiagonal in
                     the first NB columns are overwritten with the corresponding
                     elements of the reduced matrix; the elements below the k-th
                     subdiagonal, with the array TAU, represent the matrix Q as a
                     product of elementary reflectors. The other columns of A are
                     unchanged. See Further Details.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,N).

           TAU

                     TAU is COMPLEX array, dimension (NB)
                     The scalar factors of the elementary reflectors. See Further
                     Details.

           T

                     T is COMPLEX array, dimension (LDT,NB)
                     The upper triangular matrix T.

           LDT

                     LDT is INTEGER
                     The leading dimension of the array T.  LDT >= NB.

           Y

                     Y is COMPLEX array, dimension (LDY,NB)
                     The n-by-nb matrix Y.

           LDY

                     LDY is INTEGER
                     The leading dimension of the array Y. LDY >= N.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             The matrix Q is represented as a product of nb elementary reflectors

                Q = H(1) H(2) . . . H(nb).

             Each H(i) has the form

                H(i) = I - tau * v * v**H

             where tau is a complex scalar, and v is a complex vector with
             v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
             A(i+k+1:n,i), and tau in TAU(i).

             The elements of the vectors v together form the (n-k+1)-by-nb matrix
             V which is needed, with T and Y, to apply the transformation to the
             unreduced part of the matrix, using an update of the form:
             A := (I - V*T*V**H) * (A - Y*V**H).

             The contents of A on exit are illustrated by the following example
             with n = 7, k = 3 and nb = 2:

                ( a   a   a   a   a )
                ( a   a   a   a   a )
                ( a   a   a   a   a )
                ( h   h   a   a   a )
                ( v1  h   a   a   a )
                ( v1  v2  a   a   a )
                ( v1  v2  a   a   a )

             where a denotes an element of the original matrix A, h denotes a
             modified element of the upper Hessenberg matrix H, and vi denotes an
             element of the vector defining H(i).

             This subroutine is a slight modification of LAPACK-3.0's CLAHRD
             incorporating improvements proposed by Quintana-Orti and Van de
             Gejin. Note that the entries of A(1:K,2:NB) differ from those
             returned by the original LAPACK-3.0's CLAHRD routine. (This
             subroutine is not backward compatible with LAPACK-3.0's CLAHRD.)

       References:
           Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the
             performance of reduction to Hessenberg form,' ACM Transactions on Mathematical
           Software, 32(2):180-194, June 2006.

   subroutine claic1 (integer JOB, integer J, complex, dimension( j ) X, real SEST, complex,
       dimension( j ) W, complex GAMMA, real SESTPR, complex S, complex C)
       CLAIC1 applies one step of incremental condition estimation.

       Purpose:

            CLAIC1 applies one step of incremental condition estimation in
            its simplest version:

            Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
            lower triangular matrix L, such that
                     twonorm(L*x) = sest
            Then CLAIC1 computes sestpr, s, c such that
            the vector
                            [ s*x ]
                     xhat = [  c  ]
            is an approximate singular vector of
                            [ L      0  ]
                     Lhat = [ w**H gamma ]
            in the sense that
                     twonorm(Lhat*xhat) = sestpr.

            Depending on JOB, an estimate for the largest or smallest singular
            value is computed.

            Note that [s c]**H and sestpr**2 is an eigenpair of the system

                diag(sest*sest, 0) + [alpha  gamma] * [ conjg(alpha) ]
                                                      [ conjg(gamma) ]

            where  alpha =  x**H*w.

       Parameters
           JOB

                     JOB is INTEGER
                     = 1: an estimate for the largest singular value is computed.
                     = 2: an estimate for the smallest singular value is computed.

           J

                     J is INTEGER
                     Length of X and W

           X

                     X is COMPLEX array, dimension (J)
                     The j-vector x.

           SEST

                     SEST is REAL
                     Estimated singular value of j by j matrix L

           W

                     W is COMPLEX array, dimension (J)
                     The j-vector w.

           GAMMA

                     GAMMA is COMPLEX
                     The diagonal element gamma.

           SESTPR

                     SESTPR is REAL
                     Estimated singular value of (j+1) by (j+1) matrix Lhat.

           S

                     S is COMPLEX
                     Sine needed in forming xhat.

           C

                     C is COMPLEX
                     Cosine needed in forming xhat.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function clangt (character NORM, integer N, complex, dimension( * ) DL, complex,
       dimension( * ) D, complex, dimension( * ) DU)
       CLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
       absolute value of any element of a general tridiagonal matrix.

       Purpose:

            CLANGT  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the  element of  largest absolute value  of a
            complex tridiagonal matrix A.

       Returns
           CLANGT

               CLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANGT as described
                     above.

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, CLANGT is
                     set to zero.

           DL

                     DL is COMPLEX array, dimension (N-1)
                     The (n-1) sub-diagonal elements of A.

           D

                     D is COMPLEX array, dimension (N)
                     The diagonal elements of A.

           DU

                     DU is COMPLEX array, dimension (N-1)
                     The (n-1) super-diagonal elements of A.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function clanhb (character NORM, character UPLO, integer N, integer K, complex,
       dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)
       CLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
       the element of largest absolute value of a Hermitian band matrix.

       Purpose:

            CLANHB  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the element of  largest absolute value  of an
            n by n hermitian band matrix A,  with k super-diagonals.

       Returns
           CLANHB

               CLANHB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANHB as described
                     above.

           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     band matrix A is supplied.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, CLANHB is
                     set to zero.

           K

                     K is INTEGER
                     The number of super-diagonals or sub-diagonals of the
                     band matrix A.  K >= 0.

           AB

                     AB is COMPLEX array, dimension (LDAB,N)
                     The upper or lower triangle of the hermitian band matrix A,
                     stored in the first K+1 rows of AB.  The j-th column of A is
                     stored in the j-th column of the array AB as follows:
                     if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
                     Note that the imaginary parts of the diagonal elements need
                     not be set and are assumed to be zero.

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= K+1.

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)),
                     where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
                     WORK is not referenced.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function clanhp (character NORM, character UPLO, integer N, complex, dimension( * ) AP,
       real, dimension( * ) WORK)
       CLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
       the element of largest absolute value of a complex Hermitian matrix supplied in packed
       form.

       Purpose:

            CLANHP  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the  element of  largest absolute value  of a
            complex hermitian matrix A,  supplied in packed form.

       Returns
           CLANHP

               CLANHP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANHP as described
                     above.

           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     hermitian matrix A is supplied.
                     = 'U':  Upper triangular part of A is supplied
                     = 'L':  Lower triangular part of A is supplied

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, CLANHP is
                     set to zero.

           AP

                     AP is COMPLEX array, dimension (N*(N+1)/2)
                     The upper or lower triangle of the hermitian matrix A, packed
                     columnwise in a linear array.  The j-th column of A is stored
                     in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
                     Note that the  imaginary parts of the diagonal elements need
                     not be set and are assumed to be zero.

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)),
                     where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
                     WORK is not referenced.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function clanhs (character NORM, integer N, complex, dimension( lda, * ) A, integer LDA,
       real, dimension( * ) WORK)
       CLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
       absolute value of any element of an upper Hessenberg matrix.

       Purpose:

            CLANHS  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the  element of  largest absolute value  of a
            Hessenberg matrix A.

       Returns
           CLANHS

               CLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANHS as described
                     above.

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, CLANHS is
                     set to zero.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The n by n upper Hessenberg matrix A; the part of A below the
                     first sub-diagonal is not referenced.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(N,1).

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)),
                     where LWORK >= N when NORM = 'I'; otherwise, WORK is not
                     referenced.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function clanht (character NORM, integer N, real, dimension( * ) D, complex, dimension( *
       ) E)
       CLANHT returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
       the element of largest absolute value of a complex Hermitian tridiagonal matrix.

       Purpose:

            CLANHT  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the  element of  largest absolute value  of a
            complex Hermitian tridiagonal matrix A.

       Returns
           CLANHT

               CLANHT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANHT as described
                     above.

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, CLANHT is
                     set to zero.

           D

                     D is REAL array, dimension (N)
                     The diagonal elements of A.

           E

                     E is COMPLEX array, dimension (N-1)
                     The (n-1) sub-diagonal or super-diagonal elements of A.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function clansb (character NORM, character UPLO, integer N, integer K, complex,
       dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)
       CLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
       the element of largest absolute value of a symmetric band matrix.

       Purpose:

            CLANSB  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the element of  largest absolute value  of an
            n by n symmetric band matrix A,  with k super-diagonals.

       Returns
           CLANSB

               CLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANSB as described
                     above.

           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     band matrix A is supplied.
                     = 'U':  Upper triangular part is supplied
                     = 'L':  Lower triangular part is supplied

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, CLANSB is
                     set to zero.

           K

                     K is INTEGER
                     The number of super-diagonals or sub-diagonals of the
                     band matrix A.  K >= 0.

           AB

                     AB is COMPLEX array, dimension (LDAB,N)
                     The upper or lower triangle of the symmetric band matrix A,
                     stored in the first K+1 rows of AB.  The j-th column of A is
                     stored in the j-th column of the array AB as follows:
                     if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= K+1.

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)),
                     where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
                     WORK is not referenced.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function clansp (character NORM, character UPLO, integer N, complex, dimension( * ) AP,
       real, dimension( * ) WORK)
       CLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
       the element of largest absolute value of a symmetric matrix supplied in packed form.

       Purpose:

            CLANSP  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the  element of  largest absolute value  of a
            complex symmetric matrix A,  supplied in packed form.

       Returns
           CLANSP

               CLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANSP as described
                     above.

           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     symmetric matrix A is supplied.
                     = 'U':  Upper triangular part of A is supplied
                     = 'L':  Lower triangular part of A is supplied

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, CLANSP is
                     set to zero.

           AP

                     AP is COMPLEX array, dimension (N*(N+1)/2)
                     The upper or lower triangle of the symmetric matrix A, packed
                     columnwise in a linear array.  The j-th column of A is stored
                     in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)),
                     where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
                     WORK is not referenced.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function clantb (character NORM, character UPLO, character DIAG, integer N, integer K,
       complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)
       CLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
       the element of largest absolute value of a triangular band matrix.

       Purpose:

            CLANTB  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the element of  largest absolute value  of an
            n by n triangular band matrix A,  with ( k + 1 ) diagonals.

       Returns
           CLANTB

               CLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANTB as described
                     above.

           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the matrix A is upper or lower triangular.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           DIAG

                     DIAG is CHARACTER*1
                     Specifies whether or not the matrix A is unit triangular.
                     = 'N':  Non-unit triangular
                     = 'U':  Unit triangular

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, CLANTB is
                     set to zero.

           K

                     K is INTEGER
                     The number of super-diagonals of the matrix A if UPLO = 'U',
                     or the number of sub-diagonals of the matrix A if UPLO = 'L'.
                     K >= 0.

           AB

                     AB is COMPLEX array, dimension (LDAB,N)
                     The upper or lower triangular band matrix A, stored in the
                     first k+1 rows of AB.  The j-th column of A is stored
                     in the j-th column of the array AB as follows:
                     if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
                     Note that when DIAG = 'U', the elements of the array AB
                     corresponding to the diagonal elements of the matrix A are
                     not referenced, but are assumed to be one.

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= K+1.

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)),
                     where LWORK >= N when NORM = 'I'; otherwise, WORK is not
                     referenced.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function clantp (character NORM, character UPLO, character DIAG, integer N, complex,
       dimension( * ) AP, real, dimension( * ) WORK)
       CLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
       the element of largest absolute value of a triangular matrix supplied in packed form.

       Purpose:

            CLANTP  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the  element of  largest absolute value  of a
            triangular matrix A, supplied in packed form.

       Returns
           CLANTP

               CLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANTP as described
                     above.

           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the matrix A is upper or lower triangular.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           DIAG

                     DIAG is CHARACTER*1
                     Specifies whether or not the matrix A is unit triangular.
                     = 'N':  Non-unit triangular
                     = 'U':  Unit triangular

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, CLANTP is
                     set to zero.

           AP

                     AP is COMPLEX array, dimension (N*(N+1)/2)
                     The upper or lower triangular matrix A, packed columnwise in
                     a linear array.  The j-th column of A is stored in the array
                     AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
                     Note that when DIAG = 'U', the elements of the array AP
                     corresponding to the diagonal elements of the matrix A are
                     not referenced, but are assumed to be one.

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)),
                     where LWORK >= N when NORM = 'I'; otherwise, WORK is not
                     referenced.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function clantr (character NORM, character UPLO, character DIAG, integer M, integer N,
       complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)
       CLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
       the element of largest absolute value of a trapezoidal or triangular matrix.

       Purpose:

            CLANTR  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the  element of  largest absolute value  of a
            trapezoidal or triangular matrix A.

       Returns
           CLANTR

               CLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANTR as described
                     above.

           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the matrix A is upper or lower trapezoidal.
                     = 'U':  Upper trapezoidal
                     = 'L':  Lower trapezoidal
                     Note that A is triangular instead of trapezoidal if M = N.

           DIAG

                     DIAG is CHARACTER*1
                     Specifies whether or not the matrix A has unit diagonal.
                     = 'N':  Non-unit diagonal
                     = 'U':  Unit diagonal

           M

                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0, and if
                     UPLO = 'U', M <= N.  When M = 0, CLANTR is set to zero.

           N

                     N is INTEGER
                     The number of columns of the matrix A.  N >= 0, and if
                     UPLO = 'L', N <= M.  When N = 0, CLANTR is set to zero.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The trapezoidal matrix A (A is triangular if M = N).
                     If UPLO = 'U', the leading m by n upper trapezoidal part of
                     the array A contains the upper trapezoidal matrix, and the
                     strictly lower triangular part of A is not referenced.
                     If UPLO = 'L', the leading m by n lower trapezoidal part of
                     the array A contains the lower trapezoidal matrix, and the
                     strictly upper triangular part of A is not referenced.  Note
                     that when DIAG = 'U', the diagonal elements of A are not
                     referenced and are assumed to be one.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(M,1).

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)),
                     where LWORK >= M when NORM = 'I'; otherwise, WORK is not
                     referenced.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clapll (integer N, complex, dimension( * ) X, integer INCX, complex, dimension( * )
       Y, integer INCY, real SSMIN)
       CLAPLL measures the linear dependence of two vectors.

       Purpose:

            Given two column vectors X and Y, let

                                 A = ( X Y ).

            The subroutine first computes the QR factorization of A = Q*R,
            and then computes the SVD of the 2-by-2 upper triangular matrix R.
            The smaller singular value of R is returned in SSMIN, which is used
            as the measurement of the linear dependency of the vectors X and Y.

       Parameters
           N

                     N is INTEGER
                     The length of the vectors X and Y.

           X

                     X is COMPLEX array, dimension (1+(N-1)*INCX)
                     On entry, X contains the N-vector X.
                     On exit, X is overwritten.

           INCX

                     INCX is INTEGER
                     The increment between successive elements of X. INCX > 0.

           Y

                     Y is COMPLEX array, dimension (1+(N-1)*INCY)
                     On entry, Y contains the N-vector Y.
                     On exit, Y is overwritten.

           INCY

                     INCY is INTEGER
                     The increment between successive elements of Y. INCY > 0.

           SSMIN

                     SSMIN is REAL
                     The smallest singular value of the N-by-2 matrix A = ( X Y ).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clapmr (logical FORWRD, integer M, integer N, complex, dimension( ldx, * ) X,
       integer LDX, integer, dimension( * ) K)
       CLAPMR rearranges rows of a matrix as specified by a permutation vector.

       Purpose:

            CLAPMR rearranges the rows of the M by N matrix X as specified
            by the permutation K(1),K(2),...,K(M) of the integers 1,...,M.
            If FORWRD = .TRUE.,  forward permutation:

                 X(K(I),*) is moved X(I,*) for I = 1,2,...,M.

            If FORWRD = .FALSE., backward permutation:

                 X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.

       Parameters
           FORWRD

                     FORWRD is LOGICAL
                     = .TRUE., forward permutation
                     = .FALSE., backward permutation

           M

                     M is INTEGER
                     The number of rows of the matrix X. M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrix X. N >= 0.

           X

                     X is COMPLEX array, dimension (LDX,N)
                     On entry, the M by N matrix X.
                     On exit, X contains the permuted matrix X.

           LDX

                     LDX is INTEGER
                     The leading dimension of the array X, LDX >= MAX(1,M).

           K

                     K is INTEGER array, dimension (M)
                     On entry, K contains the permutation vector. K is used as
                     internal workspace, but reset to its original value on
                     output.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clapmt (logical FORWRD, integer M, integer N, complex, dimension( ldx, * ) X,
       integer LDX, integer, dimension( * ) K)
       CLAPMT performs a forward or backward permutation of the columns of a matrix.

       Purpose:

            CLAPMT rearranges the columns of the M by N matrix X as specified
            by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.
            If FORWRD = .TRUE.,  forward permutation:

                 X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.

            If FORWRD = .FALSE., backward permutation:

                 X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.

       Parameters
           FORWRD

                     FORWRD is LOGICAL
                     = .TRUE., forward permutation
                     = .FALSE., backward permutation

           M

                     M is INTEGER
                     The number of rows of the matrix X. M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrix X. N >= 0.

           X

                     X is COMPLEX array, dimension (LDX,N)
                     On entry, the M by N matrix X.
                     On exit, X contains the permuted matrix X.

           LDX

                     LDX is INTEGER
                     The leading dimension of the array X, LDX >= MAX(1,M).

