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

NAME

       doubleGEcomputational

SYNOPSIS

   Functions
       subroutine cgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
       recursive subroutine cgelqt3 (M, N, A, LDA, T, LDT, INFO)
       subroutine cgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
       subroutine dgebak (JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
           DGEBAK
       subroutine dgebal (JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
           DGEBAL
       subroutine dgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
           DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
       subroutine dgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
           DGEBRD
       subroutine dgecon (NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
           DGECON
       subroutine dgeequ (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
           DGEEQU
       subroutine dgeequb (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
           DGEEQUB
       subroutine dgehd2 (N, ILO, IHI, A, LDA, TAU, WORK, INFO)
           DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked
           algorithm.
       subroutine dgehrd (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
           DGEHRD
       subroutine dgelq2 (M, N, A, LDA, TAU, WORK, INFO)
           DGELQ2 computes the LQ factorization of a general rectangular matrix using an
           unblocked algorithm.
       subroutine dgelqf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
           DGELQF
       subroutine dgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
           DGELQT
       recursive subroutine dgelqt3 (M, N, A, LDA, T, LDT, INFO)
           DGELQT3 recursively computes a LQ factorization of a general real or complex matrix
           using the compact WY representation of Q.
       subroutine dgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
           DGEMLQT
       subroutine dgemqrt (SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT, C, LDC, WORK, INFO)
           DGEMQRT
       subroutine dgeql2 (M, N, A, LDA, TAU, WORK, INFO)
           DGEQL2 computes the QL factorization of a general rectangular matrix using an
           unblocked algorithm.
       subroutine dgeqlf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
           DGEQLF
       subroutine dgeqp3 (M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO)
           DGEQP3
       subroutine dgeqr2 (M, N, A, LDA, TAU, WORK, INFO)
           DGEQR2 computes the QR factorization of a general rectangular matrix using an
           unblocked algorithm.
       subroutine dgeqr2p (M, N, A, LDA, TAU, WORK, INFO)
           DGEQR2P computes the QR factorization of a general rectangular matrix with non-
           negative diagonal elements using an unblocked algorithm.
       subroutine dgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
           DGEQRF
       subroutine dgeqrfp (M, N, A, LDA, TAU, WORK, LWORK, INFO)
           DGEQRFP
       subroutine dgeqrt (M, N, NB, A, LDA, T, LDT, WORK, INFO)
           DGEQRT
       subroutine dgeqrt2 (M, N, A, LDA, T, LDT, INFO)
           DGEQRT2 computes a QR factorization of a general real or complex matrix using the
           compact WY representation of Q.
       recursive subroutine dgeqrt3 (M, N, A, LDA, T, LDT, INFO)
           DGEQRT3 recursively computes a QR factorization of a general real or complex matrix
           using the compact WY representation of Q.
       subroutine dgerfs (TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR,
           WORK, IWORK, INFO)
           DGERFS
       subroutine dgerfsx (TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, R, C, B, LDB, X, LDX,
           RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
           INFO)
           DGERFSX
       subroutine dgerq2 (M, N, A, LDA, TAU, WORK, INFO)
           DGERQ2 computes the RQ factorization of a general rectangular matrix using an
           unblocked algorithm.
       subroutine dgerqf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
           DGERQF
       subroutine dgesvj (JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV, WORK, LWORK, INFO)
           DGESVJ
       subroutine dgetf2 (M, N, A, LDA, IPIV, INFO)
           DGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting
           with row interchanges (unblocked algorithm).
       subroutine dgetrf (M, N, A, LDA, IPIV, INFO)
           DGETRF
       recursive subroutine dgetrf2 (M, N, A, LDA, IPIV, INFO)
           DGETRF2
       subroutine dgetri (N, A, LDA, IPIV, WORK, LWORK, INFO)
           DGETRI
       subroutine dgetrs (TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
           DGETRS
       subroutine dhgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA,
           Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
           DHGEQZ
       subroutine dla_geamv (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
           DLA_GEAMV computes a matrix-vector product using a general matrix to calculate error
           bounds.
       double precision function dla_gercond (TRANS, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO,
           WORK, IWORK)
           DLA_GERCOND estimates the Skeel condition number for a general matrix.
       subroutine dla_gerfsx_extended (PREC_TYPE, TRANS_TYPE, N, NRHS, A, LDA, AF, LDAF, IPIV,
           COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERRS_N, ERRS_C, RES, AYB, DY, Y_TAIL,
           RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
           DLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for
           general matrices by performing extra-precise iterative refinement and provides error
           bounds and backward error estimates for the solution.
       double precision function dla_gerpvgrw (N, NCOLS, A, LDA, AF, LDAF)
           DLA_GERPVGRW
       subroutine dtgevc (SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M,
           WORK, INFO)
           DTGEVC
       subroutine dtgexc (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK,
           LWORK, INFO)
           DTGEXC
       subroutine sgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
       recursive subroutine sgelqt3 (M, N, A, LDA, T, LDT, INFO)
       subroutine sgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
       subroutine zgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
           ZGELQT
       recursive subroutine zgelqt3 (M, N, A, LDA, T, LDT, INFO)
           ZGELQT3 recursively computes a LQ factorization of a general real or complex matrix
           using the compact WY representation of Q.
       subroutine zgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
           ZGEMLQT

