Provided by: liblapack-doc_3.8.0-2_all bug

NAME

       realOTHERauxiliary

SYNOPSIS

   Functions
       integer function ilaslc (M, N, A, LDA)
           ILASLC scans a matrix for its last non-zero column.
       integer function ilaslr (M, N, A, LDA)
           ILASLR scans a matrix for its last non-zero row.
       subroutine slabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
           SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
       subroutine slacn2 (N, V, X, ISGN, EST, KASE, ISAVE)
           SLACN2 estimates the 1-norm of a square matrix, using reverse communication for
           evaluating matrix-vector products.
       subroutine slacon (N, V, X, ISGN, EST, KASE)
           SLACON estimates the 1-norm of a square matrix, using reverse communication for
           evaluating matrix-vector products.
       subroutine sladiv (A, B, C, D, P, Q)
           SLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
       subroutine sladiv1 (A, B, C, D, P, Q)
       real function sladiv2 (A, B, C, D, R, T)
       subroutine slaein (RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B, LDB, WORK, EPS3, SMLNUM,
           BIGNUM, INFO)
           SLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by
           inverse iteration.
       subroutine slaexc (WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK, INFO)
           SLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur
           canonical form, by an orthogonal similarity transformation.
       subroutine slag2 (A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI)
           SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with
           scaling as necessary to avoid over-/underflow.
       subroutine slags2 (UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
           SLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A
           and B such that the rows of the transformed A and B are parallel.
       subroutine slagtm (TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB)
           SLAGTM 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 slagv2 (A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, CSR, SNR)
           SLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil
           (A,B) where B is upper triangular.
       subroutine slahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, INFO)
           SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,
           using the double-shift/single-shift QR algorithm.
       subroutine slahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
           SLAHR2 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 slaic1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
           SLAIC1 applies one step of incremental condition estimation.
       subroutine slaln2 (LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB, WR, WI, X, LDX,
           SCALE, XNORM, INFO)
           SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
       real function slangt (NORM, N, DL, D, DU)
           SLANGT 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 slanhs (NORM, N, A, LDA, WORK)
           SLANHS 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 slansb (NORM, UPLO, N, K, AB, LDAB, WORK)
           SLANSB 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 slansp (NORM, UPLO, N, AP, WORK)
           SLANSP 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 slantb (NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
           SLANTB 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 slantp (NORM, UPLO, DIAG, N, AP, WORK)
           SLANTP 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 slantr (NORM, UPLO, DIAG, M, N, A, LDA, WORK)
           SLANTR 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 slanv2 (A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN)
           SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in
           standard form.
       subroutine slapll (N, X, INCX, Y, INCY, SSMIN)
           SLAPLL measures the linear dependence of two vectors.
       subroutine slapmr (FORWRD, M, N, X, LDX, K)
           SLAPMR rearranges rows of a matrix as specified by a permutation vector.
       subroutine slapmt (FORWRD, M, N, X, LDX, K)
           SLAPMT performs a forward or backward permutation of the columns of a matrix.
       subroutine slaqp2 (M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)
           SLAQP2 computes a QR factorization with column pivoting of the matrix block.
       subroutine slaqps (M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
           SLAQPS computes a step of QR factorization with column pivoting of a real m-by-n
           matrix A by using BLAS level 3.
       subroutine slaqr0 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK,
           LWORK, INFO)
           SLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
           from the Schur decomposition.
       subroutine slaqr1 (N, H, LDH, SR1, SI1, SR2, SI2, V)
           SLAQR1 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 slaqr2 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND,
           SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
           SLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to
           detect and deflate fully converged eigenvalues from a trailing principal submatrix
           (aggressive early deflation).
       subroutine slaqr3 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND,
           SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
           SLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to
           detect and deflate fully converged eigenvalues from a trailing principal submatrix
           (aggressive early deflation).
       subroutine slaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK,
           LWORK, INFO)
           SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
           from the Schur decomposition.
       subroutine slaqr5 (WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, SR, SI, H, LDH, ILOZ,
           IHIZ, Z, LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)
           SLAQR5 performs a single small-bulge multi-shift QR sweep.
       subroutine slaqsb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
           SLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by
           spbequ.
       subroutine slaqsp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
           SLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors
           computed by sppequ.
       subroutine slaqtr (LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK, INFO)
           SLAQTR solves a real quasi-triangular system of equations, or a complex quasi-
           triangular system of special form, in real arithmetic.
       subroutine slar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT,
           ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
           SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1
           through bn of the tridiagonal matrix LDLT - λI.
       subroutine slar2v (N, X, Y, Z, INCX, C, S, INCC)
           SLAR2V applies a vector of plane rotations with real cosines and real sines from both
           sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
       subroutine slarf (SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
           SLARF applies an elementary reflector to a general rectangular matrix.
       subroutine slarfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK,
           LDWORK)
           SLARFB applies a block reflector or its transpose to a general rectangular matrix.
       subroutine slarfg (N, ALPHA, X, INCX, TAU)
           SLARFG generates an elementary reflector (Householder matrix).
       subroutine slarfgp (N, ALPHA, X, INCX, TAU)
           SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
       subroutine slarft (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
           SLARFT forms the triangular factor T of a block reflector H = I - vtvH
       subroutine slarfx (SIDE, M, N, V, TAU, C, LDC, WORK)
           SLARFX applies an elementary reflector to a general rectangular matrix, with loop
           unrolling when the reflector has order ≤ 10.
       subroutine slargv (N, X, INCX, Y, INCY, C, INCC)
           SLARGV generates a vector of plane rotations with real cosines and real sines.
       subroutine slarrv (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)
           SLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and
           the eigenvalues of L D LT.
       subroutine slartv (N, X, INCX, Y, INCY, C, S, INCC)
           SLARTV applies a vector of plane rotations with real cosines and real sines to the
           elements of a pair of vectors.
       subroutine slaswp (N, A, LDA, K1, K2, IPIV, INCX)
           SLASWP performs a series of row interchanges on a general rectangular matrix.
       subroutine slatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
           SLATBS solves a triangular banded system of equations.
       subroutine slatdf (IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)
           SLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes
           a contribution to the reciprocal Dif-estimate.
       subroutine slatps (UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
           SLATPS solves a triangular system of equations with the matrix held in packed storage.
       subroutine slatrs (UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
           SLATRS solves a triangular system of equations with the scale factor set to prevent
           overflow.
       subroutine slauu2 (UPLO, N, A, LDA, INFO)
           SLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular
           matrices (unblocked algorithm).
       subroutine slauum (UPLO, N, A, LDA, INFO)
           SLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular
           matrices (blocked algorithm).
       subroutine srscl (N, SA, SX, INCX)
           SRSCL multiplies a vector by the reciprocal of a real scalar.
       subroutine stprfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B,
           LDB, WORK, LDWORK)
           STPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real
           or complex matrix, which is composed of two blocks.

Detailed Description

       This is the group of real other auxiliary routines

Function Documentation

   integer function ilaslc (integer M, integer N, real, dimension( lda, * ) A, integer LDA)
       ILASLC scans a matrix for its last non-zero column.

       Purpose:

            ILASLC 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 REAL 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.

       Date:
           June 2017

   integer function ilaslr (integer M, integer N, real, dimension( lda, * ) A, integer LDA)
       ILASLR scans a matrix for its last non-zero row.

       Purpose:

            ILASLR 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 REAL 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.