           K

                     K is INTEGER array, dimension (N)
                     On entry, K contains the permutation vector. K is used as
                     internal workspace, but reset to its original value on
                     output.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine claqhb (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB,
       integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)
       CLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ.

       Purpose:

            CLAQHB equilibrates an Hermitian band matrix A using the scaling
            factors in the vector S.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     symmetric matrix A is stored.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           KD

                     KD is INTEGER
                     The number of super-diagonals of the matrix A if UPLO = 'U',
                     or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

           AB

                     AB is COMPLEX array, dimension (LDAB,N)
                     On entry, the upper or lower triangle of the symmetric band
                     matrix A, stored in the first KD+1 rows of the array.  The
                     j-th column of A is stored in the j-th column of the array AB
                     as follows:
                     if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

                     On exit, if INFO = 0, the triangular factor U or L from the
                     Cholesky factorization A = U**H *U or A = L*L**H of the band
                     matrix A, in the same storage format as A.

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= KD+1.

           S

                     S is REAL array, dimension (N)
                     The scale factors for A.

           SCOND

                     SCOND is REAL
                     Ratio of the smallest S(i) to the largest S(i).

           AMAX

                     AMAX is REAL
                     Absolute value of largest matrix entry.

           EQUED

                     EQUED is CHARACTER*1
                     Specifies whether or not equilibration was done.
                     = 'N':  No equilibration.
                     = 'Y':  Equilibration was done, i.e., A has been replaced by
                             diag(S) * A * diag(S).

       Internal Parameters:

             THRESH is a threshold value used to decide if scaling should be done
             based on the ratio of the scaling factors.  If SCOND < THRESH,
             scaling is done.

             LARGE and SMALL are threshold values used to decide if scaling should
             be done based on the absolute size of the largest matrix element.
             If AMAX > LARGE or AMAX < SMALL, scaling is done.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine claqhp (character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * )
       S, real SCOND, real AMAX, character EQUED)
       CLAQHP scales a Hermitian matrix stored in packed form.

       Purpose:

            CLAQHP equilibrates a Hermitian matrix A using the scaling factors
            in the vector S.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     Hermitian matrix A is stored.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           AP

                     AP is COMPLEX array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

                     On exit, the equilibrated matrix:  diag(S) * A * diag(S), in
                     the same storage format as A.

           S

                     S is REAL array, dimension (N)
                     The scale factors for A.

           SCOND

                     SCOND is REAL
                     Ratio of the smallest S(i) to the largest S(i).

           AMAX

                     AMAX is REAL
                     Absolute value of largest matrix entry.

           EQUED

                     EQUED is CHARACTER*1
                     Specifies whether or not equilibration was done.
                     = 'N':  No equilibration.
                     = 'Y':  Equilibration was done, i.e., A has been replaced by
                             diag(S) * A * diag(S).

       Internal Parameters:

             THRESH is a threshold value used to decide if scaling should be done
             based on the ratio of the scaling factors.  If SCOND < THRESH,
             scaling is done.

             LARGE and SMALL are threshold values used to decide if scaling should
             be done based on the absolute size of the largest matrix element.
             If AMAX > LARGE or AMAX < SMALL, scaling is done.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine claqp2 (integer M, integer N, integer OFFSET, complex, dimension( lda, * ) A,
       integer LDA, integer, dimension( * ) JPVT, complex, dimension( * ) TAU, real, dimension( *
       ) VN1, real, dimension( * ) VN2, complex, dimension( * ) WORK)
       CLAQP2 computes a QR factorization with column pivoting of the matrix block.

       Purpose:

            CLAQP2 computes a QR factorization with column pivoting of
            the block A(OFFSET+1:M,1:N).
            The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

       Parameters
           M

                     M is INTEGER
                     The number of rows of the matrix A. M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrix A. N >= 0.

           OFFSET

                     OFFSET is INTEGER
                     The number of rows of the matrix A that must be pivoted
                     but no factorized. OFFSET >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, the upper triangle of block A(OFFSET+1:M,1:N) is
                     the triangular factor obtained; the elements in block
                     A(OFFSET+1:M,1:N) below the diagonal, together with the
                     array TAU, represent the orthogonal matrix Q as a product of
                     elementary reflectors. Block A(1:OFFSET,1:N) has been
                     accordingly pivoted, but no factorized.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A. LDA >= max(1,M).

           JPVT

                     JPVT is INTEGER array, dimension (N)
                     On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
                     to the front of A*P (a leading column); if JPVT(i) = 0,
                     the i-th column of A is a free column.
                     On exit, if JPVT(i) = k, then the i-th column of A*P
                     was the k-th column of A.

           TAU

                     TAU is COMPLEX array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors.

           VN1

                     VN1 is REAL array, dimension (N)
                     The vector with the partial column norms.

           VN2

                     VN2 is REAL array, dimension (N)
                     The vector with the exact column norms.

           WORK

                     WORK is COMPLEX array, dimension (N)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer
           Science Dept., Duke University, USA
            Partial column norm updating strategy modified on April 2011 Z. Drmac and Z.
           Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

       References:
           LAPACK Working Note 176

   subroutine claqps (integer M, integer N, integer OFFSET, integer NB, integer KB, complex,
       dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, complex, dimension( * )
       TAU, real, dimension( * ) VN1, real, dimension( * ) VN2, complex, dimension( * ) AUXV,
       complex, dimension( ldf, * ) F, integer LDF)
       CLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A
       by using BLAS level 3.

       Purpose:

            CLAQPS computes a step of QR factorization with column pivoting
            of a complex M-by-N matrix A by using Blas-3.  It tries to factorize
            NB columns from A starting from the row OFFSET+1, and updates all
            of the matrix with Blas-3 xGEMM.

            In some cases, due to catastrophic cancellations, it cannot
            factorize NB columns.  Hence, the actual number of factorized
            columns is returned in KB.

            Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

       Parameters
           M

                     M is INTEGER
                     The number of rows of the matrix A. M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrix A. N >= 0

           OFFSET

                     OFFSET is INTEGER
                     The number of rows of A that have been factorized in
                     previous steps.

           NB

                     NB is INTEGER
                     The number of columns to factorize.

           KB

                     KB is INTEGER
                     The number of columns actually factorized.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, block A(OFFSET+1:M,1:KB) is the triangular
                     factor obtained and block A(1:OFFSET,1:N) has been
                     accordingly pivoted, but no factorized.
                     The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
                     been updated.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A. LDA >= max(1,M).

           JPVT

                     JPVT is INTEGER array, dimension (N)
                     JPVT(I) = K <==> Column K of the full matrix A has been
                     permuted into position I in AP.

           TAU

                     TAU is COMPLEX array, dimension (KB)
                     The scalar factors of the elementary reflectors.

           VN1

                     VN1 is REAL array, dimension (N)
                     The vector with the partial column norms.

           VN2

                     VN2 is REAL array, dimension (N)
                     The vector with the exact column norms.

           AUXV

                     AUXV is COMPLEX array, dimension (NB)
                     Auxiliary vector.

           F

                     F is COMPLEX array, dimension (LDF,NB)
                     Matrix  F**H = L * Y**H * A.

           LDF

                     LDF is INTEGER
                     The leading dimension of the array F. LDF >= max(1,N).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer
           Science Dept., Duke University, USA

        Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic,
       Dept. of Mathematics, University of Zagreb, Croatia.

       References:
           LAPACK Working Note 176

   subroutine claqr0 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex,
       dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, integer ILOZ, integer IHIZ,
       complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK,
       integer INFO)
       CLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
       the Schur decomposition.

       Purpose:

               CLAQR0 computes the eigenvalues of a Hessenberg matrix H
               and, optionally, the matrices T and Z from the Schur decomposition
               H = Z T Z**H, where T is an upper triangular matrix (the
               Schur form), and Z is the unitary matrix of Schur vectors.

               Optionally Z may be postmultiplied into an input unitary
               matrix Q so that this routine can give the Schur factorization
               of a matrix A which has been reduced to the Hessenberg form H
               by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.

       Parameters
           WANTT

                     WANTT is LOGICAL
                     = .TRUE. : the full Schur form T is required;
                     = .FALSE.: only eigenvalues are required.

           WANTZ

                     WANTZ is LOGICAL
                     = .TRUE. : the matrix of Schur vectors Z is required;
                     = .FALSE.: Schur vectors are not required.

           N

                     N is INTEGER
                      The order of the matrix H.  N >= 0.

           ILO

                     ILO is INTEGER

           IHI

                     IHI is INTEGER
                      It is assumed that H is already upper triangular in rows
                      and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
                      H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
                      previous call to CGEBAL, and then passed to CGEHRD when the
                      matrix output by CGEBAL is reduced to Hessenberg form.
                      Otherwise, ILO and IHI should be set to 1 and N,
                      respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
                      If N = 0, then ILO = 1 and IHI = 0.

           H

                     H is COMPLEX array, dimension (LDH,N)
                      On entry, the upper Hessenberg matrix H.
                      On exit, if INFO = 0 and WANTT is .TRUE., then H
                      contains the upper triangular matrix T from the Schur
                      decomposition (the Schur form). If INFO = 0 and WANT is
                      .FALSE., then the contents of H are unspecified on exit.
                      (The output value of H when INFO > 0 is given under the
                      description of INFO below.)

                      This subroutine may explicitly set H(i,j) = 0 for i > j and
                      j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

           LDH

                     LDH is INTEGER
                      The leading dimension of the array H. LDH >= max(1,N).

           W

                     W is COMPLEX array, dimension (N)
                      The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
                      in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
                      stored in the same order as on the diagonal of the Schur
                      form returned in H, with W(i) = H(i,i).

           ILOZ

                     ILOZ is INTEGER

           IHIZ

                     IHIZ is INTEGER
                      Specify the rows of Z to which transformations must be
                      applied if WANTZ is .TRUE..
                      1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

           Z

                     Z is COMPLEX array, dimension (LDZ,IHI)
                      If WANTZ is .FALSE., then Z is not referenced.
                      If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
                      replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
                      orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
                      (The output value of Z when INFO > 0 is given under
                      the description of INFO below.)

           LDZ

                     LDZ is INTEGER
                      The leading dimension of the array Z.  if WANTZ is .TRUE.
                      then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.

           WORK

                     WORK is COMPLEX array, dimension LWORK
                      On exit, if LWORK = -1, WORK(1) returns an estimate of
                      the optimal value for LWORK.

           LWORK

                     LWORK is INTEGER
                      The dimension of the array WORK.  LWORK >= max(1,N)
                      is sufficient, but LWORK typically as large as 6*N may
                      be required for optimal performance.  A workspace query
                      to determine the optimal workspace size is recommended.

                      If LWORK = -1, then CLAQR0 does a workspace query.
                      In this case, CLAQR0 checks the input parameters and
                      estimates the optimal workspace size for the given
                      values of N, ILO and IHI.  The estimate is returned
                      in WORK(1).  No error message related to LWORK is
                      issued by XERBLA.  Neither H nor Z are accessed.

           INFO

                     INFO is INTEGER
                        = 0:  successful exit
                        > 0:  if INFO = i, CLAQR0 failed to compute all of
                           the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                           and WI contain those eigenvalues which have been
                           successfully computed.  (Failures are rare.)

                           If INFO > 0 and WANT is .FALSE., then on exit,
                           the remaining unconverged eigenvalues are the eigen-
                           values of the upper Hessenberg matrix rows and
                           columns ILO through INFO of the final, output
                           value of H.

                           If INFO > 0 and WANTT is .TRUE., then on exit

                      (*)  (initial value of H)*U  = U*(final value of H)

                           where U is a unitary matrix.  The final
                           value of  H is upper Hessenberg and triangular in
                           rows and columns INFO+1 through IHI.

                           If INFO > 0 and WANTZ is .TRUE., then on exit

                             (final value of Z(ILO:IHI,ILOZ:IHIZ)
                              =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

                           where U is the unitary matrix in (*) (regard-
                           less of the value of WANTT.)

                           If INFO > 0 and WANTZ is .FALSE., then Z is not
                           accessed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

       References:

             K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
             Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
             Performance, SIAM Journal of Matrix Analysis, volume 23, pages
             929--947, 2002.

            K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive
           Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

   subroutine claqr1 (integer N, complex, dimension( ldh, * ) H, integer LDH, complex S1, complex
       S2, complex, dimension( * ) V)
       CLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3
       matrix H and specified shifts.

       Purpose:

                 Given a 2-by-2 or 3-by-3 matrix H, CLAQR1 sets v to a
                 scalar multiple of the first column of the product

                 (*)  K = (H - s1*I)*(H - s2*I)

                 scaling to avoid overflows and most underflows.

                 This is useful for starting double implicit shift bulges
                 in the QR algorithm.

       Parameters
           N

                     N is INTEGER
                         Order of the matrix H. N must be either 2 or 3.

           H

                     H is COMPLEX array, dimension (LDH,N)
                         The 2-by-2 or 3-by-3 matrix H in (*).

           LDH

                     LDH is INTEGER
                         The leading dimension of H as declared in
                         the calling procedure.  LDH >= N

           S1

                     S1 is COMPLEX

           S2

                     S2 is COMPLEX

                     S1 and S2 are the shifts defining K in (*) above.

           V

                     V is COMPLEX array, dimension (N)
                         A scalar multiple of the first column of the
                         matrix K in (*).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

   subroutine claqr2 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT,
       integer NW, complex, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ,
       complex, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, complex, dimension( *
       ) SH, complex, dimension( ldv, * ) V, integer LDV, integer NH, complex, dimension( ldt, *
       ) T, integer LDT, integer NV, complex, dimension( ldwv, * ) WV, integer LDWV, complex,
       dimension( * ) WORK, integer LWORK)
       CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and
       deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early
       deflation).

       Purpose:

               CLAQR2 is identical to CLAQR3 except that it avoids
               recursion by calling CLAHQR instead of CLAQR4.

               Aggressive early deflation:

               This subroutine accepts as input an upper Hessenberg matrix
               H and performs an unitary similarity transformation
               designed to detect and deflate fully converged eigenvalues from
               a trailing principal submatrix.  On output H has been over-
               written by a new Hessenberg matrix that is a perturbation of
               an unitary similarity transformation of H.  It is to be
               hoped that the final version of H has many zero subdiagonal
               entries.

       Parameters
           WANTT

                     WANTT is LOGICAL
                     If .TRUE., then the Hessenberg matrix H is fully updated
                     so that the triangular Schur factor may be
                     computed (in cooperation with the calling subroutine).
                     If .FALSE., then only enough of H is updated to preserve
                     the eigenvalues.

           WANTZ

                     WANTZ is LOGICAL
                     If .TRUE., then the unitary matrix Z is updated so
                     so that the unitary Schur factor may be computed
                     (in cooperation with the calling subroutine).
                     If .FALSE., then Z is not referenced.

           N

                     N is INTEGER
                     The order of the matrix H and (if WANTZ is .TRUE.) the
                     order of the unitary matrix Z.

           KTOP

                     KTOP is INTEGER
                     It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
                     KBOT and KTOP together determine an isolated block
                     along the diagonal of the Hessenberg matrix.

           KBOT

                     KBOT is INTEGER
                     It is assumed without a check that either
                     KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
                     determine an isolated block along the diagonal of the
                     Hessenberg matrix.

           NW

                     NW is INTEGER
                     Deflation window size.  1 <= NW <= (KBOT-KTOP+1).

           H

                     H is COMPLEX array, dimension (LDH,N)
                     On input the initial N-by-N section of H stores the
                     Hessenberg matrix undergoing aggressive early deflation.
                     On output H has been transformed by a unitary
                     similarity transformation, perturbed, and the returned
                     to Hessenberg form that (it is to be hoped) has some
                     zero subdiagonal entries.

           LDH

                     LDH is INTEGER
                     Leading dimension of H just as declared in the calling
                     subroutine.  N <= LDH

           ILOZ

                     ILOZ is INTEGER

           IHIZ

                     IHIZ is INTEGER
                     Specify the rows of Z to which transformations must be
                     applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.

           Z

                     Z is COMPLEX array, dimension (LDZ,N)
                     IF WANTZ is .TRUE., then on output, the unitary
                     similarity transformation mentioned above has been
                     accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
                     If WANTZ is .FALSE., then Z is unreferenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of Z just as declared in the
                     calling subroutine.  1 <= LDZ.

           NS

                     NS is INTEGER
                     The number of unconverged (ie approximate) eigenvalues
                     returned in SR and SI that may be used as shifts by the
                     calling subroutine.

           ND

                     ND is INTEGER
                     The number of converged eigenvalues uncovered by this
                     subroutine.

           SH

                     SH is COMPLEX array, dimension (KBOT)
                     On output, approximate eigenvalues that may
                     be used for shifts are stored in SH(KBOT-ND-NS+1)
                     through SR(KBOT-ND).  Converged eigenvalues are
                     stored in SH(KBOT-ND+1) through SH(KBOT).

           V

                     V is COMPLEX array, dimension (LDV,NW)
                     An NW-by-NW work array.

           LDV

                     LDV is INTEGER
                     The leading dimension of V just as declared in the
                     calling subroutine.  NW <= LDV

           NH

                     NH is INTEGER
                     The number of columns of T.  NH >= NW.

           T

                     T is COMPLEX array, dimension (LDT,NW)

           LDT

                     LDT is INTEGER
                     The leading dimension of T just as declared in the
                     calling subroutine.  NW <= LDT

           NV

                     NV is INTEGER
                     The number of rows of work array WV available for
                     workspace.  NV >= NW.