Detailed Description

       This is the group of double computational functions for GE matrices

Function Documentation

   subroutine cgelqt (integer M, integer N, integer MB, complex, dimension( lda, * ) A, integer
       LDA, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( * ) WORK, integer
       INFO)
       Purpose:

       CGELQT computes a blocked LQ factorization of a complex M-by-N matrix A using the compact
       WY representation of Q.

       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.

           MB

                     MB is INTEGER
                     The block size to be used in the blocked QR.  MIN(M,N) >= MB >= 1.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, the elements on and below the diagonal of the array
                     contain the M-by-MIN(M,N) lower trapezoidal matrix L (L is
                     lower triangular if M <= N); the elements above the diagonal
                     are the rows of V.

           LDA

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

           T

                     T is COMPLEX array, dimension (LDT,MIN(M,N))
                     The upper triangular block reflectors stored in compact form
                     as a sequence of upper triangular blocks.  See below
                     for further details.

           LDT

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

           WORK

                     WORK is COMPLEX array, dimension (MB*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

       Further Details:

       The matrix V stores the elementary reflectors H(i) in the i-th row above the diagonal. For
       example, if M=5 and N=3, the matrix V is

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

       where the vi's represent the vectors which define H(i), which are returned in the matrix
       A. The 1's along the diagonal of V are not stored in A. Let K=MIN(M,N). The number of
       blocks is B = ceiling(K/MB), where each block is of order MB except for the last block,
       which is of order IB = K - (B-1)*MB. For each of the B blocks, a upper triangular block
       reflector factor is computed: T1, T2, ..., TB. The MB-by-MB (and IB-by-IB for the last
       block) T's are stored in the MB-by-K matrix T as

       T = (T1 T2 ... TB).

   recursive subroutine cgelqt3 (integer M, integer N, complex, dimension( lda, * ) A, integer
       LDA, complex, dimension( ldt, * ) T, integer LDT, integer INFO)
       Purpose:

       CGELQT3 recursively computes a LQ factorization of a complex M-by-N matrix A, using the
       compact WY representation of Q.

       Based on the algorithm of Elmroth and Gustavson, IBM J. Res. Develop. Vol 44 No. 4 July
       2000.

       Parameters:
           M

                     M is INTEGER
                     The number of rows of the matrix A.  M =< N.

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the real M-by-N matrix A.  On exit, the elements on and
                     below the diagonal contain the N-by-N lower triangular matrix L; the
                     elements above the diagonal are the rows of V.  See below for
                     further details.

           LDA

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

           T

                     T is COMPLEX array, dimension (LDT,N)
                     The N-by-N upper triangular factor of the block reflector.
                     The elements on and above the diagonal contain the block
                     reflector T; the elements below the diagonal are not used.
                     See below for further details.

           LDT

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

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

       Further Details:

       The matrix V stores the elementary reflectors H(i) in the i-th row above the diagonal. For
       example, if M=5 and N=3, the matrix V is

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

       where the vi's represent the vectors which define H(i), which are returned in the matrix
       A. The 1's along the diagonal of V are not stored in A. The block reflector H is then
       given by

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

       where V**T is the transpose of V.