       Date:
           December 2016

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

       Purpose:

            SLABRD reduces the first NB rows and columns of a real general
            m by n matrix A to upper or lower bidiagonal form by an orthogonal
            transformation Q**T * 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 SGEBRD

       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 REAL 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 orthogonal
                       matrix Q as a product of elementary reflectors; and
                       elements above the diagonal in the first NB rows, with the
                       array TAUP, represent the orthogonal 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 orthogonal
                       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 orthogonal 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 REAL array, dimension (NB)
                     The scalar factors of the elementary reflectors which
                     represent the orthogonal matrix Q. See Further Details.

           TAUP

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

           X

                     X is REAL 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 REAL 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.

       Date:
           June 2017

       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**T  and G(i) = I - taup * u * u**T

             where tauq and taup are real scalars, and v and u are real 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**T 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**T - X*U**T.

             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 slacn2 (integer N, real, dimension( * ) V, real, dimension( * ) X, integer,
       dimension( * ) ISGN, real EST, integer KASE, integer, dimension( 3 ) ISAVE)
       SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating
       matrix-vector products.

       Purpose:

            SLACN2 estimates the 1-norm of a square, real 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 REAL array, dimension (N)
                    On the final return, V = A*W,  where  EST = norm(V)/norm(W)
                    (W is not returned).

           X

                     X is REAL array, dimension (N)
                    On an intermediate return, X should be overwritten by
                          A * X,   if KASE=1,
                          A**T * X,  if KASE=2,
                    and SLACN2 must be re-called with all the other parameters
                    unchanged.

           ISGN

                     ISGN is INTEGER array, dimension (N)

           EST

                     EST is REAL
                    On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
                    unchanged from the previous call to SLACN2.
                    On exit, EST is an estimate (a lower bound) for norm(A).

           KASE

                     KASE is INTEGER
                    On the initial call to SLACN2, 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**T * X.
                    On the final return from SLACN2, 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.

       Date:
           December 2016

       Further Details:

             Originally named SONEST, dated March 16, 1988.

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

                SLACON     SLACN2
                 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 slacon (integer N, real, dimension( * ) V, real, dimension( * ) X, integer,
       dimension( * ) ISGN, real EST, integer KASE)
       SLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating
       matrix-vector products.

       Purpose:

            SLACON estimates the 1-norm of a square, real 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 REAL array, dimension (N)
                    On the final return, V = A*W,  where  EST = norm(V)/norm(W)
                    (W is not returned).

           X

                     X is REAL array, dimension (N)
                    On an intermediate return, X should be overwritten by
                          A * X,   if KASE=1,
                          A**T * X,  if KASE=2,
                    and SLACON must be re-called with all the other parameters
                    unchanged.

           ISGN

                     ISGN is INTEGER array, dimension (N)

           EST

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

           KASE

                     KASE is INTEGER
                    On the initial call to SLACON, 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**T * X.
                    On the final return from SLACON, KASE will again be 0.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Contributors:
           Nick Higham, University of Manchester.
            Originally named SONEST, dated March 16, 1988.

       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 sladiv (real A, real B, real C, real D, real P, real Q)
       SLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.

       Purpose:

            SLADIV performs complex division in  real arithmetic

                                  a + i*b
                       p + i*q = ---------
                                  c + i*d

            The algorithm is due to Michael Baudin and Robert L. Smith
            and can be found in the paper
            "A Robust Complex Division in Scilab"

       Parameters:
           A

                     A is REAL

           B

                     B is REAL

           C

                     C is REAL

           D

                     D is REAL
                     The scalars a, b, c, and d in the above expression.

           P

                     P is REAL

           Q

                     Q is REAL
                     The scalars p and q in the above expression.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           January 2013

   subroutine slaein (logical RIGHTV, logical NOINIT, integer N, real, dimension( ldh, * ) H,
       integer LDH, real WR, real WI, real, dimension( * ) VR, real, dimension( * ) VI, real,
       dimension( ldb, * ) B, integer LDB, real, dimension( * ) WORK, real EPS3, real SMLNUM,
       real BIGNUM, integer INFO)
       SLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by
       inverse iteration.

       Purpose:

            SLAEIN uses inverse iteration to find a right or left eigenvector
            corresponding to the eigenvalue (WR,WI) of a real 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 (VR,VI).
                     = .FALSE.: initial vector supplied in (VR,VI).

           N

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

           H

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

           LDH

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

           WR

                     WR is REAL

           WI

                     WI is REAL
                     The real and imaginary parts of the eigenvalue of H whose
                     corresponding right or left eigenvector is to be computed.

           VR

                     VR is REAL array, dimension (N)

           VI

                     VI is REAL array, dimension (N)
                     On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain
                     a real starting vector for inverse iteration using the real
                     eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI
                     must contain the real and imaginary parts of a complex
                     starting vector for inverse iteration using the complex
                     eigenvalue (WR,WI); otherwise VR and VI need not be set.
                     On exit, if WI = 0.0 (real eigenvalue), VR contains the
                     computed real eigenvector; if WI.ne.0.0 (complex eigenvalue),
                     VR and VI contain the real and imaginary parts of the
                     computed complex eigenvector. The eigenvector is 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|.
                     VI is not referenced if WI = 0.0.

           B

                     B is REAL array, dimension (LDB,N)

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= N+1.

           WORK

                     WORK 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.

           BIGNUM

                     BIGNUM is REAL
                     A machine-dependent value close to the overflow threshold.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     = 1:  inverse iteration did not converge; VR is set to the
                           last iterate, and so is VI if WI.ne.0.0.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine slaexc (logical WANTQ, integer N, real, dimension( ldt, * ) T, integer LDT, real,
       dimension( ldq, * ) Q, integer LDQ, integer J1, integer N1, integer N2, real, dimension( *
       ) WORK, integer INFO)
       SLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur
       canonical form, by an orthogonal similarity transformation.

       Purpose:

            SLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in
            an upper quasi-triangular matrix T by an orthogonal similarity
            transformation.

            T must be in Schur canonical form, that is, block upper triangular
            with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
            has its diagonal elemnts equal and its off-diagonal elements of
            opposite sign.

       Parameters:
           WANTQ

                     WANTQ is LOGICAL
                     = .TRUE. : accumulate the transformation in the matrix Q;
                     = .FALSE.: do not accumulate the transformation.

           N

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

           T

                     T is REAL array, dimension (LDT,N)
                     On entry, the upper quasi-triangular matrix T, in Schur
                     canonical form.
                     On exit, the updated matrix T, again in Schur canonical form.

           LDT

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

           Q

                     Q is REAL array, dimension (LDQ,N)
                     On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
                     On exit, if WANTQ is .TRUE., the updated matrix Q.
                     If WANTQ is .FALSE., Q is not referenced.

           LDQ

                     LDQ is INTEGER
                     The leading dimension of the array Q.
                     LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.

           J1

                     J1 is INTEGER
                     The index of the first row of the first block T11.

           N1

                     N1 is INTEGER
                     The order of the first block T11. N1 = 0, 1 or 2.

           N2

                     N2 is INTEGER
                     The order of the second block T22. N2 = 0, 1 or 2.

           WORK

                     WORK is REAL array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     = 1: the transformed matrix T would be too far from Schur
                          form; the blocks are not swapped and T and Q are
                          unchanged.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine slag2 (real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B,
       integer LDB, real SAFMIN, real SCALE1, real SCALE2, real WR1, real WR2, real WI)
       SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as
       necessary to avoid over-/underflow.

       Purpose:

            SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
            problem  A - w B, with scaling as necessary to avoid over-/underflow.

            The scaling factor "s" results in a modified eigenvalue equation

                s A - w B

            where  s  is a non-negative scaling factor chosen so that  w,  w B,
            and  s A  do not overflow and, if possible, do not underflow, either.

       Parameters:
           A

                     A is REAL array, dimension (LDA, 2)
                     On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm
                     is less than 1/SAFMIN.  Entries less than
                     sqrt(SAFMIN)*norm(A) are subject to being treated as zero.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= 2.