           WV

                     WV is COMPLEX array, dimension (LDWV,NW)

           LDWV

                     LDWV is INTEGER
                     The leading dimension of W just as declared in the
                     calling subroutine.  NW <= LDV

           WORK

                     WORK is COMPLEX array, dimension (LWORK)
                     On exit, WORK(1) is set to an estimate of the optimal value
                     of LWORK for the given values of N, NW, KTOP and KBOT.

           LWORK

                     LWORK is INTEGER
                     The dimension of the work array WORK.  LWORK = 2*NW
                     suffices, but greater efficiency may result from larger
                     values of LWORK.

                     If LWORK = -1, then a workspace query is assumed; CLAQR2
                     only estimates the optimal workspace size for the given
                     values of N, NW, KTOP and KBOT.  The estimate is returned
                     in WORK(1).  No error message related to LWORK is issued
                     by XERBLA.  Neither H nor Z are accessed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

   subroutine claqr3 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT,
       integer NW, complex, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ,
       complex, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, complex, dimension( *
       ) SH, complex, dimension( ldv, * ) V, integer LDV, integer NH, complex, dimension( ldt, *
       ) T, integer LDT, integer NV, complex, dimension( ldwv, * ) WV, integer LDWV, complex,
       dimension( * ) WORK, integer LWORK)
       CLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and
       deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early
       deflation).

       Purpose:

               Aggressive early deflation:

               CLAQR3 accepts as input an upper Hessenberg matrix
               H and performs an unitary similarity transformation
               designed to detect and deflate fully converged eigenvalues from
               a trailing principal submatrix.  On output H has been over-
               written by a new Hessenberg matrix that is a perturbation of
               an unitary similarity transformation of H.  It is to be
               hoped that the final version of H has many zero subdiagonal
               entries.

       Parameters
           WANTT

                     WANTT is LOGICAL
                     If .TRUE., then the Hessenberg matrix H is fully updated
                     so that the triangular Schur factor may be
                     computed (in cooperation with the calling subroutine).
                     If .FALSE., then only enough of H is updated to preserve
                     the eigenvalues.

           WANTZ

                     WANTZ is LOGICAL
                     If .TRUE., then the unitary matrix Z is updated so
                     so that the unitary Schur factor may be computed
                     (in cooperation with the calling subroutine).
                     If .FALSE., then Z is not referenced.

           N

                     N is INTEGER
                     The order of the matrix H and (if WANTZ is .TRUE.) the
                     order of the unitary matrix Z.

           KTOP

                     KTOP is INTEGER
                     It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
                     KBOT and KTOP together determine an isolated block
                     along the diagonal of the Hessenberg matrix.

           KBOT

                     KBOT is INTEGER
                     It is assumed without a check that either
                     KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
                     determine an isolated block along the diagonal of the
                     Hessenberg matrix.

           NW

                     NW is INTEGER
                     Deflation window size.  1 <= NW <= (KBOT-KTOP+1).

           H

                     H is COMPLEX array, dimension (LDH,N)
                     On input the initial N-by-N section of H stores the
                     Hessenberg matrix undergoing aggressive early deflation.
                     On output H has been transformed by a unitary
                     similarity transformation, perturbed, and the returned
                     to Hessenberg form that (it is to be hoped) has some
                     zero subdiagonal entries.

           LDH

                     LDH is INTEGER
                     Leading dimension of H just as declared in the calling
                     subroutine.  N <= LDH

           ILOZ

                     ILOZ is INTEGER

           IHIZ

                     IHIZ is INTEGER
                     Specify the rows of Z to which transformations must be
                     applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.

           Z

                     Z is COMPLEX array, dimension (LDZ,N)
                     IF WANTZ is .TRUE., then on output, the unitary
                     similarity transformation mentioned above has been
                     accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
                     If WANTZ is .FALSE., then Z is unreferenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of Z just as declared in the
                     calling subroutine.  1 <= LDZ.

           NS

                     NS is INTEGER
                     The number of unconverged (ie approximate) eigenvalues
                     returned in SR and SI that may be used as shifts by the
                     calling subroutine.

           ND

                     ND is INTEGER
                     The number of converged eigenvalues uncovered by this
                     subroutine.

           SH

                     SH is COMPLEX array, dimension (KBOT)
                     On output, approximate eigenvalues that may
                     be used for shifts are stored in SH(KBOT-ND-NS+1)
                     through SR(KBOT-ND).  Converged eigenvalues are
                     stored in SH(KBOT-ND+1) through SH(KBOT).

           V

                     V is COMPLEX array, dimension (LDV,NW)
                     An NW-by-NW work array.

           LDV

                     LDV is INTEGER
                     The leading dimension of V just as declared in the
                     calling subroutine.  NW <= LDV

           NH

                     NH is INTEGER
                     The number of columns of T.  NH >= NW.

           T

                     T is COMPLEX array, dimension (LDT,NW)

           LDT

                     LDT is INTEGER
                     The leading dimension of T just as declared in the
                     calling subroutine.  NW <= LDT

           NV

                     NV is INTEGER
                     The number of rows of work array WV available for
                     workspace.  NV >= NW.

           WV

                     WV is COMPLEX array, dimension (LDWV,NW)

           LDWV

                     LDWV is INTEGER
                     The leading dimension of W just as declared in the
                     calling subroutine.  NW <= LDV

           WORK

                     WORK is COMPLEX array, dimension (LWORK)
                     On exit, WORK(1) is set to an estimate of the optimal value
                     of LWORK for the given values of N, NW, KTOP and KBOT.

           LWORK

                     LWORK is INTEGER
                     The dimension of the work array WORK.  LWORK = 2*NW
                     suffices, but greater efficiency may result from larger
                     values of LWORK.

                     If LWORK = -1, then a workspace query is assumed; CLAQR3
                     only estimates the optimal workspace size for the given
                     values of N, NW, KTOP and KBOT.  The estimate is returned
                     in WORK(1).  No error message related to LWORK is issued
                     by XERBLA.  Neither H nor Z are accessed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

   subroutine claqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex,
       dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, integer ILOZ, integer IHIZ,
       complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK,
       integer INFO)
       CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
       the Schur decomposition.

       Purpose:

               CLAQR4 implements one level of recursion for CLAQR0.
               It is a complete implementation of the small bulge multi-shift
               QR algorithm.  It may be called by CLAQR0 and, for large enough
               deflation window size, it may be called by CLAQR3.  This
               subroutine is identical to CLAQR0 except that it calls CLAQR2
               instead of CLAQR3.

               CLAQR4 computes the eigenvalues of a Hessenberg matrix H
               and, optionally, the matrices T and Z from the Schur decomposition
               H = Z T Z**H, where T is an upper triangular matrix (the
               Schur form), and Z is the unitary matrix of Schur vectors.

               Optionally Z may be postmultiplied into an input unitary
               matrix Q so that this routine can give the Schur factorization
               of a matrix A which has been reduced to the Hessenberg form H
               by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.

       Parameters
           WANTT

                     WANTT is LOGICAL
                     = .TRUE. : the full Schur form T is required;
                     = .FALSE.: only eigenvalues are required.

           WANTZ

                     WANTZ is LOGICAL
                     = .TRUE. : the matrix of Schur vectors Z is required;
                     = .FALSE.: Schur vectors are not required.

           N

                     N is INTEGER
                      The order of the matrix H.  N >= 0.

           ILO

                     ILO is INTEGER

           IHI

                     IHI is INTEGER
                      It is assumed that H is already upper triangular in rows
                      and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
                      H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
                      previous call to CGEBAL, and then passed to CGEHRD when the
                      matrix output by CGEBAL is reduced to Hessenberg form.
                      Otherwise, ILO and IHI should be set to 1 and N,
                      respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
                      If N = 0, then ILO = 1 and IHI = 0.

           H

                     H is COMPLEX array, dimension (LDH,N)
                      On entry, the upper Hessenberg matrix H.
                      On exit, if INFO = 0 and WANTT is .TRUE., then H
                      contains the upper triangular matrix T from the Schur
                      decomposition (the Schur form). If INFO = 0 and WANT is
                      .FALSE., then the contents of H are unspecified on exit.
                      (The output value of H when INFO > 0 is given under the
                      description of INFO below.)

                      This subroutine may explicitly set H(i,j) = 0 for i > j and
                      j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

           LDH

                     LDH is INTEGER
                      The leading dimension of the array H. LDH >= max(1,N).

           W

                     W is COMPLEX array, dimension (N)
                      The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
                      in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
                      stored in the same order as on the diagonal of the Schur
                      form returned in H, with W(i) = H(i,i).

           ILOZ

                     ILOZ is INTEGER

           IHIZ

                     IHIZ is INTEGER
                      Specify the rows of Z to which transformations must be
                      applied if WANTZ is .TRUE..
                      1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

           Z

                     Z is COMPLEX array, dimension (LDZ,IHI)
                      If WANTZ is .FALSE., then Z is not referenced.
                      If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
                      replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
                      orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
                      (The output value of Z when INFO > 0 is given under
                      the description of INFO below.)

           LDZ

                     LDZ is INTEGER
                      The leading dimension of the array Z.  if WANTZ is .TRUE.
                      then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.

           WORK

                     WORK is COMPLEX array, dimension LWORK
                      On exit, if LWORK = -1, WORK(1) returns an estimate of
                      the optimal value for LWORK.

           LWORK

                     LWORK is INTEGER
                      The dimension of the array WORK.  LWORK >= max(1,N)
                      is sufficient, but LWORK typically as large as 6*N may
                      be required for optimal performance.  A workspace query
                      to determine the optimal workspace size is recommended.

                      If LWORK = -1, then CLAQR4 does a workspace query.
                      In this case, CLAQR4 checks the input parameters and
                      estimates the optimal workspace size for the given
                      values of N, ILO and IHI.  The estimate is returned
                      in WORK(1).  No error message related to LWORK is
                      issued by XERBLA.  Neither H nor Z are accessed.

           INFO

                     INFO is INTEGER
                        = 0:  successful exit
                        > 0:  if INFO = i, CLAQR4 failed to compute all of
                           the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                           and WI contain those eigenvalues which have been
                           successfully computed.  (Failures are rare.)

                           If INFO > 0 and WANT is .FALSE., then on exit,
                           the remaining unconverged eigenvalues are the eigen-
                           values of the upper Hessenberg matrix rows and
                           columns ILO through INFO of the final, output
                           value of H.

                           If INFO > 0 and WANTT is .TRUE., then on exit

                      (*)  (initial value of H)*U  = U*(final value of H)

                           where U is a unitary matrix.  The final
                           value of  H is upper Hessenberg and triangular in
                           rows and columns INFO+1 through IHI.

                           If INFO > 0 and WANTZ is .TRUE., then on exit

                             (final value of Z(ILO:IHI,ILOZ:IHIZ)
                              =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

                           where U is the unitary matrix in (*) (regard-
                           less of the value of WANTT.)

                           If INFO > 0 and WANTZ is .FALSE., then Z is not
                           accessed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

       References:

             K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
             Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
             Performance, SIAM Journal of Matrix Analysis, volume 23, pages
             929--947, 2002.

            K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive
           Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

   subroutine claqr5 (logical WANTT, logical WANTZ, integer KACC22, integer N, integer KTOP,
       integer KBOT, integer NSHFTS, complex, dimension( * ) S, complex, dimension( ldh, * ) H,
       integer LDH, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ,
       complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldu, * ) U, integer LDU,
       integer NV, complex, dimension( ldwv, * ) WV, integer LDWV, integer NH, complex,
       dimension( ldwh, * ) WH, integer LDWH)
       CLAQR5 performs a single small-bulge multi-shift QR sweep.

       Purpose:

               CLAQR5 called by CLAQR0 performs a
               single small-bulge multi-shift QR sweep.

       Parameters
           WANTT

                     WANTT is LOGICAL
                        WANTT = .true. if the triangular Schur factor
                        is being computed.  WANTT is set to .false. otherwise.

           WANTZ

                     WANTZ is LOGICAL
                        WANTZ = .true. if the unitary Schur factor is being
                        computed.  WANTZ is set to .false. otherwise.

           KACC22

                     KACC22 is INTEGER with value 0, 1, or 2.
                        Specifies the computation mode of far-from-diagonal
                        orthogonal updates.
                   = 0: CLAQR5 does not accumulate reflections and does not
                        use matrix-matrix multiply to update far-from-diagonal
                        matrix entries.
                   = 1: CLAQR5 accumulates reflections and uses matrix-matrix
                        multiply to update the far-from-diagonal matrix entries.
                   = 2: Same as KACC22 = 1. This option used to enable exploiting
                        the 2-by-2 structure during matrix multiplications, but
                        this is no longer supported.

           N

                     N is INTEGER
                        N is the order of the Hessenberg matrix H upon which this
                        subroutine operates.

           KTOP

                     KTOP is INTEGER

           KBOT

                     KBOT is INTEGER
                        These are the first and last rows and columns of an
                        isolated diagonal block upon which the QR sweep is to be
                        applied. It is assumed without a check that
                                  either KTOP = 1  or   H(KTOP,KTOP-1) = 0
                        and
                                  either KBOT = N  or   H(KBOT+1,KBOT) = 0.

           NSHFTS

                     NSHFTS is INTEGER
                        NSHFTS gives the number of simultaneous shifts.  NSHFTS
                        must be positive and even.

           S

                     S is COMPLEX array, dimension (NSHFTS)
                        S contains the shifts of origin that define the multi-
                        shift QR sweep.  On output S may be reordered.

           H

                     H is COMPLEX array, dimension (LDH,N)
                        On input H contains a Hessenberg matrix.  On output a
                        multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
                        to the isolated diagonal block in rows and columns KTOP
                        through KBOT.

           LDH

                     LDH is INTEGER
                        LDH is the leading dimension of H just as declared in the
                        calling procedure.  LDH >= MAX(1,N).

           ILOZ

                     ILOZ is INTEGER

           IHIZ

                     IHIZ is INTEGER
                        Specify the rows of Z to which transformations must be
                        applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N

           Z

                     Z is COMPLEX array, dimension (LDZ,IHIZ)
                        If WANTZ = .TRUE., then the QR Sweep unitary
                        similarity transformation is accumulated into
                        Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
                        If WANTZ = .FALSE., then Z is unreferenced.

           LDZ

                     LDZ is INTEGER
                        LDA is the leading dimension of Z just as declared in
                        the calling procedure. LDZ >= N.

           V

                     V is COMPLEX array, dimension (LDV,NSHFTS/2)

           LDV

                     LDV is INTEGER
                        LDV is the leading dimension of V as declared in the
                        calling procedure.  LDV >= 3.

           U

                     U is COMPLEX array, dimension (LDU,2*NSHFTS)

           LDU

                     LDU is INTEGER
                        LDU is the leading dimension of U just as declared in the
                        in the calling subroutine.  LDU >= 2*NSHFTS.

           NV

                     NV is INTEGER
                        NV is the number of rows in WV agailable for workspace.
                        NV >= 1.

           WV

                     WV is COMPLEX array, dimension (LDWV,2*NSHFTS)

           LDWV

                     LDWV is INTEGER
                        LDWV is the leading dimension of WV as declared in the
                        in the calling subroutine.  LDWV >= NV.

           NH

                     NH is INTEGER
                        NH is the number of columns in array WH available for
                        workspace. NH >= 1.

           WH

                     WH is COMPLEX array, dimension (LDWH,NH)

           LDWH

                     LDWH is INTEGER
                        Leading dimension of WH just as declared in the
                        calling procedure.  LDWH >= 2*NSHFTS.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

       Lars Karlsson, Daniel Kressner, and Bruno Lang

       Thijs Steel, Department of Computer science, KU Leuven, Belgium

       References:
           K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining
           Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume
           23, pages 929--947, 2002.

       Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in
       multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).

   subroutine claqsb (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB,
       integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)
       CLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.

       Purpose:

            CLAQSB equilibrates a symmetric band matrix A using the scaling
            factors in the vector S.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     symmetric matrix A is stored.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           KD

                     KD is INTEGER
                     The number of super-diagonals of the matrix A if UPLO = 'U',
                     or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

           AB

                     AB is COMPLEX array, dimension (LDAB,N)
                     On entry, the upper or lower triangle of the symmetric band
                     matrix A, stored in the first KD+1 rows of the array.  The
                     j-th column of A is stored in the j-th column of the array AB
                     as follows:
                     if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

                     On exit, if INFO = 0, the triangular factor U or L from the
                     Cholesky factorization A = U**H *U or A = L*L**H of the band
                     matrix A, in the same storage format as A.

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= KD+1.

           S

                     S is REAL array, dimension (N)
                     The scale factors for A.

           SCOND

                     SCOND is REAL
                     Ratio of the smallest S(i) to the largest S(i).

           AMAX

                     AMAX is REAL
                     Absolute value of largest matrix entry.

           EQUED

                     EQUED is CHARACTER*1
                     Specifies whether or not equilibration was done.
                     = 'N':  No equilibration.
                     = 'Y':  Equilibration was done, i.e., A has been replaced by
                             diag(S) * A * diag(S).

       Internal Parameters:

             THRESH is a threshold value used to decide if scaling should be done
             based on the ratio of the scaling factors.  If SCOND < THRESH,
             scaling is done.

             LARGE and SMALL are threshold values used to decide if scaling should
             be done based on the absolute size of the largest matrix element.
             If AMAX > LARGE or AMAX < SMALL, scaling is done.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine claqsp (character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * )
       S, real SCOND, real AMAX, character EQUED)
       CLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors
       computed by sppequ.