       For details of the algorithm, see Elmroth and Gustavson (cited above).

   subroutine cgemlqt (character SIDE, character TRANS, integer M, integer N, integer K, integer
       MB, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldt, * ) T, integer
       LDT, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer
       INFO)
       Purpose:

       CGEMLQT overwrites the general real M-by-N matrix C with

       SIDE = 'L' SIDE = 'R' TRANS = 'N': Q C C Q TRANS = 'C': Q**H C C Q**H

       where Q is a complex orthogonal matrix defined as the product of K elementary reflectors:

       Q = H(1) H(2) . . . H(K) = I - V T V**H

       generated using the compact WY representation as returned by CGELQT.

       Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

       Parameters:
           SIDE

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

           TRANS

                     TRANS is CHARACTER*1
                     = 'N':  No transpose, apply Q;
                     = 'C':  Transpose, apply Q**H.

           M

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

           N

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

           K

                     K is INTEGER
                     The number of elementary reflectors whose product defines
                     the matrix Q.
                     If SIDE = 'L', M >= K >= 0;
                     if SIDE = 'R', N >= K >= 0.

           MB

                     MB is INTEGER
                     The block size used for the storage of T.  K >= MB >= 1.
                     This must be the same value of MB used to generate T
                     in DGELQT.

           V

                     V is COMPLEX array, dimension
                                          (LDV,M) if SIDE = 'L',
                                          (LDV,N) if SIDE = 'R'
                     The i-th row must contain the vector which defines the
                     elementary reflector H(i), for i = 1,2,...,k, as returned by
                     DGELQT in the first K rows of its array argument A.

           LDV

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

           T

                     T is COMPLEX array, dimension (LDT,K)
                     The upper triangular factors of the block reflectors
                     as returned by DGELQT, stored as a MB-by-K matrix.

           LDT

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

           C

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

           LDC

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

           WORK

                     WORK is COMPLEX array. The dimension of
                     WORK is N*MB if SIDE = 'L', or  M*MB if SIDE = 'R'.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

   subroutine dgebak (character JOB, character SIDE, integer N, integer ILO, integer IHI, double
       precision, dimension( * ) SCALE, integer M, double precision, dimension( ldv, * ) V,
       integer LDV, integer INFO)
       DGEBAK

       Purpose:

            DGEBAK forms the right or left eigenvectors of a real general matrix
            by backward transformation on the computed eigenvectors of the
            balanced matrix output by DGEBAL.

       Parameters:
           JOB

                     JOB is CHARACTER*1
                     Specifies the type of backward transformation required:
                     = 'N', do nothing, return immediately;
                     = 'P', do backward transformation for permutation only;
                     = 'S', do backward transformation for scaling only;
                     = 'B', do backward transformations for both permutation and
                            scaling.
                     JOB must be the same as the argument JOB supplied to DGEBAL.

           SIDE

                     SIDE is CHARACTER*1
                     = 'R':  V contains right eigenvectors;
                     = 'L':  V contains left eigenvectors.

           N

                     N is INTEGER
                     The number of rows of the matrix V.  N >= 0.

           ILO

                     ILO is INTEGER

           IHI

                     IHI is INTEGER
                     The integers ILO and IHI determined by DGEBAL.
                     1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

           SCALE

                     SCALE is DOUBLE PRECISION array, dimension (N)
                     Details of the permutation and scaling factors, as returned
                     by DGEBAL.

           M

                     M is INTEGER
                     The number of columns of the matrix V.  M >= 0.

           V

                     V is DOUBLE PRECISION array, dimension (LDV,M)
                     On entry, the matrix of right or left eigenvectors to be
                     transformed, as returned by DHSEIN or DTREVC.
                     On exit, V is overwritten by the transformed eigenvectors.

           LDV

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

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dgebal (character JOB, integer N, double precision, dimension( lda, * ) A, integer
       LDA, integer ILO, integer IHI, double precision, dimension( * ) SCALE, integer INFO)
       DGEBAL

       Purpose:

            DGEBAL balances a general real matrix A.  This involves, first,
            permuting A by a similarity transformation to isolate eigenvalues
            in the first 1 to ILO-1 and last IHI+1 to N elements on the
            diagonal; and second, applying a diagonal similarity transformation
            to rows and columns ILO to IHI to make the rows and columns as
            close in norm as possible.  Both steps are optional.