           B

                     B is REAL array, dimension (LDB, 2)
                     On entry, the 2 x 2 upper triangular matrix B.  It is
                     assumed that the one-norm of B is less than 1/SAFMIN.  The
                     diagonals should be at least sqrt(SAFMIN) times the largest
                     element of B (in absolute value); if a diagonal is smaller
                     than that, then  +/- sqrt(SAFMIN) will be used instead of
                     that diagonal.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= 2.

           SAFMIN

                     SAFMIN is REAL
                     The smallest positive number s.t. 1/SAFMIN does not
                     overflow.  (This should always be SLAMCH('S') -- it is an
                     argument in order to avoid having to call SLAMCH frequently.)

           SCALE1

                     SCALE1 is REAL
                     A scaling factor used to avoid over-/underflow in the
                     eigenvalue equation which defines the first eigenvalue.  If
                     the eigenvalues are complex, then the eigenvalues are
                     ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
                     exponent range of the machine), SCALE1=SCALE2, and SCALE1
                     will always be positive.  If the eigenvalues are real, then
                     the first (real) eigenvalue is  WR1 / SCALE1 , but this may
                     overflow or underflow, and in fact, SCALE1 may be zero or
                     less than the underflow threshold if the exact eigenvalue
                     is sufficiently large.

           SCALE2

                     SCALE2 is REAL
                     A scaling factor used to avoid over-/underflow in the
                     eigenvalue equation which defines the second eigenvalue.  If
                     the eigenvalues are complex, then SCALE2=SCALE1.  If the
                     eigenvalues are real, then the second (real) eigenvalue is
                     WR2 / SCALE2 , but this may overflow or underflow, and in
                     fact, SCALE2 may be zero or less than the underflow
                     threshold if the exact eigenvalue is sufficiently large.

           WR1

                     WR1 is REAL
                     If the eigenvalue is real, then WR1 is SCALE1 times the
                     eigenvalue closest to the (2,2) element of A B**(-1).  If the
                     eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
                     part of the eigenvalues.

           WR2

                     WR2 is REAL
                     If the eigenvalue is real, then WR2 is SCALE2 times the
                     other eigenvalue.  If the eigenvalue is complex, then
                     WR1=WR2 is SCALE1 times the real part of the eigenvalues.

           WI

                     WI is REAL
                     If the eigenvalue is real, then WI is zero.  If the
                     eigenvalue is complex, then WI is SCALE1 times the imaginary
                     part of the eigenvalues.  WI will always be non-negative.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2016

   subroutine slags2 (logical UPPER, real A1, real A2, real A3, real B1, real B2, real B3, real
       CSU, real SNU, real CSV, real SNV, real CSQ, real SNQ)
       SLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and
       B such that the rows of the transformed A and B are parallel.

       Purpose:

            SLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
            that if ( UPPER ) then

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

            or if ( .NOT.UPPER ) then

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

            The rows of the transformed A and B are parallel, where

              U = (  CSU  SNU ), V = (  CSV SNV ), Q = (  CSQ   SNQ )
                  ( -SNU  CSU )      ( -SNV CSV )      ( -SNQ   CSQ )

            Z**T denotes the transpose of Z.

       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 REAL

           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 REAL

           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 REAL
                     The desired orthogonal matrix U.

           CSV

                     CSV is REAL

           SNV

                     SNV is REAL
                     The desired orthogonal matrix V.

           CSQ

                     CSQ is REAL

           SNQ

                     SNQ is REAL
                     The desired orthogonal matrix Q.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine slagtm (character TRANS, integer N, integer NRHS, real ALPHA, real, dimension( * )
       DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( ldx, * ) X, integer
       LDX, real BETA, real, dimension( ldb, * ) B, integer LDB)
       SLAGTM 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:

            SLAGTM 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'* X + beta * B
                     = 'C':  Conjugate transpose = Transpose

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

           D

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

           DU

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

           X

                     X is REAL 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 REAL 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.

       Date:
           December 2016

   subroutine slagv2 (real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B,
       integer LDB, real, dimension( 2 ) ALPHAR, real, dimension( 2 ) ALPHAI, real, dimension( 2
       ) BETA, real CSL, real SNL, real CSR, real SNR)
       SLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B)
       where B is upper triangular.

       Purpose:

            SLAGV2 computes the Generalized Schur factorization of a real 2-by-2
            matrix pencil (A,B) where B is upper triangular. This routine
            computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
            SNR such that

            1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
               types), then

               [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
               [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

               [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
               [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],

            2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
               then

               [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
               [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

               [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
               [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]

               where b11 >= b22 > 0.

       Parameters:
           A

                     A is REAL array, dimension (LDA, 2)
                     On entry, the 2 x 2 matrix A.
                     On exit, A is overwritten by the ``A-part'' of the
                     generalized Schur form.

           LDA

                     LDA is INTEGER
                     THe leading dimension of the array A.  LDA >= 2.

           B

                     B is REAL array, dimension (LDB, 2)
                     On entry, the upper triangular 2 x 2 matrix B.
                     On exit, B is overwritten by the ``B-part'' of the
                     generalized Schur form.

           LDB

                     LDB is INTEGER
                     THe leading dimension of the array B.  LDB >= 2.

           ALPHAR

                     ALPHAR is REAL array, dimension (2)

           ALPHAI

                     ALPHAI is REAL array, dimension (2)

           BETA

                     BETA is REAL array, dimension (2)
                     (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
                     pencil (A,B), k=1,2, i = sqrt(-1).  Note that BETA(k) may
                     be zero.

           CSL

                     CSL is REAL
                     The cosine of the left rotation matrix.

           SNL

                     SNL is REAL
                     The sine of the left rotation matrix.

           CSR

                     CSR is REAL
                     The cosine of the right rotation matrix.

           SNR

                     SNR is REAL
                     The sine of the right rotation matrix.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Contributors:
           Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

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

       Purpose:

               SLAHQR is an auxiliary routine called by SHSEQR to update the
               eigenvalues and Schur decomposition already computed by SHSEQR, 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 quasi-triangular in
                     rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
                     ILO = 1). SLAHQR 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 REAL array, dimension (LDH,N)
                     On entry, the upper Hessenberg matrix H.
                     On exit, if INFO is zero and if WANTT is .TRUE., H is upper
                     quasi-triangular in rows and columns ILO:IHI, with any
                     2-by-2 diagonal blocks in standard form. If INFO is zero
                     and WANTT is .FALSE., the contents of H are unspecified on
                     exit.  The output state of H if INFO is nonzero is given
                     below under the description of INFO.

           LDH

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

           WR

                     WR is REAL array, dimension (N)

           WI

                     WI is REAL array, dimension (N)
                     The real and imaginary parts, respectively, of the computed
                     eigenvalues ILO to IHI are stored in the corresponding
                     elements of WR and WI. If two eigenvalues are computed as a
                     complex conjugate pair, they are stored in consecutive
                     elements of WR and WI, say the i-th and (i+1)th, with
                     WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
                     eigenvalues are stored in the same order as on the diagonal
                     of the Schur form returned in H, with WR(i) = H(i,i), and, if
                     H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
                     WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(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 REAL array, dimension (LDZ,N)
                     If WANTZ is .TRUE., on entry Z must contain the current
                     matrix Z of transformations accumulated by SHSEQR, 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
                     .GT. 0: If INFO = i, SLAHQR failed to compute all the
                             eigenvalues ILO to IHI in a total of 30 iterations
                             per eigenvalue; elements i+1:ihi of WR and WI
                             contain those eigenvalues which have been
                             successfully computed.

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

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

                             If INFO .GT. 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.