       Purpose:

            CLAQSP equilibrates a symmetric matrix A using the scaling factors
            in the vector S.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     symmetric matrix A is stored.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           AP

                     AP is COMPLEX array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

                     On exit, the equilibrated matrix:  diag(S) * A * diag(S), in
                     the same storage format as A.

           S

                     S is REAL array, dimension (N)
                     The scale factors for A.

           SCOND

                     SCOND is REAL
                     Ratio of the smallest S(i) to the largest S(i).

           AMAX

                     AMAX is REAL
                     Absolute value of largest matrix entry.

           EQUED

                     EQUED is CHARACTER*1
                     Specifies whether or not equilibration was done.
                     = 'N':  No equilibration.
                     = 'Y':  Equilibration was done, i.e., A has been replaced by
                             diag(S) * A * diag(S).

       Internal Parameters:

             THRESH is a threshold value used to decide if scaling should be done
             based on the ratio of the scaling factors.  If SCOND < THRESH,
             scaling is done.

             LARGE and SMALL are threshold values used to decide if scaling should
             be done based on the absolute size of the largest matrix element.
             If AMAX > LARGE or AMAX < SMALL, scaling is done.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clar1v (integer N, integer B1, integer BN, real LAMBDA, real, dimension( * ) D,
       real, dimension( * ) L, real, dimension( * ) LD, real, dimension( * ) LLD, real PIVMIN,
       real GAPTOL, complex, dimension( * ) Z, logical WANTNC, integer NEGCNT, real ZTZ, real
       MINGMA, integer R, integer, dimension( * ) ISUPPZ, real NRMINV, real RESID, real RQCORR,
       real, dimension( * ) WORK)
       CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1
       through bn of the tridiagonal matrix LDLT - λI.

       Purpose:

            CLAR1V computes the (scaled) r-th column of the inverse of
            the sumbmatrix in rows B1 through BN of the tridiagonal matrix
            L D L**T - sigma I. When sigma is close to an eigenvalue, the
            computed vector is an accurate eigenvector. Usually, r corresponds
            to the index where the eigenvector is largest in magnitude.
            The following steps accomplish this computation :
            (a) Stationary qd transform,  L D L**T - sigma I = L(+) D(+) L(+)**T,
            (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
            (c) Computation of the diagonal elements of the inverse of
                L D L**T - sigma I by combining the above transforms, and choosing
                r as the index where the diagonal of the inverse is (one of the)
                largest in magnitude.
            (d) Computation of the (scaled) r-th column of the inverse using the
                twisted factorization obtained by combining the top part of the
                the stationary and the bottom part of the progressive transform.

       Parameters
           N

                     N is INTEGER
                      The order of the matrix L D L**T.

           B1

                     B1 is INTEGER
                      First index of the submatrix of L D L**T.

           BN

                     BN is INTEGER
                      Last index of the submatrix of L D L**T.

           LAMBDA

                     LAMBDA is REAL
                      The shift. In order to compute an accurate eigenvector,
                      LAMBDA should be a good approximation to an eigenvalue
                      of L D L**T.

           L

                     L is REAL array, dimension (N-1)
                      The (n-1) subdiagonal elements of the unit bidiagonal matrix
                      L, in elements 1 to N-1.

           D

                     D is REAL array, dimension (N)
                      The n diagonal elements of the diagonal matrix D.

           LD

                     LD is REAL array, dimension (N-1)
                      The n-1 elements L(i)*D(i).

           LLD

                     LLD is REAL array, dimension (N-1)
                      The n-1 elements L(i)*L(i)*D(i).

           PIVMIN

                     PIVMIN is REAL
                      The minimum pivot in the Sturm sequence.

           GAPTOL

                     GAPTOL is REAL
                      Tolerance that indicates when eigenvector entries are negligible
                      w.r.t. their contribution to the residual.

           Z

                     Z is COMPLEX array, dimension (N)
                      On input, all entries of Z must be set to 0.
                      On output, Z contains the (scaled) r-th column of the
                      inverse. The scaling is such that Z(R) equals 1.

           WANTNC

                     WANTNC is LOGICAL
                      Specifies whether NEGCNT has to be computed.

           NEGCNT

                     NEGCNT is INTEGER
                      If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
                      in the  matrix factorization L D L**T, and NEGCNT = -1 otherwise.

           ZTZ

                     ZTZ is REAL
                      The square of the 2-norm of Z.

           MINGMA

                     MINGMA is REAL
                      The reciprocal of the largest (in magnitude) diagonal
                      element of the inverse of L D L**T - sigma I.

           R

                     R is INTEGER
                      The twist index for the twisted factorization used to
                      compute Z.
                      On input, 0 <= R <= N. If R is input as 0, R is set to
                      the index where (L D L**T - sigma I)^{-1} is largest
                      in magnitude. If 1 <= R <= N, R is unchanged.
                      On output, R contains the twist index used to compute Z.
                      Ideally, R designates the position of the maximum entry in the
                      eigenvector.

           ISUPPZ

                     ISUPPZ is INTEGER array, dimension (2)
                      The support of the vector in Z, i.e., the vector Z is
                      nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

           NRMINV

                     NRMINV is REAL
                      NRMINV = 1/SQRT( ZTZ )

           RESID

                     RESID is REAL
                      The residual of the FP vector.
                      RESID = ABS( MINGMA )/SQRT( ZTZ )

           RQCORR

                     RQCORR is REAL
                      The Rayleigh Quotient correction to LAMBDA.
                      RQCORR = MINGMA*TMP

           WORK

                     WORK is REAL array, dimension (4*N)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Beresford Parlett, University of California, Berkeley, USA
            Jim Demmel, University of California, Berkeley, USA
            Inderjit Dhillon, University of Texas, Austin, USA
            Osni Marques, LBNL/NERSC, USA
            Christof Voemel, University of California, Berkeley, USA

   subroutine clar2v (integer N, complex, dimension( * ) X, complex, dimension( * ) Y, complex,
       dimension( * ) Z, integer INCX, real, dimension( * ) C, complex, dimension( * ) S, integer
       INCC)
       CLAR2V applies a vector of plane rotations with real cosines and complex sines from both
       sides to a sequence of 2-by-2 symmetric/Hermitian matrices.

       Purpose:

            CLAR2V applies a vector of complex plane rotations with real cosines
            from both sides to a sequence of 2-by-2 complex Hermitian matrices,
            defined by the elements of the vectors x, y and z. For i = 1,2,...,n

               (       x(i)  z(i) ) :=
               ( conjg(z(i)) y(i) )

                 (  c(i) conjg(s(i)) ) (       x(i)  z(i) ) ( c(i) -conjg(s(i)) )
                 ( -s(i)       c(i)  ) ( conjg(z(i)) y(i) ) ( s(i)        c(i)  )

       Parameters
           N

                     N is INTEGER
                     The number of plane rotations to be applied.

           X

                     X is COMPLEX array, dimension (1+(N-1)*INCX)
                     The vector x; the elements of x are assumed to be real.

           Y

                     Y is COMPLEX array, dimension (1+(N-1)*INCX)
                     The vector y; the elements of y are assumed to be real.

           Z

                     Z is COMPLEX array, dimension (1+(N-1)*INCX)
                     The vector z.

           INCX

                     INCX is INTEGER
                     The increment between elements of X, Y and Z. INCX > 0.

           C

                     C is REAL array, dimension (1+(N-1)*INCC)
                     The cosines of the plane rotations.

           S

                     S is COMPLEX array, dimension (1+(N-1)*INCC)
                     The sines of the plane rotations.

           INCC

                     INCC is INTEGER
                     The increment between elements of C and S. INCC > 0.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clarcm (integer M, integer N, real, dimension( lda, * ) A, integer LDA, complex,
       dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, real,
       dimension( * ) RWORK)
       CLARCM copies all or part of a real two-dimensional array to a complex array.

       Purpose:

            CLARCM performs a very simple matrix-matrix multiplication:
                     C := A * B,
            where A is M by M and real; B is M by N and complex;
            C is M by N and complex.

       Parameters
           M

                     M is INTEGER
                     The number of rows of the matrix A and of the matrix C.
                     M >= 0.

           N

                     N is INTEGER
                     The number of columns and rows of the matrix B and
                     the number of columns of the matrix C.
                     N >= 0.

           A

                     A is REAL array, dimension (LDA, M)
                     On entry, A contains the M by M matrix A.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A. LDA >=max(1,M).

           B

                     B is COMPLEX array, dimension (LDB, N)
                     On entry, B contains the M by N matrix B.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B. LDB >=max(1,M).

           C

                     C is COMPLEX array, dimension (LDC, N)
                     On exit, C contains the M by N matrix C.

           LDC

                     LDC is INTEGER
                     The leading dimension of the array C. LDC >=max(1,M).

           RWORK

                     RWORK is REAL array, dimension (2*M*N)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clarf (character SIDE, integer M, integer N, complex, dimension( * ) V, integer
       INCV, complex TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * )
       WORK)
       CLARF applies an elementary reflector to a general rectangular matrix.

       Purpose:

            CLARF applies a complex elementary reflector H to a complex M-by-N
            matrix C, from either the left or the right. H is represented in the
            form

                  H = I - tau * v * v**H

            where tau is a complex scalar and v is a complex vector.

            If tau = 0, then H is taken to be the unit matrix.

            To apply H**H (the conjugate transpose of H), supply conjg(tau) instead
            tau.

       Parameters
           SIDE

                     SIDE is CHARACTER*1
                     = 'L': form  H * C
                     = 'R': form  C * H

           M

                     M is INTEGER
                     The number of rows of the matrix C.

           N

                     N is INTEGER
                     The number of columns of the matrix C.

           V

                     V is COMPLEX array, dimension
                                (1 + (M-1)*abs(INCV)) if SIDE = 'L'
                             or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
                     The vector v in the representation of H. V is not used if
                     TAU = 0.

           INCV

                     INCV is INTEGER
                     The increment between elements of v. INCV <> 0.

           TAU

                     TAU is COMPLEX
                     The value tau in the representation of H.

           C

                     C is COMPLEX array, dimension (LDC,N)
                     On entry, the M-by-N matrix C.
                     On exit, C is overwritten by the matrix H * C if SIDE = 'L',
                     or C * H if SIDE = 'R'.

           LDC

                     LDC is INTEGER
                     The leading dimension of the array C. LDC >= max(1,M).

           WORK

                     WORK is COMPLEX array, dimension
                                    (N) if SIDE = 'L'
                                 or (M) if SIDE = 'R'

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clarfb (character SIDE, character TRANS, character DIRECT, character STOREV,
       integer M, integer N, integer K, complex, dimension( ldv, * ) V, integer LDV, complex,
       dimension( ldt, * ) T, integer LDT, complex, dimension( ldc, * ) C, integer LDC, complex,
       dimension( ldwork, * ) WORK, integer LDWORK)
       CLARFB applies a block reflector or its conjugate-transpose to a general rectangular
       matrix.

       Purpose:

            CLARFB applies a complex block reflector H or its transpose H**H to a
            complex M-by-N matrix C, from either the left or the right.

       Parameters
           SIDE

                     SIDE is CHARACTER*1
                     = 'L': apply H or H**H from the Left
                     = 'R': apply H or H**H from the Right

           TRANS

                     TRANS is CHARACTER*1
                     = 'N': apply H (No transpose)
                     = 'C': apply H**H (Conjugate transpose)

           DIRECT

                     DIRECT is CHARACTER*1
                     Indicates how H is formed from a product of elementary
                     reflectors
                     = 'F': H = H(1) H(2) . . . H(k) (Forward)
                     = 'B': H = H(k) . . . H(2) H(1) (Backward)

           STOREV

                     STOREV is CHARACTER*1
                     Indicates how the vectors which define the elementary
                     reflectors are stored:
                     = 'C': Columnwise
                     = 'R': Rowwise

           M

                     M is INTEGER
                     The number of rows of the matrix C.

           N

                     N is INTEGER
                     The number of columns of the matrix C.

           K

                     K is INTEGER
                     The order of the matrix T (= the number of elementary
                     reflectors whose product defines the block reflector).
                     If SIDE = 'L', M >= K >= 0;
                     if SIDE = 'R', N >= K >= 0.

           V

                     V is COMPLEX array, dimension
                                           (LDV,K) if STOREV = 'C'
                                           (LDV,M) if STOREV = 'R' and SIDE = 'L'
                                           (LDV,N) if STOREV = 'R' and SIDE = 'R'
                     The matrix V. See Further Details.

           LDV

                     LDV is INTEGER
                     The leading dimension of the array V.
                     If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
                     if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
                     if STOREV = 'R', LDV >= K.

           T

                     T is COMPLEX array, dimension (LDT,K)
                     The triangular K-by-K matrix T in the representation of the
                     block reflector.

           LDT

                     LDT is INTEGER
                     The leading dimension of the array T. LDT >= K.

           C

                     C is COMPLEX array, dimension (LDC,N)
                     On entry, the M-by-N matrix C.
                     On exit, C is overwritten by H*C or H**H*C or C*H or C*H**H.

           LDC

                     LDC is INTEGER
                     The leading dimension of the array C. LDC >= max(1,M).

           WORK

                     WORK is COMPLEX array, dimension (LDWORK,K)

           LDWORK

                     LDWORK is INTEGER
                     The leading dimension of the array WORK.
                     If SIDE = 'L', LDWORK >= max(1,N);
                     if SIDE = 'R', LDWORK >= max(1,M).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             The shape of the matrix V and the storage of the vectors which define
             the H(i) is best illustrated by the following example with n = 5 and
             k = 3. The elements equal to 1 are not stored; the corresponding
             array elements are modified but restored on exit. The rest of the
             array is not used.

             DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':

                          V = (  1       )                 V = (  1 v1 v1 v1 v1 )
                              ( v1  1    )                     (     1 v2 v2 v2 )
                              ( v1 v2  1 )                     (        1 v3 v3 )
                              ( v1 v2 v3 )
                              ( v1 v2 v3 )

             DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':

                          V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
                              ( v1 v2 v3 )                     ( v2 v2 v2  1    )
                              (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                              (     1 v3 )
                              (        1 )

   subroutine clarfb_gett (character IDENT, integer M, integer N, integer K, complex, dimension(
       ldt, * ) T, integer LDT, complex, dimension( lda, * ) A, integer LDA, complex, dimension(
       ldb, * ) B, integer LDB, complex, dimension( ldwork, * ) WORK, integer LDWORK)
       CLARFB_GETT

       Purpose:

            CLARFB_GETT applies a complex Householder block reflector H from the
            left to a complex (K+M)-by-N  'triangular-pentagonal' matrix
            composed of two block matrices: an upper trapezoidal K-by-N matrix A
            stored in the array A, and a rectangular M-by-(N-K) matrix B, stored
            in the array B. The block reflector H is stored in a compact
            WY-representation, where the elementary reflectors are in the
            arrays A, B and T. See Further Details section.

       Parameters
           IDENT

                     IDENT is CHARACTER*1
                     If IDENT = not 'I', or not 'i', then V1 is unit
                        lower-triangular and stored in the left K-by-K block of
                        the input matrix A,
                     If IDENT = 'I' or 'i', then  V1 is an identity matrix and
                        not stored.
                     See Further Details section.

           M

                     M is INTEGER
                     The number of rows of the matrix B.
                     M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrices A and B.
                     N >= 0.

           K

                     K is INTEGER
                     The number or rows of the matrix A.
                     K is also order of the matrix T, i.e. the number of
                     elementary reflectors whose product defines the block
                     reflector. 0 <= K <= N.

           T

                     T is COMPLEX array, dimension (LDT,K)
                     The upper-triangular K-by-K matrix T in the representation
                     of the block reflector.

           LDT

                     LDT is INTEGER
                     The leading dimension of the array T. LDT >= K.

           A

                     A is COMPLEX array, dimension (LDA,N)

                     On entry:
                      a) In the K-by-N upper-trapezoidal part A: input matrix A.
                      b) In the columns below the diagonal: columns of V1
                         (ones are not stored on the diagonal).

                     On exit:
                       A is overwritten by rectangular K-by-N product H*A.

                     See Further Details section.

           LDA

                     LDB is INTEGER
                     The leading dimension of the array A. LDA >= max(1,K).

           B

                     B is COMPLEX array, dimension (LDB,N)

                     On entry:
                       a) In the M-by-(N-K) right block: input matrix B.
                       b) In the M-by-N left block: columns of V2.

                     On exit:
                       B is overwritten by rectangular M-by-N product H*B.

                     See Further Details section.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,M).

           WORK

                     WORK is COMPLEX array,
                     dimension (LDWORK,max(K,N-K))

           LDWORK

                     LDWORK is INTEGER
                     The leading dimension of the array WORK. LDWORK>=max(1,K).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:

            November 2020, Igor Kozachenko,
                           Computer Science Division,
                           University of California, Berkeley

       Further Details:

               (1) Description of the Algebraic Operation.

               The matrix A is a K-by-N matrix composed of two column block
               matrices, A1, which is K-by-K, and A2, which is K-by-(N-K):
               A = ( A1, A2 ).
               The matrix B is an M-by-N matrix composed of two column block
               matrices, B1, which is M-by-K, and B2, which is M-by-(N-K):
               B = ( B1, B2 ).