            Balancing may reduce the 1-norm of the matrix, and improve the
            accuracy of the computed eigenvalues and/or eigenvectors.

       Parameters:
           JOB

                     JOB is CHARACTER*1
                     Specifies the operations to be performed on A:
                     = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(I) = 1.0
                             for i = 1,...,N;
                     = 'P':  permute only;
                     = 'S':  scale only;
                     = 'B':  both permute and scale.

           N

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the input matrix A.
                     On exit,  A is overwritten by the balanced matrix.
                     If JOB = 'N', A is not referenced.
                     See Further Details.

           LDA

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

           ILO

                     ILO is INTEGER

           IHI

                     IHI is INTEGER
                     ILO and IHI are set to integers such that on exit
                     A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
                     If JOB = 'N' or 'S', ILO = 1 and IHI = N.

           SCALE

                     SCALE is DOUBLE PRECISION array, dimension (N)
                     Details of the permutations and scaling factors applied to
                     A.  If P(j) is the index of the row and column interchanged
                     with row and column j and D(j) is the scaling factor
                     applied to row and column j, then
                     SCALE(j) = P(j)    for j = 1,...,ILO-1
                              = D(j)    for j = ILO,...,IHI
                              = P(j)    for j = IHI+1,...,N.
                     The order in which the interchanges are made is N to IHI+1,
                     then 1 to ILO-1.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

       Further Details:

             The permutations consist of row and column interchanges which put
             the matrix in the form

                        ( T1   X   Y  )
                P A P = (  0   B   Z  )
                        (  0   0   T2 )

             where T1 and T2 are upper triangular matrices whose eigenvalues lie
             along the diagonal.  The column indices ILO and IHI mark the starting
             and ending columns of the submatrix B. Balancing consists of applying
             a diagonal similarity transformation inv(D) * B * D to make the
             1-norms of each row of B and its corresponding column nearly equal.
             The output matrix is

                ( T1     X*D          Y    )
                (  0  inv(D)*B*D  inv(D)*Z ).
                (  0      0           T2   )

             Information about the permutations P and the diagonal matrix D is
             returned in the vector SCALE.

             This subroutine is based on the EISPACK routine BALANC.

             Modified by Tzu-Yi Chen, Computer Science Division, University of
               California at Berkeley, USA

   subroutine dgebd2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) D, double precision, dimension( * ) E, double precision,
       dimension( * ) TAUQ, double precision, dimension( * ) TAUP, double precision, dimension( *
       ) WORK, integer INFO)
       DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

       Purpose:

            DGEBD2 reduces a real general m by n matrix A to upper or lower
            bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

            If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

       Parameters:
           M

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

           N

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the m by n general matrix to be reduced.
                     On exit,
                     if m >= n, the diagonal and the first superdiagonal are
                       overwritten with the upper bidiagonal matrix B; the
                       elements below the diagonal, with the array TAUQ, represent
                       the orthogonal matrix Q as a product of elementary
                       reflectors, and the elements above the first superdiagonal,
                       with the array TAUP, represent the orthogonal matrix P as
                       a product of elementary reflectors;
                     if m < n, the diagonal and the first subdiagonal are
                       overwritten with the lower bidiagonal matrix B; the
                       elements below the first subdiagonal, with the array TAUQ,
                       represent the orthogonal matrix Q as a product of
                       elementary reflectors, and the elements above the diagonal,
                       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 DOUBLE PRECISION array, dimension (min(M,N))
                     The diagonal elements of the bidiagonal matrix B:
                     D(i) = A(i,i).

           E

                     E is DOUBLE PRECISION array, dimension (min(M,N)-1)
                     The off-diagonal elements of the bidiagonal matrix B:
                     if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
                     if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

           TAUQ

                     TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors which
                     represent the orthogonal matrix Q. See Further Details.

           TAUP

                     TAUP is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors which
                     represent the orthogonal matrix P. See Further Details.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (max(M,N))

           INFO

                     INFO is INTEGER
                     = 0: successful exit.
                     < 0: if INFO = -i, the i-th argument had an illegal value.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

       Further Details:

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

             If m >= n,

                Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

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

             If m < n,

                Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

             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;
             v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
             u(1:i-1) = 0, u(i) = 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).