       Date:
           December 2016

       Further Details:

                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 SLAHQR 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 slahr2 (integer N, integer K, integer NB, real, dimension( lda, * ) A, integer LDA,
       real, dimension( nb ) TAU, real, dimension( ldt, nb ) T, integer LDT, real, dimension(
       ldy, nb ) Y, integer LDY)
       SLAHR2 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:

            SLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
            matrix A so that elements below the k-th subdiagonal are zero. The
            reduction is performed by an orthogonal similarity transformation
            Q**T * A * Q. The routine returns the matrices V and T which determine
            Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.

            This is an auxiliary routine called by SGEHRD.

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

           T

                     T is REAL 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 REAL 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.

       Date:
           December 2016

       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**T

             where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).

             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 DLAHRD
             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 DLAHRD routine. (This
             subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)

       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 slaic1 (integer JOB, integer J, real, dimension( j ) X, real SEST, real, dimension(
       j ) W, real GAMMA, real SESTPR, real S, real C)
       SLAIC1 applies one step of incremental condition estimation.

       Purpose:

            SLAIC1 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 SLAIC1 computes sestpr, s, c such that
            the vector
                            [ s*x ]
                     xhat = [  c  ]
            is an approximate singular vector of
                            [ L      0  ]
                     Lhat = [ w**T 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]**T and sestpr**2 is an eigenpair of the system

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

            where  alpha =  x**T*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 REAL array, dimension (J)
                     The j-vector x.

           SEST

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

           W

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

           GAMMA

                     GAMMA is REAL
                     The diagonal element gamma.

           SESTPR

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

           S

                     S is REAL
                     Sine needed in forming xhat.

           C

                     C is REAL
                     Cosine needed in forming xhat.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine slaln2 (logical LTRANS, integer NA, integer NW, real SMIN, real CA, real,
       dimension( lda, * ) A, integer LDA, real D1, real D2, real, dimension( ldb, * ) B, integer
       LDB, real WR, real WI, real, dimension( ldx, * ) X, integer LDX, real SCALE, real XNORM,
       integer INFO)
       SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.

       Purpose:

            SLALN2 solves a system of the form  (ca A - w D ) X = s B
            or (ca A**T - w D) X = s B   with possible scaling ("s") and
            perturbation of A.  (A**T means A-transpose.)

            A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
            real diagonal matrix, w is a real or complex value, and X and B are
            NA x 1 matrices -- real if w is real, complex if w is complex.  NA
            may be 1 or 2.

            If w is complex, X and B are represented as NA x 2 matrices,
            the first column of each being the real part and the second
            being the imaginary part.

            "s" is a scaling factor (.LE. 1), computed by SLALN2, which is
            so chosen that X can be computed without overflow.  X is further
            scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
            than overflow.

            If both singular values of (ca A - w D) are less than SMIN,
            SMIN*identity will be used instead of (ca A - w D).  If only one
            singular value is less than SMIN, one element of (ca A - w D) will be
            perturbed enough to make the smallest singular value roughly SMIN.
            If both singular values are at least SMIN, (ca A - w D) will not be
            perturbed.  In any case, the perturbation will be at most some small
            multiple of max( SMIN, ulp*norm(ca A - w D) ).  The singular values
            are computed by infinity-norm approximations, and thus will only be
            correct to a factor of 2 or so.

            Note: all input quantities are assumed to be smaller than overflow
            by a reasonable factor.  (See BIGNUM.)

       Parameters:
           LTRANS

                     LTRANS is LOGICAL
                     =.TRUE.:  A-transpose will be used.
                     =.FALSE.: A will be used (not transposed.)

           NA

                     NA is INTEGER
                     The size of the matrix A.  It may (only) be 1 or 2.

           NW

                     NW is INTEGER
                     1 if "w" is real, 2 if "w" is complex.  It may only be 1
                     or 2.

           SMIN

                     SMIN is REAL
                     The desired lower bound on the singular values of A.  This
                     should be a safe distance away from underflow or overflow,
                     say, between (underflow/machine precision) and  (machine
                     precision * overflow ).  (See BIGNUM and ULP.)

           CA

                     CA is REAL
                     The coefficient c, which A is multiplied by.

           A

                     A is REAL array, dimension (LDA,NA)
                     The NA x NA matrix A.

           LDA

                     LDA is INTEGER
                     The leading dimension of A.  It must be at least NA.

           D1

                     D1 is REAL
                     The 1,1 element in the diagonal matrix D.

           D2

                     D2 is REAL
                     The 2,2 element in the diagonal matrix D.  Not used if NA=1.

           B

                     B is REAL array, dimension (LDB,NW)
                     The NA x NW matrix B (right-hand side).  If NW=2 ("w" is
                     complex), column 1 contains the real part of B and column 2
                     contains the imaginary part.

           LDB

                     LDB is INTEGER
                     The leading dimension of B.  It must be at least NA.

           WR

                     WR is REAL
                     The real part of the scalar "w".

           WI

                     WI is REAL
                     The imaginary part of the scalar "w".  Not used if NW=1.

           X

                     X is REAL array, dimension (LDX,NW)
                     The NA x NW matrix X (unknowns), as computed by SLALN2.
                     If NW=2 ("w" is complex), on exit, column 1 will contain
                     the real part of X and column 2 will contain the imaginary
                     part.

           LDX

                     LDX is INTEGER
                     The leading dimension of X.  It must be at least NA.

           SCALE

                     SCALE is REAL
                     The scale factor that B must be multiplied by to insure
                     that overflow does not occur when computing X.  Thus,
                     (ca A - w D) X  will be SCALE*B, not B (ignoring
                     perturbations of A.)  It will be at most 1.

           XNORM

                     XNORM is REAL
                     The infinity-norm of X, when X is regarded as an NA x NW
                     real matrix.

           INFO

                     INFO is INTEGER
                     An error flag.  It will be set to zero if no error occurs,
                     a negative number if an argument is in error, or a positive
                     number if  ca A - w D  had to be perturbed.
                     The possible values are:
                     = 0: No error occurred, and (ca A - w D) did not have to be
                            perturbed.
                     = 1: (ca A - w D) had to be perturbed to make its smallest
                          (or only) singular value greater than SMIN.
                     NOTE: In the interests of speed, this routine does not
                           check the inputs for errors.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   real function slangt (character NORM, integer N, real, dimension( * ) DL, real, dimension( * )
       D, real, dimension( * ) DU)
       SLANGT 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:

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

       Returns:
           SLANGT

               SLANGT = ( 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 SLANGT as described
                     above.

           N

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

           DL

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

           D

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

           DU

                     DU is REAL 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.

       Date:
           December 2016

   real function slanhs (character NORM, integer N, real, dimension( lda, * ) A, integer LDA,
       real, dimension( * ) WORK)
       SLANHS 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:

            SLANHS  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:
           SLANHS

               SLANHS = ( 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 SLANHS as described
                     above.

           N

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

           A

                     A is REAL 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.

       Date:
           December 2016

   real function slansb (character NORM, character UPLO, integer N, integer K, real, dimension(
       ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)
       SLANSB 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:

            SLANSB  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:
           SLANSB

               SLANSB = ( 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 SLANSB 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, SLANSB 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 REAL 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.

       Date:
           December 2016

   real function slansp (character NORM, character UPLO, integer N, real, dimension( * ) AP,
       real, dimension( * ) WORK)
       SLANSP 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:

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

       Returns:
           SLANSP

               SLANSP = ( 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 SLANSP 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, SLANSP is
                     set to zero.

           AP

                     AP is REAL 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.