               Perform the operation:

                  ( A_out ) := H * ( A_in ) = ( I - V * T * V**H ) * ( A_in ) =
                  ( B_out )        ( B_in )                          ( B_in )
                             = ( I - ( V1 ) * T * ( V1**H, V2**H ) ) * ( A_in )
                                     ( V2 )                            ( B_in )
                On input:

               a) ( A_in )  consists of two block columns:
                  ( B_in )

                  ( A_in ) = (( A1_in ) ( A2_in )) = (( A1_in ) ( A2_in ))
                  ( B_in )   (( B1_in ) ( B2_in ))   ((     0 ) ( B2_in )),

                  where the column blocks are:

                  (  A1_in )  is a K-by-K upper-triangular matrix stored in the
                              upper triangular part of the array A(1:K,1:K).
                  (  B1_in )  is an M-by-K rectangular ZERO matrix and not stored.

                  ( A2_in )  is a K-by-(N-K) rectangular matrix stored
                             in the array A(1:K,K+1:N).
                  ( B2_in )  is an M-by-(N-K) rectangular matrix stored
                             in the array B(1:M,K+1:N).

               b) V = ( V1 )
                      ( V2 )

                  where:
                  1) if IDENT == 'I',V1 is a K-by-K identity matrix, not stored;
                  2) if IDENT != 'I',V1 is a K-by-K unit lower-triangular matrix,
                     stored in the lower-triangular part of the array
                     A(1:K,1:K) (ones are not stored),
                  and V2 is an M-by-K rectangular stored the array B(1:M,1:K),
                            (because on input B1_in is a rectangular zero
                             matrix that is not stored and the space is
                             used to store V2).

               c) T is a K-by-K upper-triangular matrix stored
                  in the array T(1:K,1:K).

               On output:

               a) ( A_out ) consists of two  block columns:
                  ( B_out )

                  ( A_out ) = (( A1_out ) ( A2_out ))
                  ( B_out )   (( B1_out ) ( B2_out )),

                  where the column blocks are:

                  ( A1_out )  is a K-by-K square matrix, or a K-by-K
                              upper-triangular matrix, if V1 is an
                              identity matrix. AiOut is stored in
                              the array A(1:K,1:K).
                  ( B1_out )  is an M-by-K rectangular matrix stored
                              in the array B(1:M,K:N).

                  ( A2_out )  is a K-by-(N-K) rectangular matrix stored
                              in the array A(1:K,K+1:N).
                  ( B2_out )  is an M-by-(N-K) rectangular matrix stored
                              in the array B(1:M,K+1:N).

               The operation above can be represented as the same operation
               on each block column:

                  ( A1_out ) := H * ( A1_in ) = ( I - V * T * V**H ) * ( A1_in )
                  ( B1_out )        (     0 )                          (     0 )

                  ( A2_out ) := H * ( A2_in ) = ( I - V * T * V**H ) * ( A2_in )
                  ( B2_out )        ( B2_in )                          ( B2_in )

               If IDENT != 'I':

                  The computation for column block 1:

                  A1_out: = A1_in - V1*T*(V1**H)*A1_in

                  B1_out: = - V2*T*(V1**H)*A1_in

                  The computation for column block 2, which exists if N > K:

                  A2_out: = A2_in - V1*T*( (V1**H)*A2_in + (V2**H)*B2_in )

                  B2_out: = B2_in - V2*T*( (V1**H)*A2_in + (V2**H)*B2_in )

               If IDENT == 'I':

                  The operation for column block 1:

                  A1_out: = A1_in - V1*T*A1_in

                  B1_out: = - V2*T*A1_in

                  The computation for column block 2, which exists if N > K:

                  A2_out: = A2_in - T*( A2_in + (V2**H)*B2_in )

                  B2_out: = B2_in - V2*T*( A2_in + (V2**H)*B2_in )

               (2) Description of the Algorithmic Computation.

               In the first step, we compute column block 2, i.e. A2 and B2.
               Here, we need to use the K-by-(N-K) rectangular workspace
               matrix W2 that is of the same size as the matrix A2.
               W2 is stored in the array WORK(1:K,1:(N-K)).

               In the second step, we compute column block 1, i.e. A1 and B1.
               Here, we need to use the K-by-K square workspace matrix W1
               that is of the same size as the as the matrix A1.
               W1 is stored in the array WORK(1:K,1:K).

               NOTE: Hence, in this routine, we need the workspace array WORK
               only of size WORK(1:K,1:max(K,N-K)) so it can hold both W2 from
               the first step and W1 from the second step.

               Case (A), when V1 is unit lower-triangular, i.e. IDENT != 'I',
               more computations than in the Case (B).

               if( IDENT != 'I' ) then
                if ( N > K ) then
                  (First Step - column block 2)
                  col2_(1) W2: = A2
                  col2_(2) W2: = (V1**H) * W2 = (unit_lower_tr_of_(A1)**H) * W2
                  col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2
                  col2_(4) W2: = T * W2
                  col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2
                  col2_(6) W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2
                  col2_(7) A2: = A2 - W2
                else
                  (Second Step - column block 1)
                  col1_(1) W1: = A1
                  col1_(2) W1: = (V1**H) * W1 = (unit_lower_tr_of_(A1)**H) * W1
                  col1_(3) W1: = T * W1
                  col1_(4) B1: = - V2 * W1 = - B1 * W1
                  col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1
                  col1_(6) square A1: = A1 - W1
                end if
               end if

               Case (B), when V1 is an identity matrix, i.e. IDENT == 'I',
               less computations than in the Case (A)

               if( IDENT == 'I' ) then
                if ( N > K ) then
                  (First Step - column block 2)
                  col2_(1) W2: = A2
                  col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2
                  col2_(4) W2: = T * W2
                  col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2
                  col2_(7) A2: = A2 - W2
                else
                  (Second Step - column block 1)
                  col1_(1) W1: = A1
                  col1_(3) W1: = T * W1
                  col1_(4) B1: = - V2 * W1 = - B1 * W1
                  col1_(6) upper-triangular_of_(A1): = A1 - W1
                end if
               end if

               Combine these cases (A) and (B) together, this is the resulting
               algorithm:

               if ( N > K ) then

                 (First Step - column block 2)

                 col2_(1)  W2: = A2
                 if( IDENT != 'I' ) then
                   col2_(2)  W2: = (V1**H) * W2
                                 = (unit_lower_tr_of_(A1)**H) * W2
                 end if
                 col2_(3)  W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2]
                 col2_(4)  W2: = T * W2
                 col2_(5)  B2: = B2 - V2 * W2 = B2 - B1 * W2
                 if( IDENT != 'I' ) then
                   col2_(6)    W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2
                 end if
                 col2_(7) A2: = A2 - W2

               else

               (Second Step - column block 1)

                 col1_(1) W1: = A1
                 if( IDENT != 'I' ) then
                   col1_(2) W1: = (V1**H) * W1
                               = (unit_lower_tr_of_(A1)**H) * W1
                 end if
                 col1_(3) W1: = T * W1
                 col1_(4) B1: = - V2 * W1 = - B1 * W1
                 if( IDENT != 'I' ) then
                   col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1
                   col1_(6_a) below_diag_of_(A1): =  - below_diag_of_(W1)
                 end if
                 col1_(6_b) up_tr_of_(A1): = up_tr_of_(A1) - up_tr_of_(W1)

               end if

   subroutine clarfg (integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex
       TAU)
       CLARFG generates an elementary reflector (Householder matrix).

       Purpose:

            CLARFG generates a complex elementary reflector H of order n, such
            that

                  H**H * ( alpha ) = ( beta ),   H**H * H = I.
                         (   x   )   (   0  )

            where alpha and beta are scalars, with beta real, and x is an
            (n-1)-element complex vector. H is represented in the form

                  H = I - tau * ( 1 ) * ( 1 v**H ) ,
                                ( v )

            where tau is a complex scalar and v is a complex (n-1)-element
            vector. Note that H is not hermitian.

            If the elements of x are all zero and alpha is real, then tau = 0
            and H is taken to be the unit matrix.

            Otherwise  1 <= real(tau) <= 2  and  abs(tau-1) <= 1 .

       Parameters
           N

                     N is INTEGER
                     The order of the elementary reflector.

           ALPHA

                     ALPHA is COMPLEX
                     On entry, the value alpha.
                     On exit, it is overwritten with the value beta.

           X

                     X is COMPLEX array, dimension
                                    (1+(N-2)*abs(INCX))
                     On entry, the vector x.
                     On exit, it is overwritten with the vector v.

           INCX

                     INCX is INTEGER
                     The increment between elements of X. INCX > 0.

           TAU

                     TAU is COMPLEX
                     The value tau.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clarfgp (integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex
       TAU)
       CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

       Purpose:

            CLARFGP generates a complex elementary reflector H of order n, such
            that

                  H**H * ( alpha ) = ( beta ),   H**H * H = I.
                         (   x   )   (   0  )

            where alpha and beta are scalars, beta is real and non-negative, and
            x is an (n-1)-element complex vector.  H is represented in the form

                  H = I - tau * ( 1 ) * ( 1 v**H ) ,
                                ( v )

            where tau is a complex scalar and v is a complex (n-1)-element
            vector. Note that H is not hermitian.

            If the elements of x are all zero and alpha is real, then tau = 0
            and H is taken to be the unit matrix.

       Parameters
           N

                     N is INTEGER
                     The order of the elementary reflector.

           ALPHA

                     ALPHA is COMPLEX
                     On entry, the value alpha.
                     On exit, it is overwritten with the value beta.

           X

                     X is COMPLEX array, dimension
                                    (1+(N-2)*abs(INCX))
                     On entry, the vector x.
                     On exit, it is overwritten with the vector v.

           INCX

                     INCX is INTEGER
                     The increment between elements of X. INCX > 0.

           TAU

                     TAU is COMPLEX
                     The value tau.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clarft (character DIRECT, character STOREV, integer N, integer K, complex,
       dimension( ldv, * ) V, integer LDV, complex, dimension( * ) TAU, complex, dimension( ldt,
       * ) T, integer LDT)
       CLARFT forms the triangular factor T of a block reflector H = I - vtvH

       Purpose:

            CLARFT forms the triangular factor T of a complex block reflector H
            of order n, which is defined as a product of k elementary reflectors.

            If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;

            If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.

            If STOREV = 'C', the vector which defines the elementary reflector
            H(i) is stored in the i-th column of the array V, and

               H  =  I - V * T * V**H

            If STOREV = 'R', the vector which defines the elementary reflector
            H(i) is stored in the i-th row of the array V, and

               H  =  I - V**H * T * V

       Parameters
           DIRECT

                     DIRECT is CHARACTER*1
                     Specifies the order in which the elementary reflectors are
                     multiplied to form the block reflector:
                     = 'F': H = H(1) H(2) . . . H(k) (Forward)
                     = 'B': H = H(k) . . . H(2) H(1) (Backward)

           STOREV

                     STOREV is CHARACTER*1
                     Specifies how the vectors which define the elementary
                     reflectors are stored (see also Further Details):
                     = 'C': columnwise
                     = 'R': rowwise

           N

                     N is INTEGER
                     The order of the block reflector H. N >= 0.

           K

                     K is INTEGER
                     The order of the triangular factor T (= the number of
                     elementary reflectors). K >= 1.

           V

                     V is COMPLEX array, dimension
                                          (LDV,K) if STOREV = 'C'
                                          (LDV,N) if STOREV = 'R'
                     The matrix V. See further details.

           LDV

                     LDV is INTEGER
                     The leading dimension of the array V.
                     If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

           TAU

                     TAU is COMPLEX array, dimension (K)
                     TAU(i) must contain the scalar factor of the elementary
                     reflector H(i).

           T

                     T is COMPLEX array, dimension (LDT,K)
                     The k by k triangular factor T of the block reflector.
                     If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
                     lower triangular. The rest of the array is not used.

           LDT

                     LDT is INTEGER
                     The leading dimension of the array T. LDT >= K.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             The shape of the matrix V and the storage of the vectors which define
             the H(i) is best illustrated by the following example with n = 5 and
             k = 3. The elements equal to 1 are not stored.

             DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':

                          V = (  1       )                 V = (  1 v1 v1 v1 v1 )
                              ( v1  1    )                     (     1 v2 v2 v2 )
                              ( v1 v2  1 )                     (        1 v3 v3 )
                              ( v1 v2 v3 )
                              ( v1 v2 v3 )

             DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':

                          V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
                              ( v1 v2 v3 )                     ( v2 v2 v2  1    )
                              (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                              (     1 v3 )
                              (        1 )

   subroutine clarfx (character SIDE, integer M, integer N, complex, dimension( * ) V, complex
       TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK)
       CLARFX applies an elementary reflector to a general rectangular matrix, with loop
       unrolling when the reflector has order ≤ 10.

       Purpose:

            CLARFX applies a complex elementary reflector H to a complex m by n
            matrix C, from either the left or the right. H is represented in the
            form

                  H = I - tau * v * v**H

            where tau is a complex scalar and v is a complex vector.

            If tau = 0, then H is taken to be the unit matrix

            This version uses inline code if H has order < 11.

       Parameters
           SIDE

                     SIDE is CHARACTER*1
                     = 'L': form  H * C
                     = 'R': form  C * H

           M

                     M is INTEGER
                     The number of rows of the matrix C.

           N

                     N is INTEGER
                     The number of columns of the matrix C.

           V

                     V is COMPLEX array, dimension (M) if SIDE = 'L'
                                                   or (N) if SIDE = 'R'
                     The vector v in the representation of H.

           TAU

                     TAU is COMPLEX
                     The value tau in the representation of H.

           C

                     C is COMPLEX array, dimension (LDC,N)
                     On entry, the m by n matrix C.
                     On exit, C is overwritten by the matrix H * C if SIDE = 'L',
                     or C * H if SIDE = 'R'.

           LDC

                     LDC is INTEGER
                     The leading dimension of the array C. LDC >= max(1,M).

           WORK

                     WORK is COMPLEX array, dimension (N) if SIDE = 'L'
                                                       or (M) if SIDE = 'R'
                     WORK is not referenced if H has order < 11.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clarfy (character UPLO, integer N, complex, dimension( * ) V, integer INCV, complex
       TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK)
       CLARFY

       Purpose:

            CLARFY applies an elementary reflector, or Householder matrix, H,
            to an n x n Hermitian matrix C, from both the left and the right.

            H is represented in the form

               H = I - tau * v * v'

            where  tau  is a scalar and  v  is a vector.

            If  tau  is  zero, then  H  is taken to be the unit matrix.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     Hermitian matrix C is stored.
                     = 'U':  Upper triangle
                     = 'L':  Lower triangle

           N

                     N is INTEGER
                     The number of rows and columns of the matrix C.  N >= 0.

           V

                     V is COMPLEX array, dimension
                             (1 + (N-1)*abs(INCV))
                     The vector v as described above.

           INCV

                     INCV is INTEGER
                     The increment between successive elements of v.  INCV must
                     not be zero.

           TAU

                     TAU is COMPLEX
                     The value tau as described above.

           C

                     C is COMPLEX array, dimension (LDC, N)
                     On entry, the matrix C.
                     On exit, C is overwritten by H * C * H'.

           LDC

                     LDC is INTEGER
                     The leading dimension of the array C.  LDC >= max( 1, N ).

           WORK

                     WORK is COMPLEX array, dimension (N)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clargv (integer N, complex, dimension( * ) X, integer INCX, complex, dimension( * )
       Y, integer INCY, real, dimension( * ) C, integer INCC)
       CLARGV generates a vector of plane rotations with real cosines and complex sines.

       Purpose:

            CLARGV generates a vector of complex plane rotations with real
            cosines, determined by elements of the complex vectors x and y.
            For i = 1,2,...,n

               (        c(i)   s(i) ) ( x(i) ) = ( r(i) )
               ( -conjg(s(i))  c(i) ) ( y(i) ) = (   0  )

               where c(i)**2 + ABS(s(i))**2 = 1

            The following conventions are used (these are the same as in CLARTG,
            but differ from the BLAS1 routine CROTG):
               If y(i)=0, then c(i)=1 and s(i)=0.
               If x(i)=0, then c(i)=0 and s(i) is chosen so that r(i) is real.

       Parameters
           N

                     N is INTEGER
                     The number of plane rotations to be generated.

           X

                     X is COMPLEX array, dimension (1+(N-1)*INCX)
                     On entry, the vector x.
                     On exit, x(i) is overwritten by r(i), for i = 1,...,n.

           INCX

                     INCX is INTEGER
                     The increment between elements of X. INCX > 0.

           Y

                     Y is COMPLEX array, dimension (1+(N-1)*INCY)
                     On entry, the vector y.
                     On exit, the sines of the plane rotations.

           INCY

                     INCY is INTEGER
                     The increment between elements of Y. INCY > 0.

           C

                     C is REAL array, dimension (1+(N-1)*INCC)
                     The cosines of the plane rotations.

           INCC

                     INCC is INTEGER
                     The increment between elements of C. INCC > 0.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             6-6-96 - Modified with a new algorithm by W. Kahan and J. Demmel

             This version has a few statements commented out for thread safety
             (machine parameters are computed on each entry). 10 feb 03, SJH.

   subroutine clarnv (integer IDIST, integer, dimension( 4 ) ISEED, integer N, complex,
       dimension( * ) X)
       CLARNV returns a vector of random numbers from a uniform or normal distribution.

       Purpose:

            CLARNV returns a vector of n random complex numbers from a uniform or
            normal distribution.

       Parameters
           IDIST

                     IDIST is INTEGER
                     Specifies the distribution of the random numbers:
                     = 1:  real and imaginary parts each uniform (0,1)
                     = 2:  real and imaginary parts each uniform (-1,1)
                     = 3:  real and imaginary parts each normal (0,1)
                     = 4:  uniformly distributed on the disc abs(z) < 1
                     = 5:  uniformly distributed on the circle abs(z) = 1

           ISEED

                     ISEED is INTEGER array, dimension (4)
                     On entry, the seed of the random number generator; the array
                     elements must be between 0 and 4095, and ISEED(4) must be
                     odd.
                     On exit, the seed is updated.