             The contents of A on exit are illustrated by the following examples:

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

               (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
               (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
               (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
               (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
               (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
               (  v1  v2  v3  v4  v5 )

             where d and e denote diagonal and off-diagonal elements of B, vi
             denotes an element of the vector defining H(i), and ui an element of
             the vector defining G(i).

   subroutine dgebrd (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) D, double precision, dimension( * ) E, double precision,
       dimension( * ) TAUQ, double precision, dimension( * ) TAUP, double precision, dimension( *
       ) WORK, integer LWORK, integer INFO)
       DGEBRD

       Purpose:

            DGEBRD reduces a general real M-by-N matrix A to upper or lower
            bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

            If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

       Parameters:
           M

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

           N

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N general matrix to be reduced.
                     On exit,
                     if m >= n, the diagonal and the first superdiagonal are
                       overwritten with the upper bidiagonal matrix B; the
                       elements below the diagonal, with the array TAUQ, represent
                       the orthogonal matrix Q as a product of elementary
                       reflectors, and the elements above the first superdiagonal,
                       with the array TAUP, represent the orthogonal matrix P as
                       a product of elementary reflectors;
                     if m < n, the diagonal and the first subdiagonal are
                       overwritten with the lower bidiagonal matrix B; the
                       elements below the first subdiagonal, with the array TAUQ,
                       represent the orthogonal matrix Q as a product of
                       elementary reflectors, and the elements above the diagonal,
                       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 DOUBLE PRECISION array, dimension (min(M,N))
                     The diagonal elements of the bidiagonal matrix B:
                     D(i) = A(i,i).

           E

                     E is DOUBLE PRECISION array, dimension (min(M,N)-1)
                     The off-diagonal elements of the bidiagonal matrix B:
                     if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
                     if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

           TAUQ

                     TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors which
                     represent the orthogonal matrix Q. See Further Details.

           TAUP

                     TAUP is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors which
                     represent the orthogonal matrix P. See Further Details.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The length of the array WORK.  LWORK >= max(1,M,N).
                     For optimum performance LWORK >= (M+N)*NB, where NB
                     is the optimal blocksize.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

       Further Details:

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

             If m >= n,

                Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

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

             If m < n,

                Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

             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;
             v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
             u(1:i-1) = 0, u(i) = 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).

             The contents of A on exit are illustrated by the following examples:

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

               (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
               (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
               (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
               (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
               (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
               (  v1  v2  v3  v4  v5 )

             where d and e denote diagonal and off-diagonal elements of B, vi
             denotes an element of the vector defining H(i), and ui an element of
             the vector defining G(i).

   subroutine dgecon (character NORM, integer N, double precision, dimension( lda, * ) A, integer
       LDA, double precision ANORM, double precision RCOND, double precision, dimension( * )
       WORK, integer, dimension( * ) IWORK, integer INFO)
       DGECON

       Purpose:

            DGECON estimates the reciprocal of the condition number of a general
            real matrix A, in either the 1-norm or the infinity-norm, using
            the LU factorization computed by DGETRF.

            An estimate is obtained for norm(inv(A)), and the reciprocal of the
            condition number is computed as
               RCOND = 1 / ( norm(A) * norm(inv(A)) ).

       Parameters:
           NORM

                     NORM is CHARACTER*1
                     Specifies whether the 1-norm condition number or the
                     infinity-norm condition number is required:
                     = '1' or 'O':  1-norm;
                     = 'I':         Infinity-norm.

           N

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     The factors L and U from the factorization A = P*L*U
                     as computed by DGETRF.

           LDA

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

           ANORM

                     ANORM is DOUBLE PRECISION
                     If NORM = '1' or 'O', the 1-norm of the original matrix A.
                     If NORM = 'I', the infinity-norm of the original matrix A.

           RCOND

                     RCOND is DOUBLE PRECISION
                     The reciprocal of the condition number of the matrix A,
                     computed as RCOND = 1/(norm(A) * norm(inv(A))).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (4*N)

           IWORK

                     IWORK is INTEGER array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dgeequ (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) R, double precision, dimension( * ) C, double precision
       ROWCND, double precision COLCND, double precision AMAX, integer INFO)
       DGEEQU

       Purpose:

            DGEEQU computes row and column scalings intended to equilibrate an
            M-by-N matrix A and reduce its condition number.  R returns the row
            scale factors and C the column scale factors, chosen to try to make
            the largest element in each row and column of the matrix B with
            elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

            R(i) and C(j) are restricted to be between SMLNUM = smallest safe
            number and BIGNUM = largest safe number.  Use of these scaling
            factors is not guaranteed to reduce the condition number of A but
            works well in practice.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     The M-by-N matrix whose equilibration factors are
                     to be computed.