       Date:
           December 2016

   real function slantb (character NORM, character UPLO, character DIAG, integer N, integer K,
       real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)
       SLANTB 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:

            SLANTB  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:
           SLANTB

               SLANTB = ( 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 SLANTB 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, SLANTB 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 REAL 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.

       Date:
           December 2016

   real function slantp (character NORM, character UPLO, character DIAG, integer N, real,
       dimension( * ) AP, real, dimension( * ) WORK)
       SLANTP 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:

            SLANTP  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:
           SLANTP

               SLANTP = ( 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 SLANTP 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, SLANTP is
                     set to zero.

           AP

                     AP is REAL 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.

       Date:
           December 2016

   real function slantr (character NORM, character UPLO, character DIAG, integer M, integer N,
       real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)
       SLANTR 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:

            SLANTR  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:
           SLANTR

               SLANTR = ( 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 SLANTR 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, SLANTR 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, SLANTR is set to zero.

           A

                     A is REAL 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.

       Date:
           December 2016

   subroutine slanv2 (real A, real B, real C, real D, real RT1R, real RT1I, real RT2R, real RT2I,
       real CS, real SN)
       SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard
       form.

       Purpose:

            SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
            matrix in standard form:

                 [ A  B ] = [ CS -SN ] [ AA  BB ] [ CS  SN ]
                 [ C  D ]   [ SN  CS ] [ CC  DD ] [-SN  CS ]

            where either
            1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
            2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
            conjugate eigenvalues.

       Parameters:
           A

                     A is REAL

           B

                     B is REAL

           C

                     C is REAL

           D

                     D is REAL
                     On entry, the elements of the input matrix.
                     On exit, they are overwritten by the elements of the
                     standardised Schur form.

           RT1R

                     RT1R is REAL

           RT1I

                     RT1I is REAL

           RT2R

                     RT2R is REAL

           RT2I

                     RT2I is REAL
                     The real and imaginary parts of the eigenvalues. If the
                     eigenvalues are a complex conjugate pair, RT1I > 0.

           CS

                     CS is REAL

           SN

                     SN is REAL
                     Parameters of the rotation matrix.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             Modified by V. Sima, Research Institute for Informatics, Bucharest,
             Romania, to reduce the risk of cancellation errors,
             when computing real eigenvalues, and to ensure, if possible, that
             abs(RT1R) >= abs(RT2R).

   subroutine slapll (integer N, real, dimension( * ) X, integer INCX, real, dimension( * ) Y,
       integer INCY, real SSMIN)
       SLAPLL 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 REAL 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 REAL 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.

       Date:
           December 2016

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

       Purpose:

            SLAPMR 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 REAL 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.

       Date:
           December 2016

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

       Purpose:

            SLAPMT 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 REAL 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.

       Date:
           December 2016

   subroutine slaqp2 (integer M, integer N, integer OFFSET, real, dimension( lda, * ) A, integer
       LDA, integer, dimension( * ) JPVT, real, dimension( * ) TAU, real, dimension( * ) VN1,
       real, dimension( * ) VN2, real, dimension( * ) WORK)
       SLAQP2 computes a QR factorization with column pivoting of the matrix block.

       Purpose:

            SLAQP2 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 REAL 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 REAL 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 REAL array, dimension (N)

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

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

       Purpose:

            SLAQPS computes a step of QR factorization with column pivoting
            of a real 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 REAL 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 REAL 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 REAL array, dimension (NB)
                     Auxiliar vector.

           F

                     F is REAL array, dimension (LDF,NB)
                     Matrix F**T = L*Y**T*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.

       Date:
           December 2016

       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 slaqr0 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, real,
       dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI,
       integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * )
       WORK, integer LWORK, integer INFO)
       SLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
       the Schur decomposition.

       Purpose:

               SLAQR0 computes the eigenvalues of a Hessenberg matrix H
               and, optionally, the matrices T and Z from the Schur decomposition
               H = Z T Z**T, where T is an upper quasi-triangular matrix (the
               Schur form), and Z is the orthogonal matrix of Schur vectors.

               Optionally Z may be postmultiplied into an input orthogonal
               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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

       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 .GE. 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.GT.1,
                      H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
                      previous call to SGEBAL, and then passed to SGEHRD when the
                      matrix output by SGEBAL is reduced to Hessenberg form.
                      Otherwise, ILO and IHI should be set to 1 and N,
                      respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
                      If N = 0, then ILO = 1 and IHI = 0.

           H

                     H is REAL 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 quasi-triangular matrix T from the Schur
                      decomposition (the Schur form); 2-by-2 diagonal blocks
                      (corresponding to complex conjugate pairs of eigenvalues)
                      are returned in standard form, with H(i,i) = H(i+1,i+1)
                      and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
                      .FALSE., then the contents of H are unspecified on exit.
                      (The output value of H when INFO.GT.0 is given under the
                      description of INFO below.)

                      This subroutine may explicitly set H(i,j) = 0 for i.GT.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 .GE. max(1,N).

           WR

                     WR is REAL array, dimension (IHI)

           WI

                     WI is REAL array, dimension (IHI)
                      The real and imaginary parts, respectively, of the computed
                      eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
                      and WI(ILO:IHI). If two eigenvalues are computed as a
                      complex conjugate pair, they are stored in consecutive
                      elements of WR and WI, say the i-th and (i+1)th, with
                      WI(i) .GT. 0 and WI(i+1) .LT. 0. 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
                      WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
                      block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
                      WI(i+1) = -WI(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 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

           Z

                     Z is REAL 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.GT.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.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

           WORK

                     WORK is REAL 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 .GE. 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 SLAQR0 does a workspace query.
                      In this case, SLAQR0 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
                      .GT. 0:  if INFO = i, SLAQR0 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 .GT. 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 .GT. 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 quasi-triangular
                           in rows and columns INFO+1 through IHI.

                           If INFO .GT. 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 orthogonal matrix in (*) (regard-
                           less of the value of WANTT.)

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       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 slaqr1 (integer N, real, dimension( ldh, * ) H, integer LDH, real SR1, real SI1,
       real SR2, real SI2, real, dimension( * ) V)
       SLAQR1 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, SLAQR1 sets v to a
                 scalar multiple of the first column of the product

                 (*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)

                 scaling to avoid overflows and most underflows. It
                 is assumed that either

                         1) sr1 = sr2 and si1 = -si2
                     or
                         2) si1 = si2 = 0.

                 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 REAL 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.GE.N

           SR1

                     SR1 is REAL

           SI1

                     SI1 is REAL

           SR2

                     SR2 is REAL

           SI2

                     SI2 is REAL
                         The shifts in (*).

           V

                     V is REAL 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.

       Date:
           June 2017

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

   subroutine slaqr2 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT,
       integer NW, real, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, real,
       dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, real, dimension( * ) SR, real,
       dimension( * ) SI, real, dimension( ldv, * ) V, integer LDV, integer NH, real, dimension(
       ldt, * ) T, integer LDT, integer NV, real, dimension( ldwv, * ) WV, integer LDWV, real,
       dimension( * ) WORK, integer LWORK)
       SLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect
       and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive
       early deflation).

       Purpose:

               SLAQR2 is identical to SLAQR3 except that it avoids
               recursion by calling SLAHQR instead of SLAQR4.

               Aggressive early deflation:

               This subroutine accepts as input an upper Hessenberg matrix
               H and performs an orthogonal 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 orthogonal 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 quasi-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 orthogonal matrix Z is updated so
                     so that the orthogonal 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 orthogonal 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 .LE. NW .LE. (KBOT-KTOP+1).

           H

                     H is REAL 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 an orthogonal
                     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 .LE. 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 .LE. ILOZ .LE. IHIZ .LE. N.