           N

                     N is INTEGER
                     The number of random numbers to be generated.

           X

                     X is COMPLEX array, dimension (N)
                     The generated random numbers.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             This routine calls the auxiliary routine SLARUV to generate random
             real numbers from a uniform (0,1) distribution, in batches of up to
             128 using vectorisable code. The Box-Muller method is used to
             transform numbers from a uniform to a normal distribution.

   subroutine clarrv (integer N, real VL, real VU, real, dimension( * ) D, real, dimension( * )
       L, real PIVMIN, integer, dimension( * ) ISPLIT, integer M, integer DOL, integer DOU, real
       MINRGP, real RTOL1, real RTOL2, real, dimension( * ) W, real, dimension( * ) WERR, real,
       dimension( * ) WGAP, integer, dimension( * ) IBLOCK, integer, dimension( * ) INDEXW, real,
       dimension( * ) GERS, complex, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * )
       ISUPPZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
       CLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the
       eigenvalues of L D LT.

       Purpose:

            CLARRV computes the eigenvectors of the tridiagonal matrix
            T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
            The input eigenvalues should have been computed by SLARRE.

       Parameters
           N

                     N is INTEGER
                     The order of the matrix.  N >= 0.

           VL

                     VL is REAL
                     Lower bound of the interval that contains the desired
                     eigenvalues. VL < VU. Needed to compute gaps on the left or right
                     end of the extremal eigenvalues in the desired RANGE.

           VU

                     VU is REAL
                     Upper bound of the interval that contains the desired
                     eigenvalues. VL < VU. Needed to compute gaps on the left or right
                     end of the extremal eigenvalues in the desired RANGE.

           D

                     D is REAL array, dimension (N)
                     On entry, the N diagonal elements of the diagonal matrix D.
                     On exit, D may be overwritten.

           L

                     L is REAL array, dimension (N)
                     On entry, the (N-1) subdiagonal elements of the unit
                     bidiagonal matrix L are in elements 1 to N-1 of L
                     (if the matrix is not split.) At the end of each block
                     is stored the corresponding shift as given by SLARRE.
                     On exit, L is overwritten.

           PIVMIN

                     PIVMIN is REAL
                     The minimum pivot allowed in the Sturm sequence.

           ISPLIT

                     ISPLIT is INTEGER array, dimension (N)
                     The splitting points, at which T breaks up into blocks.
                     The first block consists of rows/columns 1 to
                     ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
                     through ISPLIT( 2 ), etc.

           M

                     M is INTEGER
                     The total number of input eigenvalues.  0 <= M <= N.

           DOL

                     DOL is INTEGER

           DOU

                     DOU is INTEGER
                     If the user wants to compute only selected eigenvectors from all
                     the eigenvalues supplied, he can specify an index range DOL:DOU.
                     Or else the setting DOL=1, DOU=M should be applied.
                     Note that DOL and DOU refer to the order in which the eigenvalues
                     are stored in W.
                     If the user wants to compute only selected eigenpairs, then
                     the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
                     computed eigenvectors. All other columns of Z are set to zero.

           MINRGP

                     MINRGP is REAL

           RTOL1

                     RTOL1 is REAL

           RTOL2

                     RTOL2 is REAL
                      Parameters for bisection.
                      An interval [LEFT,RIGHT] has converged if
                      RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

           W

                     W is REAL array, dimension (N)
                     The first M elements of W contain the APPROXIMATE eigenvalues for
                     which eigenvectors are to be computed.  The eigenvalues
                     should be grouped by split-off block and ordered from
                     smallest to largest within the block ( The output array
                     W from SLARRE is expected here ). Furthermore, they are with
                     respect to the shift of the corresponding root representation
                     for their block. On exit, W holds the eigenvalues of the
                     UNshifted matrix.

           WERR

                     WERR is REAL array, dimension (N)
                     The first M elements contain the semiwidth of the uncertainty
                     interval of the corresponding eigenvalue in W

           WGAP

                     WGAP is REAL array, dimension (N)
                     The separation from the right neighbor eigenvalue in W.

           IBLOCK

                     IBLOCK is INTEGER array, dimension (N)
                     The indices of the blocks (submatrices) associated with the
                     corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
                     W(i) belongs to the first block from the top, =2 if W(i)
                     belongs to the second block, etc.

           INDEXW

                     INDEXW is INTEGER array, dimension (N)
                     The indices of the eigenvalues within each block (submatrix);
                     for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
                     i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.

           GERS

                     GERS is REAL array, dimension (2*N)
                     The N Gerschgorin intervals (the i-th Gerschgorin interval
                     is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
                     be computed from the original UNshifted matrix.

           Z

                     Z is COMPLEX array, dimension (LDZ, max(1,M) )
                     If INFO = 0, the first M columns of Z contain the
                     orthonormal eigenvectors of the matrix T
                     corresponding to the input eigenvalues, with the i-th
                     column of Z holding the eigenvector associated with W(i).
                     Note: the user must ensure that at least max(1,M) columns are
                     supplied in the array Z.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           ISUPPZ

                     ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
                     The support of the eigenvectors in Z, i.e., the indices
                     indicating the nonzero elements in Z. The I-th eigenvector
                     is nonzero only in elements ISUPPZ( 2*I-1 ) through
                     ISUPPZ( 2*I ).

           WORK

                     WORK is REAL array, dimension (12*N)

           IWORK

                     IWORK is INTEGER array, dimension (7*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit

                     > 0:  A problem occurred in CLARRV.
                     < 0:  One of the called subroutines signaled an internal problem.
                           Needs inspection of the corresponding parameter IINFO
                           for further information.

                     =-1:  Problem in SLARRB when refining a child's eigenvalues.
                     =-2:  Problem in SLARRF when computing the RRR of a child.
                           When a child is inside a tight cluster, it can be difficult
                           to find an RRR. A partial remedy from the user's point of
                           view is to make the parameter MINRGP smaller and recompile.
                           However, as the orthogonality of the computed vectors is
                           proportional to 1/MINRGP, the user should be aware that
                           he might be trading in precision when he decreases MINRGP.
                     =-3:  Problem in SLARRB when refining a single eigenvalue
                           after the Rayleigh correction was rejected.
                     = 5:  The Rayleigh Quotient Iteration failed to converge to
                           full accuracy in MAXITR steps.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Beresford Parlett, University of California, Berkeley, USA
            Jim Demmel, University of California, Berkeley, USA
            Inderjit Dhillon, University of Texas, Austin, USA
            Osni Marques, LBNL/NERSC, USA
            Christof Voemel, University of California, Berkeley, USA

   subroutine clartv (integer N, complex, dimension( * ) X, integer INCX, complex, dimension( * )
       Y, integer INCY, real, dimension( * ) C, complex, dimension( * ) S, integer INCC)
       CLARTV applies a vector of plane rotations with real cosines and complex sines to the
       elements of a pair of vectors.

       Purpose:

            CLARTV applies a vector of complex plane rotations with real cosines
            to elements of the complex vectors x and y. For i = 1,2,...,n

               ( x(i) ) := (        c(i)   s(i) ) ( x(i) )
               ( y(i) )    ( -conjg(s(i))  c(i) ) ( y(i) )

       Parameters
           N

                     N is INTEGER
                     The number of plane rotations to be applied.

           X

                     X is COMPLEX array, dimension (1+(N-1)*INCX)
                     The vector x.

           INCX

                     INCX is INTEGER
                     The increment between elements of X. INCX > 0.

           Y

                     Y is COMPLEX array, dimension (1+(N-1)*INCY)
                     The vector y.

           INCY

                     INCY is INTEGER
                     The increment between elements of Y. INCY > 0.

           C

                     C is REAL array, dimension (1+(N-1)*INCC)
                     The cosines of the plane rotations.

           S

                     S is COMPLEX array, dimension (1+(N-1)*INCC)
                     The sines of the plane rotations.

           INCC

                     INCC is INTEGER
                     The increment between elements of C and S. INCC > 0.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clascl (character TYPE, integer KL, integer KU, real CFROM, real CTO, integer M,
       integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)
       CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.

       Purpose:

            CLASCL multiplies the M by N complex matrix A by the real scalar
            CTO/CFROM.  This is done without over/underflow as long as the final
            result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that
            A may be full, upper triangular, lower triangular, upper Hessenberg,
            or banded.

       Parameters
           TYPE

                     TYPE is CHARACTER*1
                     TYPE indices the storage type of the input matrix.
                     = 'G':  A is a full matrix.
                     = 'L':  A is a lower triangular matrix.
                     = 'U':  A is an upper triangular matrix.
                     = 'H':  A is an upper Hessenberg matrix.
                     = 'B':  A is a symmetric band matrix with lower bandwidth KL
                             and upper bandwidth KU and with the only the lower
                             half stored.
                     = 'Q':  A is a symmetric band matrix with lower bandwidth KL
                             and upper bandwidth KU and with the only the upper
                             half stored.
                     = 'Z':  A is a band matrix with lower bandwidth KL and upper
                             bandwidth KU. See CGBTRF for storage details.

           KL

                     KL is INTEGER
                     The lower bandwidth of A.  Referenced only if TYPE = 'B',
                     'Q' or 'Z'.

           KU

                     KU is INTEGER
                     The upper bandwidth of A.  Referenced only if TYPE = 'B',
                     'Q' or 'Z'.

           CFROM

                     CFROM is REAL

           CTO

                     CTO is REAL

                     The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
                     without over/underflow if the final result CTO*A(I,J)/CFROM
                     can be represented without over/underflow.  CFROM must be
                     nonzero.

           M

                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrix A.  N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The matrix to be multiplied by CTO/CFROM.  See TYPE for the
                     storage type.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.
                     If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M);
                        TYPE = 'B', LDA >= KL+1;
                        TYPE = 'Q', LDA >= KU+1;
                        TYPE = 'Z', LDA >= 2*KL+KU+1.

           INFO

                     INFO is INTEGER
                     0  - successful exit
                     <0 - if INFO = -i, the i-th argument had an illegal value.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine claset (character UPLO, integer M, integer N, complex ALPHA, complex BETA, complex,
       dimension( lda, * ) A, integer LDA)
       CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to
       given values.

       Purpose:

            CLASET initializes a 2-D array A to BETA on the diagonal and
            ALPHA on the offdiagonals.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies the part of the matrix A to be set.
                     = 'U':      Upper triangular part is set. The lower triangle
                                 is unchanged.
                     = 'L':      Lower triangular part is set. The upper triangle
                                 is unchanged.
                     Otherwise:  All of the matrix A is set.

           M

                     M is INTEGER
                     On entry, M specifies the number of rows of A.

           N

                     N is INTEGER
                     On entry, N specifies the number of columns of A.

           ALPHA

                     ALPHA is COMPLEX
                     All the offdiagonal array elements are set to ALPHA.

           BETA

                     BETA is COMPLEX
                     All the diagonal array elements are set to BETA.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the m by n matrix A.
                     On exit, A(i,j) = ALPHA, 1 <= i <= m, 1 <= j <= n, i.ne.j;
                              A(i,i) = BETA , 1 <= i <= min(m,n)

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,M).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clasr (character SIDE, character PIVOT, character DIRECT, integer M, integer N,
       real, dimension( * ) C, real, dimension( * ) S, complex, dimension( lda, * ) A, integer
       LDA)
       CLASR applies a sequence of plane rotations to a general rectangular matrix.

       Purpose:

            CLASR applies a sequence of real plane rotations to a complex matrix
            A, from either the left or the right.

            When SIDE = 'L', the transformation takes the form

               A := P*A

            and when SIDE = 'R', the transformation takes the form

               A := A*P**T

            where P is an orthogonal matrix consisting of a sequence of z plane
            rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
            and P**T is the transpose of P.

            When DIRECT = 'F' (Forward sequence), then

               P = P(z-1) * ... * P(2) * P(1)

            and when DIRECT = 'B' (Backward sequence), then

               P = P(1) * P(2) * ... * P(z-1)

            where P(k) is a plane rotation matrix defined by the 2-by-2 rotation

               R(k) = (  c(k)  s(k) )
                    = ( -s(k)  c(k) ).

            When PIVOT = 'V' (Variable pivot), the rotation is performed
            for the plane (k,k+1), i.e., P(k) has the form

               P(k) = (  1                                            )
                      (       ...                                     )
                      (              1                                )
                      (                   c(k)  s(k)                  )
                      (                  -s(k)  c(k)                  )
                      (                                1              )
                      (                                     ...       )
                      (                                            1  )

            where R(k) appears as a rank-2 modification to the identity matrix in
            rows and columns k and k+1.

            When PIVOT = 'T' (Top pivot), the rotation is performed for the
            plane (1,k+1), so P(k) has the form

               P(k) = (  c(k)                    s(k)                 )
                      (         1                                     )
                      (              ...                              )
                      (                     1                         )
                      ( -s(k)                    c(k)                 )
                      (                                 1             )
                      (                                      ...      )
                      (                                             1 )

            where R(k) appears in rows and columns 1 and k+1.

            Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
            performed for the plane (k,z), giving P(k) the form

               P(k) = ( 1                                             )
                      (      ...                                      )
                      (             1                                 )
                      (                  c(k)                    s(k) )
                      (                         1                     )
                      (                              ...              )
                      (                                     1         )
                      (                 -s(k)                    c(k) )

            where R(k) appears in rows and columns k and z.  The rotations are
            performed without ever forming P(k) explicitly.

       Parameters
           SIDE

                     SIDE is CHARACTER*1
                     Specifies whether the plane rotation matrix P is applied to
                     A on the left or the right.
                     = 'L':  Left, compute A := P*A
                     = 'R':  Right, compute A:= A*P**T

           PIVOT

                     PIVOT is CHARACTER*1
                     Specifies the plane for which P(k) is a plane rotation
                     matrix.
                     = 'V':  Variable pivot, the plane (k,k+1)
                     = 'T':  Top pivot, the plane (1,k+1)
                     = 'B':  Bottom pivot, the plane (k,z)

           DIRECT

                     DIRECT is CHARACTER*1
                     Specifies whether P is a forward or backward sequence of
                     plane rotations.
                     = 'F':  Forward, P = P(z-1)*...*P(2)*P(1)
                     = 'B':  Backward, P = P(1)*P(2)*...*P(z-1)

           M

                     M is INTEGER
                     The number of rows of the matrix A.  If m <= 1, an immediate
                     return is effected.

           N

                     N is INTEGER
                     The number of columns of the matrix A.  If n <= 1, an
                     immediate return is effected.

           C

                     C is REAL array, dimension
                             (M-1) if SIDE = 'L'
                             (N-1) if SIDE = 'R'
                     The cosines c(k) of the plane rotations.

           S

                     S is REAL array, dimension
                             (M-1) if SIDE = 'L'
                             (N-1) if SIDE = 'R'
                     The sines s(k) of the plane rotations.  The 2-by-2 plane
                     rotation part of the matrix P(k), R(k), has the form
                     R(k) = (  c(k)  s(k) )
                            ( -s(k)  c(k) ).

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The M-by-N matrix A.  On exit, A is overwritten by P*A if
                     SIDE = 'R' or by A*P**T if SIDE = 'L'.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,M).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine claswp (integer N, complex, dimension( lda, * ) A, integer LDA, integer K1, integer
       K2, integer, dimension( * ) IPIV, integer INCX)
       CLASWP performs a series of row interchanges on a general rectangular matrix.

       Purpose:

            CLASWP performs a series of row interchanges on the matrix A.
            One row interchange is initiated for each of rows K1 through K2 of A.

       Parameters
           N

                     N is INTEGER
                     The number of columns of the matrix A.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the matrix of column dimension N to which the row
                     interchanges will be applied.
                     On exit, the permuted matrix.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.

           K1

                     K1 is INTEGER
                     The first element of IPIV for which a row interchange will
                     be done.

           K2

                     K2 is INTEGER
                     (K2-K1+1) is the number of elements of IPIV for which a row
                     interchange will be done.

           IPIV

                     IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
                     The vector of pivot indices. Only the elements in positions
                     K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed.
                     IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be
                     interchanged.

           INCX

                     INCX is INTEGER
                     The increment between successive values of IPIV. If INCX
                     is negative, the pivots are applied in reverse order.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             Modified by
              R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA

   subroutine clatbs (character UPLO, character TRANS, character DIAG, character NORMIN, integer
       N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( * ) X,
       real SCALE, real, dimension( * ) CNORM, integer INFO)
       CLATBS solves a triangular banded system of equations.

       Purpose:

            CLATBS solves one of the triangular systems

               A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,

            with scaling to prevent overflow, where A is an upper or lower
            triangular band matrix.  Here A**T denotes the transpose of A, x and b
            are n-element vectors, and s is a scaling factor, usually less than
            or equal to 1, chosen so that the components of x will be less than
            the overflow threshold.  If the unscaled problem will not cause
            overflow, the Level 2 BLAS routine CTBSV is called.  If the matrix A
            is singular (A(j,j) = 0 for some j), then s is set to 0 and a
            non-trivial solution to A*x = 0 is returned.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the matrix A is upper or lower triangular.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           TRANS

                     TRANS is CHARACTER*1
                     Specifies the operation applied to A.
                     = 'N':  Solve A * x = s*b     (No transpose)
                     = 'T':  Solve A**T * x = s*b  (Transpose)
                     = 'C':  Solve A**H * x = s*b  (Conjugate transpose)

           DIAG

                     DIAG is CHARACTER*1
                     Specifies whether or not the matrix A is unit triangular.
                     = 'N':  Non-unit triangular
                     = 'U':  Unit triangular

           NORMIN

                     NORMIN is CHARACTER*1
                     Specifies whether CNORM has been set or not.
                     = 'Y':  CNORM contains the column norms on entry
                     = 'N':  CNORM is not set on entry.  On exit, the norms will
                             be computed and stored in CNORM.