           LDA

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

           R

                     R is DOUBLE PRECISION array, dimension (M)
                     If INFO = 0 or INFO > M, R contains the row scale factors
                     for A.

           C

                     C is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0,  C contains the column scale factors for A.

           ROWCND

                     ROWCND is DOUBLE PRECISION
                     If INFO = 0 or INFO > M, ROWCND contains the ratio of the
                     smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
                     AMAX is neither too large nor too small, it is not worth
                     scaling by R.

           COLCND

                     COLCND is DOUBLE PRECISION
                     If INFO = 0, COLCND contains the ratio of the smallest
                     C(i) to the largest C(i).  If COLCND >= 0.1, it is not
                     worth scaling by C.

           AMAX

                     AMAX is DOUBLE PRECISION
                     Absolute value of largest matrix element.  If AMAX is very
                     close to overflow or very close to underflow, the matrix
                     should be scaled.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i,  and i is
                           <= M:  the i-th row of A is exactly zero
                           >  M:  the (i-M)-th column of A is exactly zero

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dgeequb (integer M, integer N, double precision, dimension( lda, * ) A, integer
       LDA, double precision, dimension( * ) R, double precision, dimension( * ) C, double
       precision ROWCND, double precision COLCND, double precision AMAX, integer INFO)
       DGEEQUB

       Purpose:

            DGEEQUB computes row and column scalings intended to equilibrate an
            M-by-N matrix A and reduce its condition number.  R returns the row
            scale factors and C the column scale factors, chosen to try to make
            the largest element in each row and column of the matrix B with
            elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
            the radix.

            R(i) and C(j) are restricted to be a power of the radix between
            SMLNUM = smallest safe number and BIGNUM = largest safe number.  Use
            of these scaling factors is not guaranteed to reduce the condition
            number of A but works well in practice.

            This routine differs from DGEEQU by restricting the scaling factors
            to a power of the radix.  Barring over- and underflow, scaling by
            these factors introduces no additional rounding errors.  However, the
            scaled entries' magnitudes are no longer approximately 1 but lie
            between sqrt(radix) and 1/sqrt(radix).

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     The M-by-N matrix whose equilibration factors are
                     to be computed.

           LDA

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

           R

                     R is DOUBLE PRECISION array, dimension (M)
                     If INFO = 0 or INFO > M, R contains the row scale factors
                     for A.

           C

                     C is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0,  C contains the column scale factors for A.

           ROWCND

                     ROWCND is DOUBLE PRECISION
                     If INFO = 0 or INFO > M, ROWCND contains the ratio of the
                     smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
                     AMAX is neither too large nor too small, it is not worth
                     scaling by R.

           COLCND

                     COLCND is DOUBLE PRECISION
                     If INFO = 0, COLCND contains the ratio of the smallest
                     C(i) to the largest C(i).  If COLCND >= 0.1, it is not
                     worth scaling by C.

           AMAX

                     AMAX is DOUBLE PRECISION
                     Absolute value of largest matrix element.  If AMAX is very
                     close to overflow or very close to underflow, the matrix
                     should be scaled.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i,  and i is
                           <= M:  the i-th row of A is exactly zero
                           >  M:  the (i-M)-th column of A is exactly zero

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dgehd2 (integer N, integer ILO, integer IHI, double precision, dimension( lda, * )
       A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * )
       WORK, integer INFO)
       DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked
       algorithm.

       Purpose:

            DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
            an orthogonal similarity transformation:  Q**T * A * Q = H .

       Parameters:
           N

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

           ILO

                     ILO is INTEGER

           IHI

                     IHI is INTEGER

                     It is assumed that A is already upper triangular in rows
                     and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
                     set by a previous call to DGEBAL; otherwise they should be
                     set to 1 and N respectively. See Further Details.
                     1 <= ILO <= IHI <= max(1,N).