           Z

                     Z is REAL array, dimension (LDZ,N)
                     IF WANTZ is .TRUE., then on output, the orthogonal
                     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 .LE. 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.

           SR

                     SR is REAL array, dimension (KBOT)

           SI

                     SI is REAL array, dimension (KBOT)
                     On output, the real and imaginary parts of approximate
                     eigenvalues that may be used for shifts are stored in
                     SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
                     SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
                     The real and imaginary parts of converged eigenvalues
                     are stored in SR(KBOT-ND+1) through SR(KBOT) and
                     SI(KBOT-ND+1) through SI(KBOT), respectively.

           V

                     V is REAL 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 .LE. LDV

           NH

                     NH is INTEGER
                     The number of columns of T.  NH.GE.NW.

           T

                     T is REAL array, dimension (LDT,NW)

           LDT

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

           NV

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

           WV

                     WV is REAL array, dimension (LDWV,NW)

           LDWV

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

           WORK

                     WORK is REAL 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; SLAQR2
                     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.

       Date:
           June 2017

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

   subroutine slaqr3 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT,
       integer NW, real, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, real,
       dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, real, dimension( * ) SR, real,
       dimension( * ) SI, real, dimension( ldv, * ) V, integer LDV, integer NH, real, dimension(
       ldt, * ) T, integer LDT, integer NV, real, dimension( ldwv, * ) WV, integer LDWV, real,
       dimension( * ) WORK, integer LWORK)
       SLAQR3 performs the orthogonal 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:

               SLAQR3 accepts as input an upper Hessenberg matrix
               H and performs an orthogonal 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 orthogonal 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 quasi-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 orthogonal matrix Z is updated so
                     so that the orthogonal 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 orthogonal 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 .LE. NW .LE. (KBOT-KTOP+1).

           H

                     H is REAL 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 an orthogonal
                     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 .LE. 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 .LE. ILOZ .LE. IHIZ .LE. N.

           Z

                     Z is REAL array, dimension (LDZ,N)
                     IF WANTZ is .TRUE., then on output, the orthogonal
                     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 .LE. 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.

           SR

                     SR is REAL array, dimension (KBOT)

           SI

                     SI is REAL array, dimension (KBOT)
                     On output, the real and imaginary parts of approximate
                     eigenvalues that may be used for shifts are stored in
                     SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
                     SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
                     The real and imaginary parts of converged eigenvalues
                     are stored in SR(KBOT-ND+1) through SR(KBOT) and
                     SI(KBOT-ND+1) through SI(KBOT), respectively.

           V

                     V is REAL 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 .LE. LDV

           NH

                     NH is INTEGER
                     The number of columns of T.  NH.GE.NW.

           T

                     T is REAL array, dimension (LDT,NW)

           LDT

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

           NV

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

           WV

                     WV is REAL array, dimension (LDWV,NW)

           LDWV

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

           WORK

                     WORK is REAL 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; SLAQR3
                     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.

       Date:
           June 2016

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

   subroutine slaqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, real,
       dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI,
       integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * )
       WORK, integer LWORK, integer INFO)
       SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
       the Schur decomposition.

       Purpose:

               SLAQR4 implements one level of recursion for SLAQR0.
               It is a complete implementation of the small bulge multi-shift
               QR algorithm.  It may be called by SLAQR0 and, for large enough
               deflation window size, it may be called by SLAQR3.  This
               subroutine is identical to SLAQR0 except that it calls SLAQR2
               instead of SLAQR3.

               SLAQR4 computes the eigenvalues of a Hessenberg matrix H
               and, optionally, the matrices T and Z from the Schur decomposition
               H = Z T Z**T, where T is an upper quasi-triangular matrix (the
               Schur form), and Z is the orthogonal matrix of Schur vectors.

               Optionally Z may be postmultiplied into an input orthogonal
               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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

       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 .GE. 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.GT.1,
                      H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
                      previous call to SGEBAL, and then passed to SGEHRD when the
                      matrix output by SGEBAL is reduced to Hessenberg form.
                      Otherwise, ILO and IHI should be set to 1 and N,
                      respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
                      If N = 0, then ILO = 1 and IHI = 0.

           H

                     H is REAL 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 quasi-triangular matrix T from the Schur
                      decomposition (the Schur form); 2-by-2 diagonal blocks
                      (corresponding to complex conjugate pairs of eigenvalues)
                      are returned in standard form, with H(i,i) = H(i+1,i+1)
                      and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
                      .FALSE., then the contents of H are unspecified on exit.
                      (The output value of H when INFO.GT.0 is given under the
                      description of INFO below.)

                      This subroutine may explicitly set H(i,j) = 0 for i.GT.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 .GE. max(1,N).

           WR

                     WR is REAL array, dimension (IHI)

           WI

                     WI is REAL array, dimension (IHI)
                      The real and imaginary parts, respectively, of the computed
                      eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
                      and WI(ILO:IHI). If two eigenvalues are computed as a
                      complex conjugate pair, they are stored in consecutive
                      elements of WR and WI, say the i-th and (i+1)th, with
                      WI(i) .GT. 0 and WI(i+1) .LT. 0. 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
                      WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
                      block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
                      WI(i+1) = -WI(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 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

           Z

                     Z is REAL 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.GT.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.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

           WORK

                     WORK is REAL 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 .GE. 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 SLAQR4 does a workspace query.
                      In this case, SLAQR4 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
            batim
                     INFO is INTEGER
                        =  0:  successful exit
                      .GT. 0:  if INFO = i, SLAQR4 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 .GT. 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 .GT. 0 and WANTT is .TRUE., then on exit

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

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

                           If INFO .GT. 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 orthogonal matrix in (*) (regard-
                           less of the value of WANTT.)

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       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 slaqr5 (logical WANTT, logical WANTZ, integer KACC22, integer N, integer KTOP,
       integer KBOT, integer NSHFTS, real, dimension( * ) SR, real, dimension( * ) SI, real,
       dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, real, dimension( ldz, * )
       Z, integer LDZ, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldu, * ) U,
       integer LDU, integer NV, real, dimension( ldwv, * ) WV, integer LDWV, integer NH, real,
       dimension( ldwh, * ) WH, integer LDWH)
       SLAQR5 performs a single small-bulge multi-shift QR sweep.

       Purpose:

               SLAQR5, called by SLAQR0, performs a
               single small-bulge multi-shift QR sweep.

       Parameters:
           WANTT

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

           WANTZ

                     WANTZ is LOGICAL
                        WANTZ = .true. if the orthogonal 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: SLAQR5 does not accumulate reflections and does not
                        use matrix-matrix multiply to update far-from-diagonal
                        matrix entries.
                   = 1: SLAQR5 accumulates reflections and uses matrix-matrix
                        multiply to update the far-from-diagonal matrix entries.
                   = 2: SLAQR5 accumulates reflections, uses matrix-matrix
                        multiply to update the far-from-diagonal matrix entries,
                        and takes advantage of 2-by-2 block structure during
                        matrix multiplies.

           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.

           SR

                     SR is REAL array, dimension (NSHFTS)

           SI

                     SI is REAL array, dimension (NSHFTS)
                        SR contains the real parts and SI contains the imaginary
                        parts of the NSHFTS shifts of origin that define the
                        multi-shift QR sweep.  On output SR and SI may be
                        reordered.

           H

                     H is REAL 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.GE.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 .LE. ILOZ .LE. IHIZ .LE. N

           Z

                     Z is REAL array, dimension (LDZ,IHIZ)
                        If WANTZ = .TRUE., then the QR Sweep orthogonal
                        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.GE.N.

           V

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

           LDV

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

           U

                     U is REAL array, dimension (LDU,3*NSHFTS-3)

           LDU

                     LDU is INTEGER
                        LDU is the leading dimension of U just as declared in the
                        in the calling subroutine.  LDU.GE.3*NSHFTS-3.