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           KD

                     KD is INTEGER
                     The number of subdiagonals or superdiagonals in the
                     triangular matrix A.  KD >= 0.

           AB

                     AB is COMPLEX array, dimension (LDAB,N)
                     The upper or lower triangular band matrix A, stored in the
                     first KD+1 rows of the array. The j-th column of A is stored
                     in the j-th column of the array AB as follows:
                     if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= KD+1.

           X

                     X is COMPLEX array, dimension (N)
                     On entry, the right hand side b of the triangular system.
                     On exit, X is overwritten by the solution vector x.

           SCALE

                     SCALE is REAL
                     The scaling factor s for the triangular system
                        A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
                     If SCALE = 0, the matrix A is singular or badly scaled, and
                     the vector x is an exact or approximate solution to A*x = 0.

           CNORM

                     CNORM is REAL array, dimension (N)

                     If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
                     contains the norm of the off-diagonal part of the j-th column
                     of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
                     to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
                     must be greater than or equal to the 1-norm.

                     If NORMIN = 'N', CNORM is an output argument and CNORM(j)
                     returns the 1-norm of the offdiagonal part of the j-th column
                     of A.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -k, the k-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             A rough bound on x is computed; if that is less than overflow, CTBSV
             is called, otherwise, specific code is used which checks for possible
             overflow or divide-by-zero at every operation.

             A columnwise scheme is used for solving A*x = b.  The basic algorithm
             if A is lower triangular is

                  x[1:n] := b[1:n]
                  for j = 1, ..., n
                       x(j) := x(j) / A(j,j)
                       x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
                  end

             Define bounds on the components of x after j iterations of the loop:
                M(j) = bound on x[1:j]
                G(j) = bound on x[j+1:n]
             Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

             Then for iteration j+1 we have
                M(j+1) <= G(j) / | A(j+1,j+1) |
                G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                       <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

             where CNORM(j+1) is greater than or equal to the infinity-norm of
             column j+1 of A, not counting the diagonal.  Hence

                G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                             1<=i<=j
             and

                |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                              1<=i< j

             Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTBSV if the
             reciprocal of the largest M(j), j=1,..,n, is larger than
             max(underflow, 1/overflow).

             The bound on x(j) is also used to determine when a step in the
             columnwise method can be performed without fear of overflow.  If
             the computed bound is greater than a large constant, x is scaled to
             prevent overflow, but if the bound overflows, x is set to 0, x(j) to
             1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

             Similarly, a row-wise scheme is used to solve A**T *x = b  or
             A**H *x = b.  The basic algorithm for A upper triangular is

                  for j = 1, ..., n
                       x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
                  end

             We simultaneously compute two bounds
                  G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
                  M(j) = bound on x(i), 1<=i<=j

             The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
             add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
             Then the bound on x(j) is

                  M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

                       <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                                 1<=i<=j

             and we can safely call CTBSV if 1/M(n) and 1/G(n) are both greater
             than max(underflow, 1/overflow).

   subroutine clatdf (integer IJOB, integer N, complex, dimension( ldz, * ) Z, integer LDZ,
       complex, dimension( * ) RHS, real RDSUM, real RDSCAL, integer, dimension( * ) IPIV,
       integer, dimension( * ) JPIV)
       CLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a
       contribution to the reciprocal Dif-estimate.

       Purpose:

            CLATDF computes the contribution to the reciprocal Dif-estimate
            by solving for x in Z * x = b, where b is chosen such that the norm
            of x is as large as possible. It is assumed that LU decomposition
            of Z has been computed by CGETC2. On entry RHS = f holds the
            contribution from earlier solved sub-systems, and on return RHS = x.

            The factorization of Z returned by CGETC2 has the form
            Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
            triangular with unit diagonal elements and U is upper triangular.

       Parameters
           IJOB

                     IJOB is INTEGER
                     IJOB = 2: First compute an approximative null-vector e
                         of Z using CGECON, e is normalized and solve for
                         Zx = +-e - f with the sign giving the greater value of
                         2-norm(x).  About 5 times as expensive as Default.
                     IJOB .ne. 2: Local look ahead strategy where
                         all entries of the r.h.s. b is chosen as either +1 or
                         -1.  Default.

           N

                     N is INTEGER
                     The number of columns of the matrix Z.

           Z

                     Z is COMPLEX array, dimension (LDZ, N)
                     On entry, the LU part of the factorization of the n-by-n
                     matrix Z computed by CGETC2:  Z = P * L * U * Q

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDA >= max(1, N).

           RHS

                     RHS is COMPLEX array, dimension (N).
                     On entry, RHS contains contributions from other subsystems.
                     On exit, RHS contains the solution of the subsystem with
                     entries according to the value of IJOB (see above).

           RDSUM

                     RDSUM is REAL
                     On entry, the sum of squares of computed contributions to
                     the Dif-estimate under computation by CTGSYL, where the
                     scaling factor RDSCAL (see below) has been factored out.
                     On exit, the corresponding sum of squares updated with the
                     contributions from the current sub-system.
                     If TRANS = 'T' RDSUM is not touched.
                     NOTE: RDSUM only makes sense when CTGSY2 is called by CTGSYL.

           RDSCAL

                     RDSCAL is REAL
                     On entry, scaling factor used to prevent overflow in RDSUM.
                     On exit, RDSCAL is updated w.r.t. the current contributions
                     in RDSUM.
                     If TRANS = 'T', RDSCAL is not touched.
                     NOTE: RDSCAL only makes sense when CTGSY2 is called by
                     CTGSYL.

           IPIV

                     IPIV is INTEGER array, dimension (N).
                     The pivot indices; for 1 <= i <= N, row i of the
                     matrix has been interchanged with row IPIV(i).

           JPIV

                     JPIV is INTEGER array, dimension (N).
                     The pivot indices; for 1 <= j <= N, column j of the
                     matrix has been interchanged with column JPIV(j).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:
           This routine is a further developed implementation of algorithm BSOLVE in [1] using
           complete pivoting in the LU factorization.

       Contributors:
           Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901
           87 Umea, Sweden.

       References:
           [1] Bo Kagstrom and Lars Westin, Generalized Schur Methods with Condition Estimators
           for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic
           Control, Vol. 34, No. 7, July 1989, pp 745-751.

       [2] Peter Poromaa, On Efficient and Robust Estimators for the Separation between two
       Regular Matrix Pairs with Applications in Condition Estimation. Report UMINF-95.05,
       Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.

   subroutine clatps (character UPLO, character TRANS, character DIAG, character NORMIN, integer
       N, complex, dimension( * ) AP, complex, dimension( * ) X, real SCALE, real, dimension( * )
       CNORM, integer INFO)
       CLATPS solves a triangular system of equations with the matrix held in packed storage.

       Purpose:

            CLATPS solves one of the triangular systems

               A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,

            with scaling to prevent overflow, where A is an upper or lower
            triangular matrix stored in packed form.  Here A**T denotes the
            transpose of A, A**H denotes the conjugate transpose of A, x and b
            are n-element vectors, and s is a scaling factor, usually less than
            or equal to 1, chosen so that the components of x will be less than
            the overflow threshold.  If the unscaled problem will not cause
            overflow, the Level 2 BLAS routine CTPSV is called. If the matrix A
            is singular (A(j,j) = 0 for some j), then s is set to 0 and a
            non-trivial solution to A*x = 0 is returned.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the matrix A is upper or lower triangular.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           TRANS

                     TRANS is CHARACTER*1
                     Specifies the operation applied to A.
                     = 'N':  Solve A * x = s*b     (No transpose)
                     = 'T':  Solve A**T * x = s*b  (Transpose)
                     = 'C':  Solve A**H * x = s*b  (Conjugate transpose)

           DIAG

                     DIAG is CHARACTER*1
                     Specifies whether or not the matrix A is unit triangular.
                     = 'N':  Non-unit triangular
                     = 'U':  Unit triangular

           NORMIN

                     NORMIN is CHARACTER*1
                     Specifies whether CNORM has been set or not.
                     = 'Y':  CNORM contains the column norms on entry
                     = 'N':  CNORM is not set on entry.  On exit, the norms will
                             be computed and stored in CNORM.

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           AP

                     AP is COMPLEX array, dimension (N*(N+1)/2)
                     The upper or lower triangular matrix A, packed columnwise in
                     a linear array.  The j-th column of A is stored in the array
                     AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

           X

                     X is COMPLEX array, dimension (N)
                     On entry, the right hand side b of the triangular system.
                     On exit, X is overwritten by the solution vector x.

           SCALE

                     SCALE is REAL
                     The scaling factor s for the triangular system
                        A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
                     If SCALE = 0, the matrix A is singular or badly scaled, and
                     the vector x is an exact or approximate solution to A*x = 0.

           CNORM

                     CNORM is REAL array, dimension (N)

                     If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
                     contains the norm of the off-diagonal part of the j-th column
                     of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
                     to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
                     must be greater than or equal to the 1-norm.

                     If NORMIN = 'N', CNORM is an output argument and CNORM(j)
                     returns the 1-norm of the offdiagonal part of the j-th column
                     of A.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -k, the k-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             A rough bound on x is computed; if that is less than overflow, CTPSV
             is called, otherwise, specific code is used which checks for possible
             overflow or divide-by-zero at every operation.

             A columnwise scheme is used for solving A*x = b.  The basic algorithm
             if A is lower triangular is

                  x[1:n] := b[1:n]
                  for j = 1, ..., n
                       x(j) := x(j) / A(j,j)
                       x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
                  end

             Define bounds on the components of x after j iterations of the loop:
                M(j) = bound on x[1:j]
                G(j) = bound on x[j+1:n]
             Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

             Then for iteration j+1 we have
                M(j+1) <= G(j) / | A(j+1,j+1) |
                G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                       <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

             where CNORM(j+1) is greater than or equal to the infinity-norm of
             column j+1 of A, not counting the diagonal.  Hence

                G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                             1<=i<=j
             and

                |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                              1<=i< j

             Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTPSV if the
             reciprocal of the largest M(j), j=1,..,n, is larger than
             max(underflow, 1/overflow).

             The bound on x(j) is also used to determine when a step in the
             columnwise method can be performed without fear of overflow.  If
             the computed bound is greater than a large constant, x is scaled to
             prevent overflow, but if the bound overflows, x is set to 0, x(j) to
             1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

             Similarly, a row-wise scheme is used to solve A**T *x = b  or
             A**H *x = b.  The basic algorithm for A upper triangular is

                  for j = 1, ..., n
                       x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
                  end

             We simultaneously compute two bounds
                  G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
                  M(j) = bound on x(i), 1<=i<=j

             The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
             add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
             Then the bound on x(j) is

                  M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

                       <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                                 1<=i<=j

             and we can safely call CTPSV if 1/M(n) and 1/G(n) are both greater
             than max(underflow, 1/overflow).

   subroutine clatrd (character UPLO, integer N, integer NB, complex, dimension( lda, * ) A,
       integer LDA, real, dimension( * ) E, complex, dimension( * ) TAU, complex, dimension( ldw,
       * ) W, integer LDW)
       CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real
       tridiagonal form by an unitary similarity transformation.

       Purpose:

            CLATRD reduces NB rows and columns of a complex Hermitian matrix A to
            Hermitian tridiagonal form by a unitary similarity
            transformation Q**H * A * Q, and returns the matrices V and W which are
            needed to apply the transformation to the unreduced part of A.

            If UPLO = 'U', CLATRD reduces the last NB rows and columns of a
            matrix, of which the upper triangle is supplied;
            if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
            matrix, of which the lower triangle is supplied.

            This is an auxiliary routine called by CHETRD.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     Hermitian matrix A is stored:
                     = 'U': Upper triangular
                     = 'L': Lower triangular

           N

                     N is INTEGER
                     The order of the matrix A.

           NB

                     NB is INTEGER
                     The number of rows and columns to be reduced.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
                     n-by-n upper triangular part of A contains the upper
                     triangular part of the matrix A, and the strictly lower
                     triangular part of A is not referenced.  If UPLO = 'L', the
                     leading n-by-n lower triangular part of A contains the lower
                     triangular part of the matrix A, and the strictly upper
                     triangular part of A is not referenced.
                     On exit:
                     if UPLO = 'U', the last NB columns have been reduced to
                       tridiagonal form, with the diagonal elements overwriting
                       the diagonal elements of A; the elements above the diagonal
                       with the array TAU, represent the unitary matrix Q as a
                       product of elementary reflectors;
                     if UPLO = 'L', the first NB columns have been reduced to
                       tridiagonal form, with the diagonal elements overwriting
                       the diagonal elements of A; the elements below the diagonal
                       with the array TAU, represent the  unitary matrix Q as a
                       product of elementary reflectors.
                     See Further Details.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,N).

           E

                     E is REAL array, dimension (N-1)
                     If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
                     elements of the last NB columns of the reduced matrix;
                     if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
                     the first NB columns of the reduced matrix.

           TAU

                     TAU is COMPLEX array, dimension (N-1)
                     The scalar factors of the elementary reflectors, stored in
                     TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
                     See Further Details.

           W

                     W is COMPLEX array, dimension (LDW,NB)
                     The n-by-nb matrix W required to update the unreduced part
                     of A.

           LDW

                     LDW is INTEGER
                     The leading dimension of the array W. LDW >= max(1,N).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             If UPLO = 'U', the matrix Q is represented as a product of elementary
             reflectors

                Q = H(n) H(n-1) . . . H(n-nb+1).

             Each H(i) has the form

                H(i) = I - tau * v * v**H

             where tau is a complex scalar, and v is a complex vector with
             v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
             and tau in TAU(i-1).

             If UPLO = 'L', the matrix Q is represented as a product of elementary
             reflectors

                Q = H(1) H(2) . . . H(nb).

             Each H(i) has the form

                H(i) = I - tau * v * v**H

             where tau is a complex scalar, and v is a complex vector with
             v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
             and tau in TAU(i).

             The elements of the vectors v together form the n-by-nb matrix V
             which is needed, with W, to apply the transformation to the unreduced
             part of the matrix, using a Hermitian rank-2k update of the form:
             A := A - V*W**H - W*V**H.

             The contents of A on exit are illustrated by the following examples
             with n = 5 and nb = 2:

             if UPLO = 'U':                       if UPLO = 'L':

               (  a   a   a   v4  v5 )              (  d                  )
               (      a   a   v4  v5 )              (  1   d              )
               (          a   1   v5 )              (  v1  1   a          )
               (              d   1  )              (  v1  v2  a   a      )
               (                  d  )              (  v1  v2  a   a   a  )

             where d denotes a diagonal element of the reduced matrix, a denotes
             an element of the original matrix that is unchanged, and vi denotes
             an element of the vector defining H(i).

   subroutine clatrs (character UPLO, character TRANS, character DIAG, character NORMIN, integer
       N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) X, real SCALE,
       real, dimension( * ) CNORM, integer INFO)
       CLATRS solves a triangular system of equations with the scale factor set to prevent
       overflow.

       Purpose:

            CLATRS solves one of the triangular systems

               A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,

            with scaling to prevent overflow.  Here A is an upper or lower
            triangular matrix, A**T denotes the transpose of A, A**H denotes the
            conjugate transpose of A, x and b are n-element vectors, and s is a
            scaling factor, usually less than or equal to 1, chosen so that the
            components of x will be less than the overflow threshold.  If the
            unscaled problem will not cause overflow, the Level 2 BLAS routine
            CTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
            then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the matrix A is upper or lower triangular.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           TRANS

                     TRANS is CHARACTER*1
                     Specifies the operation applied to A.
                     = 'N':  Solve A * x = s*b     (No transpose)
                     = 'T':  Solve A**T * x = s*b  (Transpose)
                     = 'C':  Solve A**H * x = s*b  (Conjugate transpose)

           DIAG

                     DIAG is CHARACTER*1
                     Specifies whether or not the matrix A is unit triangular.
                     = 'N':  Non-unit triangular
                     = 'U':  Unit triangular

           NORMIN

                     NORMIN is CHARACTER*1
                     Specifies whether CNORM has been set or not.
                     = 'Y':  CNORM contains the column norms on entry
                     = 'N':  CNORM is not set on entry.  On exit, the norms will
                             be computed and stored in CNORM.

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The triangular matrix A.  If UPLO = 'U', the leading n by n
                     upper triangular part of the array A contains the upper
                     triangular matrix, and the strictly lower triangular part of
                     A is not referenced.  If UPLO = 'L', the leading n by n lower
                     triangular part of the array A contains the lower triangular
                     matrix, and the strictly upper triangular part of A is not
                     referenced.  If DIAG = 'U', the diagonal elements of A are
                     also not referenced and are assumed to be 1.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max (1,N).

           X

                     X is COMPLEX array, dimension (N)
                     On entry, the right hand side b of the triangular system.
                     On exit, X is overwritten by the solution vector x.

           SCALE

                     SCALE is REAL
                     The scaling factor s for the triangular system
                        A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
                     If SCALE = 0, the matrix A is singular or badly scaled, and
                     the vector x is an exact or approximate solution to A*x = 0.