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the n by n general matrix to be reduced.
                     On exit, the upper triangle and the first subdiagonal of A
                     are overwritten with the upper Hessenberg matrix H, and the
                     elements below the first subdiagonal, with the array TAU,
                     represent the orthogonal matrix Q as a product of elementary
                     reflectors. See Further Details.

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (N-1)
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of (ihi-ilo) elementary
             reflectors

                Q = H(ilo) H(ilo+1) . . . H(ihi-1).

             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) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
             exit in A(i+2:ihi,i), and tau in TAU(i).

             The contents of A are illustrated by the following example, with
             n = 7, ilo = 2 and ihi = 6:

             on entry,                        on exit,

             ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
             (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
             (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
             (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
             (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
             (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
             (                         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).

   subroutine dgehrd (integer N, integer ILO, integer IHI, double precision, dimension( lda, * )
       A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * )
       WORK, integer LWORK, integer INFO)
       DGEHRD

       Purpose:

            DGEHRD reduces a real general matrix A to upper Hessenberg form H by
            an orthogonal similarity transformation:  Q**T * A * Q = H .

       Parameters:
           N

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

           ILO

                     ILO is INTEGER

           IHI

                     IHI is INTEGER

                     It is assumed that A is already upper triangular in rows
                     and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
                     set by a previous call to DGEBAL; otherwise they should be
                     set to 1 and N respectively. See Further Details.
                     1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the N-by-N general matrix to be reduced.
                     On exit, the upper triangle and the first subdiagonal of A
                     are overwritten with the upper Hessenberg matrix H, and the
                     elements below the first subdiagonal, with the array TAU,
                     represent the orthogonal matrix Q as a product of elementary
                     reflectors. See Further Details.

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (N-1)
                     The scalar factors of the elementary reflectors (see Further
                     Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
                     zero.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (LWORK)
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The length of the array WORK.  LWORK >= max(1,N).
                     For good performance, LWORK should generally be larger.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of (ihi-ilo) elementary
             reflectors

                Q = H(ilo) H(ilo+1) . . . H(ihi-1).

             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) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
             exit in A(i+2:ihi,i), and tau in TAU(i).

             The contents of A are illustrated by the following example, with
             n = 7, ilo = 2 and ihi = 6:

             on entry,                        on exit,

             ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
             (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
             (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
             (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
             (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
             (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
             (                         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 file is a slight modification of LAPACK-3.0's DGEHRD
             subroutine incorporating improvements proposed by Quintana-Orti and
             Van de Geijn (2006). (See DLAHR2.)

   subroutine dgelq2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO)
       DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked
       algorithm.

       Purpose:

            DGELQ2 computes an LQ factorization of a real m by n matrix A:
            A = L * Q.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the m by n matrix A.
                     On exit, the elements on and below the diagonal of the array
                     contain the m by min(m,n) lower trapezoidal matrix L (L is
                     lower triangular if m <= n); the elements above the diagonal,
                     with the array TAU, represent the orthogonal matrix Q as a
                     product of elementary reflectors (see Further Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (M)

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(k) . . . H(2) H(1), where k = min(m,n).

             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-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
             and tau in TAU(i).

   subroutine dgelqf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer
       LWORK, integer INFO)
       DGELQF

       Purpose:

            DGELQF computes an LQ factorization of a real M-by-N matrix A:
            A = L * Q.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, the elements on and below the diagonal of the array
                     contain the m-by-min(m,n) lower trapezoidal matrix L (L is
                     lower triangular if m <= n); the elements above the diagonal,
                     with the array TAU, represent the orthogonal matrix Q as a
                     product of elementary reflectors (see Further Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= max(1,M).
                     For optimum performance LWORK >= M*NB, where NB is the
                     optimal blocksize.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(k) . . . H(2) H(1), where k = min(m,n).

             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-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
             and tau in TAU(i).

   subroutine dgelqt (integer M, integer N, integer MB, double precision, dimension( lda, * ) A,
       integer LDA, double precision, dimension( ldt, * ) T, integer LDT, double precision,
       dimension( * ) WORK, integer INFO)
       DGELQT

       Purpose:

            DGELQT computes a blocked LQ factorization of a real M-by-N matrix A
            using the compact WY representation of Q.

       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.