           NH

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

           WH

                     WH is REAL array, dimension (LDWH,NH)

           LDWH

                     LDWH is INTEGER
                        Leading dimension of WH just as declared in the
                        calling procedure.  LDWH.GE.3*NSHFTS-3.

           NV

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

           WV

                     WV is REAL array, dimension (LDWV,3*NSHFTS-3)

           LDWV

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2016

       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.

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

       Purpose:

            SLAQSB 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 REAL 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**T*U or A = L*L**T 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.

       Date:
           December 2016

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

       Purpose:

            SLAQSP 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 REAL 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.

       Date:
           December 2016

   subroutine slaqtr (logical LTRAN, logical LREAL, integer N, real, dimension( ldt, * ) T,
       integer LDT, real, dimension( * ) B, real W, real SCALE, real, dimension( * ) X, real,
       dimension( * ) WORK, integer INFO)
       SLAQTR solves a real quasi-triangular system of equations, or a complex quasi-triangular
       system of special form, in real arithmetic.

       Purpose:

            SLAQTR solves the real quasi-triangular system

                         op(T)*p = scale*c,               if LREAL = .TRUE.

            or the complex quasi-triangular systems

                       op(T + iB)*(p+iq) = scale*(c+id),  if LREAL = .FALSE.

            in real arithmetic, where T is upper quasi-triangular.
            If LREAL = .FALSE., then the first diagonal block of T must be
            1 by 1, B is the specially structured matrix

                           B = [ b(1) b(2) ... b(n) ]
                               [       w            ]
                               [           w        ]
                               [              .     ]
                               [                 w  ]

            op(A) = A or A**T, A**T denotes the transpose of
            matrix A.

            On input, X = [ c ].  On output, X = [ p ].
                          [ d ]                  [ q ]

            This subroutine is designed for the condition number estimation
            in routine STRSNA.

       Parameters:
           LTRAN

                     LTRAN is LOGICAL
                     On entry, LTRAN specifies the option of conjugate transpose:
                        = .FALSE.,    op(T+i*B) = T+i*B,
                        = .TRUE.,     op(T+i*B) = (T+i*B)**T.

           LREAL

                     LREAL is LOGICAL
                     On entry, LREAL specifies the input matrix structure:
                        = .FALSE.,    the input is complex
                        = .TRUE.,     the input is real

           N

                     N is INTEGER
                     On entry, N specifies the order of T+i*B. N >= 0.

           T

                     T is REAL array, dimension (LDT,N)
                     On entry, T contains a matrix in Schur canonical form.
                     If LREAL = .FALSE., then the first diagonal block of T must
                     be 1 by 1.

           LDT

                     LDT is INTEGER
                     The leading dimension of the matrix T. LDT >= max(1,N).

           B

                     B is REAL array, dimension (N)
                     On entry, B contains the elements to form the matrix
                     B as described above.
                     If LREAL = .TRUE., B is not referenced.

           W

                     W is REAL
                     On entry, W is the diagonal element of the matrix B.
                     If LREAL = .TRUE., W is not referenced.

           SCALE

                     SCALE is REAL
                     On exit, SCALE is the scale factor.

           X

                     X is REAL array, dimension (2*N)
                     On entry, X contains the right hand side of the system.
                     On exit, X is overwritten by the solution.

           WORK

                     WORK is REAL array, dimension (N)

           INFO

                     INFO is INTEGER
                     On exit, INFO is set to
                        0: successful exit.
                          1: the some diagonal 1 by 1 block has been perturbed by
                             a small number SMIN to keep nonsingularity.
                          2: the some diagonal 2 by 2 block has been perturbed by
                             a small number in SLALN2 to keep nonsingularity.
                     NOTE: In the interests of speed, this routine does not
                           check the inputs for errors.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine slar1v (integer N, integer B1, integer BN, real LAMBDA, real, dimension( * ) D,
       real, dimension( * ) L, real, dimension( * ) LD, real, dimension( * ) LLD, real PIVMIN,
       real GAPTOL, real, dimension( * ) Z, logical WANTNC, integer NEGCNT, real ZTZ, real
       MINGMA, integer R, integer, dimension( * ) ISUPPZ, real NRMINV, real RESID, real RQCORR,
       real, dimension( * ) WORK)
       SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1
       through bn of the tridiagonal matrix LDLT - λI.

       Purpose:

            SLAR1V 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 REAL 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.

       Date:
           December 2016

       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 slar2v (integer N, real, dimension( * ) X, real, dimension( * ) Y, real, dimension(
       * ) Z, integer INCX, real, dimension( * ) C, real, dimension( * ) S, integer INCC)
       SLAR2V applies a vector of plane rotations with real cosines and real sines from both
       sides to a sequence of 2-by-2 symmetric/Hermitian matrices.

       Purpose:

            SLAR2V applies a vector of real plane rotations from both sides to
            a sequence of 2-by-2 real symmetric matrices, defined by the elements
            of the vectors x, y and z. For i = 1,2,...,n

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

       Parameters:
           N

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

           X

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

           Y

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

           Z

                     Z is REAL 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 REAL 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.

       Date:
           December 2016

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

       Purpose:

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

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

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

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

       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 REAL 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 REAL
                     The value tau in the representation of H.

           C

                     C is REAL 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 REAL 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.

       Date:
           December 2016

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

       Purpose:

            SLARFB applies a real block reflector H or its transpose H**T to a
            real m by n matrix C, from either the left or the right.

       Parameters:
           SIDE

                     SIDE is CHARACTER*1
                     = 'L': apply H or H**T from the Left
                     = 'R': apply H or H**T from the Right

           TRANS

                     TRANS is CHARACTER*1
                     = 'N': apply H (No transpose)
                     = 'T': apply H**T (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).

           V

                     V is REAL 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 REAL 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 REAL array, dimension (LDC,N)
                     On entry, the m by n matrix C.
                     On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.

           LDC

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

           WORK

                     WORK is REAL 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.

       Date:
           June 2013

       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 slarfg (integer N, real ALPHA, real, dimension( * ) X, integer INCX, real TAU)
       SLARFG generates an elementary reflector (Householder matrix).

       Purpose:

            SLARFG generates a real elementary reflector H of order n, such
            that

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

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

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

            where tau is a real scalar and v is a real (n-1)-element
            vector.

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

            Otherwise  1 <= tau <= 2.

       Parameters:
           N

                     N is INTEGER
                     The order of the elementary reflector.

           ALPHA

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

           X

                     X is REAL 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 REAL
                     The value tau.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

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

       Purpose:

            SLARFGP generates a real elementary reflector H of order n, such
            that

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

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

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

            where tau is a real scalar and v is a real (n-1)-element
            vector.

            If the elements of x are all zero, 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 REAL
                     On entry, the value alpha.
                     On exit, it is overwritten with the value beta.

           X

                     X is REAL 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 REAL
                     The value tau.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

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

       Purpose:

            SLARFT forms the triangular factor T of a real 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**T

            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**T * 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 REAL 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 REAL array, dimension (K)
                     TAU(i) must contain the scalar factor of the elementary
                     reflector H(i).

           T

                     T is REAL 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.

       Date:
           December 2016

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

       Purpose:

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

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

            where tau is a real scalar and v is a real 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 REAL array, dimension (M) if SIDE = 'L'
                                                or (N) if SIDE = 'R'
                     The vector v in the representation of H.

           TAU

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

           C

                     C is REAL 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. LDA >= (1,M).

           WORK

                     WORK is REAL 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.