           CNORM

                     CNORM is REAL array, dimension (N)

                     If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
                     contains the norm of the off-diagonal part of the j-th column
                     of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
                     to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
                     must be greater than or equal to the 1-norm.

                     If NORMIN = 'N', CNORM is an output argument and CNORM(j)
                     returns the 1-norm of the offdiagonal part of the j-th column
                     of A.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -k, the k-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             A rough bound on x is computed; if that is less than overflow, CTRSV
             is called, otherwise, specific code is used which checks for possible
             overflow or divide-by-zero at every operation.

             A columnwise scheme is used for solving A*x = b.  The basic algorithm
             if A is lower triangular is

                  x[1:n] := b[1:n]
                  for j = 1, ..., n
                       x(j) := x(j) / A(j,j)
                       x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
                  end

             Define bounds on the components of x after j iterations of the loop:
                M(j) = bound on x[1:j]
                G(j) = bound on x[j+1:n]
             Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

             Then for iteration j+1 we have
                M(j+1) <= G(j) / | A(j+1,j+1) |
                G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                       <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

             where CNORM(j+1) is greater than or equal to the infinity-norm of
             column j+1 of A, not counting the diagonal.  Hence

                G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                             1<=i<=j
             and

                |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                              1<=i< j

             Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTRSV if the
             reciprocal of the largest M(j), j=1,..,n, is larger than
             max(underflow, 1/overflow).

             The bound on x(j) is also used to determine when a step in the
             columnwise method can be performed without fear of overflow.  If
             the computed bound is greater than a large constant, x is scaled to
             prevent overflow, but if the bound overflows, x is set to 0, x(j) to
             1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

             Similarly, a row-wise scheme is used to solve A**T *x = b  or
             A**H *x = b.  The basic algorithm for A upper triangular is

                  for j = 1, ..., n
                       x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
                  end

             We simultaneously compute two bounds
                  G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
                  M(j) = bound on x(i), 1<=i<=j

             The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
             add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
             Then the bound on x(j) is

                  M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

                       <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                                 1<=i<=j

             and we can safely call CTRSV if 1/M(n) and 1/G(n) are both greater
             than max(underflow, 1/overflow).

   subroutine clauu2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       integer INFO)
       CLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular
       matrices (unblocked algorithm).

       Purpose:

            CLAUU2 computes the product U * U**H or L**H * L, where the triangular
            factor U or L is stored in the upper or lower triangular part of
            the array A.

            If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
            overwriting the factor U in A.
            If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
            overwriting the factor L in A.

            This is the unblocked form of the algorithm, calling Level 2 BLAS.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the triangular factor stored in the array A
                     is upper or lower triangular:
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           N

                     N is INTEGER
                     The order of the triangular factor U or L.  N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the triangular factor U or L.
                     On exit, if UPLO = 'U', the upper triangle of A is
                     overwritten with the upper triangle of the product U * U**H;
                     if UPLO = 'L', the lower triangle of A is overwritten with
                     the lower triangle of the product L**H * L.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,N).

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -k, the k-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine clauum (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       integer INFO)
       CLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular
       matrices (blocked algorithm).

       Purpose:

            CLAUUM computes the product U * U**H or L**H * L, where the triangular
            factor U or L is stored in the upper or lower triangular part of
            the array A.

            If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
            overwriting the factor U in A.
            If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
            overwriting the factor L in A.

            This is the blocked form of the algorithm, calling Level 3 BLAS.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the triangular factor stored in the array A
                     is upper or lower triangular:
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           N

                     N is INTEGER
                     The order of the triangular factor U or L.  N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the triangular factor U or L.
                     On exit, if UPLO = 'U', the upper triangle of A is
                     overwritten with the upper triangle of the product U * U**H;
                     if UPLO = 'L', the lower triangle of A is overwritten with
                     the lower triangle of the product L**H * L.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,N).

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -k, the k-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine crot (integer N, complex, dimension( * ) CX, integer INCX, complex, dimension( * )
       CY, integer INCY, real C, complex S)
       CROT applies a plane rotation with real cosine and complex sine to a pair of complex
       vectors.

       Purpose:

            CROT   applies a plane rotation, where the cos (C) is real and the
            sin (S) is complex, and the vectors CX and CY are complex.

       Parameters
           N

                     N is INTEGER
                     The number of elements in the vectors CX and CY.

           CX

                     CX is COMPLEX array, dimension (N)
                     On input, the vector X.
                     On output, CX is overwritten with C*X + S*Y.

           INCX

                     INCX is INTEGER
                     The increment between successive values of CX.  INCX <> 0.

           CY

                     CY is COMPLEX array, dimension (N)
                     On input, the vector Y.
                     On output, CY is overwritten with -CONJG(S)*X + C*Y.

           INCY

                     INCY is INTEGER
                     The increment between successive values of CY.  INCX <> 0.

           C

                     C is REAL

           S

                     S is COMPLEX
                     C and S define a rotation
                        [  C          S  ]
                        [ -conjg(S)   C  ]
                     where C*C + S*CONJG(S) = 1.0.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cspmv (character UPLO, integer N, complex ALPHA, complex, dimension( * ) AP,
       complex, dimension( * ) X, integer INCX, complex BETA, complex, dimension( * ) Y, integer
       INCY)
       CSPMV computes a matrix-vector product for complex vectors using a complex symmetric
       packed matrix

       Purpose:

            CSPMV  performs the matrix-vector operation

               y := alpha*A*x + beta*y,

            where alpha and beta are scalars, x and y are n element vectors and
            A is an n by n symmetric matrix, supplied in packed form.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the upper or lower
                      triangular part of the matrix A is supplied in the packed
                      array AP as follows:

                         UPLO = 'U' or 'u'   The upper triangular part of A is
                                             supplied in AP.

                         UPLO = 'L' or 'l'   The lower triangular part of A is
                                             supplied in AP.

                      Unchanged on exit.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.
                      Unchanged on exit.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.
                      Unchanged on exit.

           AP

                     AP is COMPLEX array, dimension at least
                      ( ( N*( N + 1 ) )/2 ).
                      Before entry, with UPLO = 'U' or 'u', the array AP must
                      contain the upper triangular part of the symmetric matrix
                      packed sequentially, column by column, so that AP( 1 )
                      contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
                      and a( 2, 2 ) respectively, and so on.
                      Before entry, with UPLO = 'L' or 'l', the array AP must
                      contain the lower triangular part of the symmetric matrix
                      packed sequentially, column by column, so that AP( 1 )
                      contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
                      and a( 3, 1 ) respectively, and so on.
                      Unchanged on exit.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( N - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the N-
                      element vector x.
                      Unchanged on exit.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.
                      Unchanged on exit.

           BETA

                     BETA is COMPLEX
                      On entry, BETA specifies the scalar beta. When BETA is
                      supplied as zero then Y need not be set on input.
                      Unchanged on exit.

           Y

                     Y is COMPLEX array, dimension at least
                      ( 1 + ( N - 1 )*abs( INCY ) ).
                      Before entry, the incremented array Y must contain the n
                      element vector y. On exit, Y is overwritten by the updated
                      vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.
                      Unchanged on exit.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cspr (character UPLO, integer N, complex ALPHA, complex, dimension( * ) X, integer
       INCX, complex, dimension( * ) AP)
       CSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.

       Purpose:

            CSPR    performs the symmetric rank 1 operation

               A := alpha*x*x**H + A,

            where alpha is a complex scalar, x is an n element vector and A is an
            n by n symmetric matrix, supplied in packed form.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the upper or lower
                      triangular part of the matrix A is supplied in the packed
                      array AP as follows:

                         UPLO = 'U' or 'u'   The upper triangular part of A is
                                             supplied in AP.

                         UPLO = 'L' or 'l'   The lower triangular part of A is
                                             supplied in AP.

                      Unchanged on exit.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.
                      Unchanged on exit.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.
                      Unchanged on exit.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( N - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the N-
                      element vector x.
                      Unchanged on exit.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.
                      Unchanged on exit.

           AP

                     AP is COMPLEX array, dimension at least
                      ( ( N*( N + 1 ) )/2 ).
                      Before entry, with  UPLO = 'U' or 'u', the array AP must
                      contain the upper triangular part of the symmetric matrix
                      packed sequentially, column by column, so that AP( 1 )
                      contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
                      and a( 2, 2 ) respectively, and so on. On exit, the array
                      AP is overwritten by the upper triangular part of the
                      updated matrix.
                      Before entry, with UPLO = 'L' or 'l', the array AP must
                      contain the lower triangular part of the symmetric matrix
                      packed sequentially, column by column, so that AP( 1 )
                      contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
                      and a( 3, 1 ) respectively, and so on. On exit, the array
                      AP is overwritten by the lower triangular part of the
                      updated matrix.
                      Note that the imaginary parts of the diagonal elements need
                      not be set, they are assumed to be zero, and on exit they
                      are set to zero.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine csrscl (integer N, real SA, complex, dimension( * ) SX, integer INCX)
       CSRSCL multiplies a vector by the reciprocal of a real scalar.

       Purpose:

            CSRSCL multiplies an n-element complex vector x by the real scalar
            1/a.  This is done without overflow or underflow as long as
            the final result x/a does not overflow or underflow.

       Parameters
           N

                     N is INTEGER
                     The number of components of the vector x.

           SA

                     SA is REAL
                     The scalar a which is used to divide each component of x.
                     SA must be >= 0, or the subroutine will divide by zero.

           SX

                     SX is COMPLEX array, dimension
                                    (1+(N-1)*abs(INCX))
                     The n-element vector x.

           INCX

                     INCX is INTEGER
                     The increment between successive values of the vector SX.
                     > 0:  SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i),     1< i<= n

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine ctprfb (character SIDE, character TRANS, character DIRECT, character STOREV,
       integer M, integer N, integer K, integer L, complex, dimension( ldv, * ) V, integer LDV,
       complex, dimension( ldt, * ) T, integer LDT, complex, dimension( lda, * ) A, integer LDA,
       complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldwork, * ) WORK, integer
       LDWORK)
       CTPRFB applies a complex 'triangular-pentagonal' block reflector to a complex matrix,
       which is composed of two blocks.

       Purpose:

            CTPRFB applies a complex 'triangular-pentagonal' block reflector H or its
            conjugate transpose H**H to a complex matrix C, which is composed of two
            blocks A and B, either from the left or right.

       Parameters
           SIDE

                     SIDE is CHARACTER*1
                     = 'L': apply H or H**H from the Left
                     = 'R': apply H or H**H from the Right

           TRANS

                     TRANS is CHARACTER*1
                     = 'N': apply H (No transpose)
                     = 'C': apply H**H (Conjugate transpose)

           DIRECT

                     DIRECT is CHARACTER*1
                     Indicates how H is formed from a product of elementary
                     reflectors
                     = 'F': H = H(1) H(2) . . . H(k) (Forward)
                     = 'B': H = H(k) . . . H(2) H(1) (Backward)

           STOREV

                     STOREV is CHARACTER*1
                     Indicates how the vectors which define the elementary
                     reflectors are stored:
                     = 'C': Columns
                     = 'R': Rows

           M

                     M is INTEGER
                     The number of rows of the matrix B.
                     M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrix B.
                     N >= 0.

           K

                     K is INTEGER
                     The order of the matrix T, i.e. the number of elementary
                     reflectors whose product defines the block reflector.
                     K >= 0.

           L

                     L is INTEGER
                     The order of the trapezoidal part of V.
                     K >= L >= 0.  See Further Details.

           V

                     V is COMPLEX array, dimension
                                           (LDV,K) if STOREV = 'C'
                                           (LDV,M) if STOREV = 'R' and SIDE = 'L'
                                           (LDV,N) if STOREV = 'R' and SIDE = 'R'
                     The pentagonal matrix V, which contains the elementary reflectors
                     H(1), H(2), ..., H(K).  See Further Details.

           LDV

                     LDV is INTEGER
                     The leading dimension of the array V.
                     If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
                     if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
                     if STOREV = 'R', LDV >= K.

           T

                     T is COMPLEX array, dimension (LDT,K)
                     The triangular K-by-K matrix T in the representation of the
                     block reflector.

           LDT

                     LDT is INTEGER
                     The leading dimension of the array T.
                     LDT >= K.

           A

                     A is COMPLEX array, dimension
                     (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'
                     On entry, the K-by-N or M-by-K matrix A.
                     On exit, A is overwritten by the corresponding block of
                     H*C or H**H*C or C*H or C*H**H.  See Further Details.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.
                     If SIDE = 'L', LDA >= max(1,K);
                     If SIDE = 'R', LDA >= max(1,M).

           B

                     B is COMPLEX array, dimension (LDB,N)
                     On entry, the M-by-N matrix B.
                     On exit, B is overwritten by the corresponding block of
                     H*C or H**H*C or C*H or C*H**H.  See Further Details.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.
                     LDB >= max(1,M).

           WORK

                     WORK is COMPLEX array, dimension
                     (LDWORK,N) if SIDE = 'L',
                     (LDWORK,K) if SIDE = 'R'.

           LDWORK

                     LDWORK is INTEGER
                     The leading dimension of the array WORK.
                     If SIDE = 'L', LDWORK >= K;
                     if SIDE = 'R', LDWORK >= M.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             The matrix C is a composite matrix formed from blocks A and B.
             The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K,
             and if SIDE = 'L', A is of size K-by-N.

             If SIDE = 'R' and DIRECT = 'F', C = [A B].

             If SIDE = 'L' and DIRECT = 'F', C = [A]
                                                 [B].

             If SIDE = 'R' and DIRECT = 'B', C = [B A].

             If SIDE = 'L' and DIRECT = 'B', C = [B]
                                                 [A].

             The pentagonal matrix V is composed of a rectangular block V1 and a
             trapezoidal block V2.  The size of the trapezoidal block is determined by
             the parameter L, where 0<=L<=K.  If L=K, the V2 block of V is triangular;
             if L=0, there is no trapezoidal block, thus V = V1 is rectangular.

             If DIRECT = 'F' and STOREV = 'C':  V = [V1]
                                                    [V2]
                - V2 is upper trapezoidal (first L rows of K-by-K upper triangular)

             If DIRECT = 'F' and STOREV = 'R':  V = [V1 V2]

                - V2 is lower trapezoidal (first L columns of K-by-K lower triangular)

             If DIRECT = 'B' and STOREV = 'C':  V = [V2]
                                                    [V1]
                - V2 is lower trapezoidal (last L rows of K-by-K lower triangular)

             If DIRECT = 'B' and STOREV = 'R':  V = [V2 V1]

                - V2 is upper trapezoidal (last L columns of K-by-K upper triangular)

             If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K.

             If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K.

             If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L.

             If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.

   integer function icmax1 (integer N, complex, dimension(*) CX, integer INCX)
       ICMAX1 finds the index of the first vector element of maximum absolute value.

       Purpose:

            ICMAX1 finds the index of the first vector element of maximum absolute value.

            Based on ICAMAX from Level 1 BLAS.
            The change is to use the 'genuine' absolute value.

       Parameters
           N

                     N is INTEGER
                     The number of elements in the vector CX.

           CX

                     CX is COMPLEX array, dimension (N)
                     The vector CX. The ICMAX1 function returns the index of its first
                     element of maximum absolute value.

           INCX

                     INCX is INTEGER
                     The spacing between successive values of CX.  INCX >= 1.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Nick Higham for use with CLACON.

   integer function ilaclc (integer M, integer N, complex, dimension( lda, * ) A, integer LDA)
       ILACLC scans a matrix for its last non-zero column.

       Purpose:

            ILACLC scans A for its last non-zero column.

       Parameters
           M

                     M is INTEGER
                     The number of rows of the matrix A.

           N

                     N is INTEGER
                     The number of columns of the matrix A.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The m by n matrix A.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A. LDA >= max(1,M).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   integer function ilaclr (integer M, integer N, complex, dimension( lda, * ) A, integer LDA)
       ILACLR scans a matrix for its last non-zero row.

       Purpose:

            ILACLR scans A for its last non-zero row.

       Parameters
           M

                     M is INTEGER
                     The number of rows of the matrix A.

           N

                     N is INTEGER
                     The number of columns of the matrix A.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The m by n matrix A.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A. LDA >= max(1,M).

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   integer function izmax1 (integer N, complex*16, dimension(*) ZX, integer INCX)
       IZMAX1 finds the index of the first vector element of maximum absolute value.

       Purpose:

            IZMAX1 finds the index of the first vector element of maximum absolute value.

            Based on IZAMAX from Level 1 BLAS.
            The change is to use the 'genuine' absolute value.

       Parameters
           N

                     N is INTEGER
                     The number of elements in the vector ZX.

           ZX

                     ZX is COMPLEX*16 array, dimension (N)
                     The vector ZX. The IZMAX1 function returns the index of its first
                     element of maximum absolute value.

           INCX

                     INCX is INTEGER
                     The spacing between successive values of ZX.  INCX >= 1.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Nick Higham for use with ZLACON.

   real function scsum1 (integer N, complex, dimension( * ) CX, integer INCX)
       SCSUM1 forms the 1-norm of the complex vector using the true absolute value.

       Purpose:

            SCSUM1 takes the sum of the absolute values of a complex
            vector and returns a single precision result.

            Based on SCASUM from the Level 1 BLAS.
            The change is to use the 'genuine' absolute value.

       Parameters
           N

                     N is INTEGER
                     The number of elements in the vector CX.

           CX

                     CX is COMPLEX array, dimension (N)
                     The vector whose elements will be summed.

           INCX

                     INCX is INTEGER
                     The spacing between successive values of CX.  INCX > 0.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Nick Higham for use with CLACON.

Author

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