       Date:
           December 2016

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

       Purpose:

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

               (  c(i)  s(i) ) ( x(i) ) = ( a(i) )
               ( -s(i)  c(i) ) ( y(i) ) = (   0  )

       Parameters:
           N

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

           X

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

           INCX

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

           Y

                     Y is REAL 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.

       Date:
           December 2016

   subroutine slarrv (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, real, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * )
       ISUPPZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
       SLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the
       eigenvalues of L D LT.

       Purpose:

            SLARRV 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.
                     Note: VU is currently not used by this implementation of SLARRV, VU is
                     passed to SLARRV because it could be used compute gaps on the right end
                     of the extremal eigenvalues. However, with not much initial accuracy in
                     LAMBDA and VU, the formula can lead to an overestimation of the right gap
                     and thus to inadequately early RQI 'convergence'. This is currently
                     prevented this by forcing a small right gap. And so it turns out that VU
                     is currently not used by this implementation of SLARRV.

           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.LT.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 REAL 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 SLARRV.
                     < 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.

       Date:
           June 2016

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

       Purpose:

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

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

       Parameters:
           N

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

           X

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

           INCX

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

           Y

                     Y is REAL 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 REAL 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.

       Date:
           December 2016

   subroutine slaswp (integer N, real, dimension( lda, * ) A, integer LDA, integer K1, integer
       K2, integer, dimension( * ) IPIV, integer INCX)
       SLASWP performs a series of row interchanges on a general rectangular matrix.

       Purpose:

            SLASWP 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 REAL 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.

       Date:
           June 2017

       Further Details:

             Modified by
              R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA

   subroutine slatbs (character UPLO, character TRANS, character DIAG, character NORMIN, integer
       N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) X, real
       SCALE, real, dimension( * ) CNORM, integer INFO)
       SLATBS solves a triangular banded system of equations.

       Purpose:

            SLATBS solves one of the triangular systems

               A *x = s*b  or  A**T*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 STBSV 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**T* x = s*b  (Conjugate transpose = 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 REAL 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 REAL 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  or  A**T* 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.

       Date:
           December 2016

       Further Details:

             A rough bound on x is computed; if that is less than overflow, STBSV
             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 STBSV 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.  The basic
             algorithm for A upper triangular is

                  for j = 1, ..., n
                       x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
                  end

             We simultaneously compute two bounds
                  G(j) = bound on ( b(i) - A[1:i-1,i]**T * 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 STBSV if 1/M(n) and 1/G(n) are both greater
             than max(underflow, 1/overflow).

   subroutine slatdf (integer IJOB, integer N, real, dimension( ldz, * ) Z, integer LDZ, real,
       dimension( * ) RHS, real RDSUM, real RDSCAL, integer, dimension( * ) IPIV, integer,
       dimension( * ) JPIV)
       SLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a
       contribution to the reciprocal Dif-estimate.

       Purpose:

            SLATDF uses the LU factorization of the n-by-n matrix Z computed by
            SGETC2 and computes a contribution to the reciprocal Dif-estimate
            by solving Z * x = b for x, and choosing the r.h.s. b such that
            the norm of x is as large as possible. On entry RHS = b holds the
            contribution from earlier solved sub-systems, and on return RHS = x.

            The factorization of Z returned by SGETC2 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 SGECON, 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 REAL array, dimension (LDZ, N)
                     On entry, the LU part of the factorization of the n-by-n
                     matrix Z computed by SGETC2:  Z = P * L * U * Q

           LDZ

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

           RHS

                     RHS is REAL array, dimension N.
                     On entry, RHS contains contributions from other subsystems.
                     On exit, RHS contains the solution of the subsystem with
                     entries acoording 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 STGSYL, 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 STGSY2 is called by STGSYL.

           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 STGSY2 is called by
                           STGSYL.

           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.

       Date:
           June 2016

       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 IMINF-95.05, Departement of
                 Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.

   subroutine slatps (character UPLO, character TRANS, character DIAG, character NORMIN, integer
       N, real, dimension( * ) AP, real, dimension( * ) X, real SCALE, real, dimension( * )
       CNORM, integer INFO)
       SLATPS solves a triangular system of equations with the matrix held in packed storage.

       Purpose:

            SLATPS solves one of the triangular systems

               A *x = s*b  or  A**T*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, 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
            STPSV 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**T* x = s*b  (Conjugate transpose = 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 REAL 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 REAL 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  or  A**T* 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.

       Date:
           December 2016

       Further Details:

             A rough bound on x is computed; if that is less than overflow, STPSV
             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 STPSV 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.  The basic
             algorithm for A upper triangular is

                  for j = 1, ..., n
                       x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
                  end

             We simultaneously compute two bounds
                  G(j) = bound on ( b(i) - A[1:i-1,i]**T * 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 STPSV if 1/M(n) and 1/G(n) are both greater
             than max(underflow, 1/overflow).

   subroutine slatrs (character UPLO, character TRANS, character DIAG, character NORMIN, integer
       N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) X, real SCALE, real,
       dimension( * ) CNORM, integer INFO)
       SLATRS solves a triangular system of equations with the scale factor set to prevent
       overflow.

       Purpose:

            SLATRS solves one of the triangular systems

               A *x = s*b  or  A**T*x = s*b

            with scaling to prevent overflow.  Here A is an upper or lower
            triangular matrix, 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 STRSV 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**T* x = s*b  (Conjugate transpose = 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 REAL 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 REAL 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  or  A**T* 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.

       Date:
           December 2016

       Further Details:

             A rough bound on x is computed; if that is less than overflow, STRSV
             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 STRSV 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.  The basic
             algorithm for A upper triangular is

                  for j = 1, ..., n
                       x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
                  end

             We simultaneously compute two bounds
                  G(j) = bound on ( b(i) - A[1:i-1,i]**T * 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 STRSV if 1/M(n) and 1/G(n) are both greater
             than max(underflow, 1/overflow).

   subroutine slauu2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA,
       integer INFO)
       SLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular
       matrices (unblocked algorithm).

       Purpose:

            SLAUU2 computes the product U * U**T or L**T * 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 REAL 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**T;
                     if UPLO = 'L', the lower triangle of A is overwritten with
                     the lower triangle of the product L**T * 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.

       Date:
           December 2016

   subroutine slauum (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA,
       integer INFO)
       SLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular
       matrices (blocked algorithm).

       Purpose:

            SLAUUM computes the product U * U**T or L**T * 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 REAL 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**T;
                     if UPLO = 'L', the lower triangle of A is overwritten with
                     the lower triangle of the product L**T * 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.

       Date:
           December 2016

   subroutine srscl (integer N, real SA, real, dimension( * ) SX, integer INCX)
       SRSCL multiplies a vector by the reciprocal of a real scalar.

       Purpose:

            SRSCL multiplies an n-element real 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 REAL 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.

       Date:
           December 2016

   subroutine stprfb (character SIDE, character TRANS, character DIRECT, character STOREV,
       integer M, integer N, integer K, integer L, real, dimension( ldv, * ) V, integer LDV,
       real, dimension( ldt, * ) T, integer LDT, real, dimension( lda, * ) A, integer LDA, real,
       dimension( ldb, * ) B, integer LDB, real, dimension( ldwork, * ) WORK, integer LDWORK)
       STPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or
       complex matrix, which is composed of two blocks.

       Purpose:

            STPRFB applies a real "triangular-pentagonal" block reflector H or its
            conjugate transpose H^H to a real 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 REAL 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 REAL 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 REAL 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', LDC >= max(1,K);
                     If SIDE = 'R', LDC >= max(1,M).

           B

                     B is REAL 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 REAL 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.

       Date:
           December 2016

       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.

Author

       Generated automatically by Doxygen for LAPACK from the source code.