           MB

                     MB is INTEGER
                     The block size to be used in the blocked QR.  MIN(M,N) >= MB >= 1.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, the elements on and below the diagonal of the array
                     contain the M-by-MIN(M,N) lower trapezoidal matrix L (L is
                     lower triangular if M <= N); the elements above the diagonal
                     are the rows of V.

           LDA

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

           T

                     T is DOUBLE PRECISION array, dimension (LDT,MIN(M,N))
                     The upper triangular block reflectors stored in compact form
                     as a sequence of upper triangular blocks.  See below
                     for further details.

           LDT

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

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MB*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

       Further Details:

             The matrix V stores the elementary reflectors H(i) in the i-th row
             above the diagonal. For example, if M=5 and N=3, the matrix V is

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

             where the vi's represent the vectors which define H(i), which are returned
             in the matrix A.  The 1's along the diagonal of V are not stored in A.
             Let K=MIN(M,N).  The number of blocks is B = ceiling(K/MB), where each
             block is of order MB except for the last block, which is of order
             IB = K - (B-1)*MB.  For each of the B blocks, a upper triangular block
             reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
             for the last block) T's are stored in the MB-by-K matrix T as

                          T = (T1 T2 ... TB).

   recursive subroutine dgelqt3 (integer M, integer N, double precision, dimension( lda, * ) A,
       integer LDA, double precision, dimension( ldt, * ) T, integer LDT, integer INFO)
       DGELQT3 recursively computes a LQ factorization of a general real or complex matrix using
       the compact WY representation of Q.

       Purpose:

            DGELQT3 recursively computes a LQ factorization of a real M-by-N
            matrix A, using the compact WY representation of Q.

            Based on the algorithm of Elmroth and Gustavson,
            IBM J. Res. Develop. Vol 44 No. 4 July 2000.

       Parameters:
           M

                     M is INTEGER
                     The number of rows of the matrix A.  M =< N.

           N

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the real M-by-N matrix A.  On exit, the elements on and
                     below the diagonal contain the N-by-N lower triangular matrix L; the
                     elements above the diagonal are the rows of V.  See below for
                     further details.

           LDA

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

           T

                     T is DOUBLE PRECISION array, dimension (LDT,N)
                     The N-by-N upper triangular factor of the block reflector.
                     The elements on and above the diagonal contain the block
                     reflector T; the elements below the diagonal are not used.
                     See below for further details.

           LDT

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

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

       Further Details:

             The matrix V stores the elementary reflectors H(i) in the i-th row
             above the diagonal. For example, if M=5 and N=3, the matrix V is

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

             where the vi's represent the vectors which define H(i), which are returned
             in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
             block reflector H is then given by

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

             where V**T is the transpose of V.

             For details of the algorithm, see Elmroth and Gustavson (cited above).

   subroutine dgemlqt (character SIDE, character TRANS, integer M, integer N, integer K, integer
       MB, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension(
       ldt, * ) T, integer LDT, double precision, dimension( ldc, * ) C, integer LDC, double
       precision, dimension( * ) WORK, integer INFO)
       DGEMLQT

       Purpose:

            DGEMLQT overwrites the general real M-by-N matrix C with

                            SIDE = 'L'     SIDE = 'R'
            TRANS = 'N':      Q C            C Q
            TRANS = 'T':   Q**T C            C Q**T

            where Q is a real orthogonal matrix defined as the product of K
            elementary reflectors:

                  Q = H(1) H(2) . . . H(K) = I - V T V**T

            generated using the compact WY representation as returned by DGELQT.

            Q is of order M if SIDE = 'L' and of order N  if SIDE = 'R'.

       Parameters:
           SIDE

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

           TRANS

                     TRANS is CHARACTER*1
                     = 'N':  No transpose, apply Q;
                     = 'C':  Transpose, apply Q**T.

           M

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

           N

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

           K

                     K is INTEGER
                     The number of elementary reflectors whose product defines
                     the matrix Q.
                     If SIDE = 'L', M >= K >= 0;
                     if SIDE = 'R', N >= K >= 0.

           MB

                     MB is INTEGER
                     The block size used for the storage of T.  K >= MB >= 1.
                     This must be the same value of MB used to generate T
                     in DGELQT.

           V

                     V is DOUBLE PRECISION array, dimension
                                          (LDV,M) if SIDE = 'L',
                                          (LDV,N) if SIDE = 'R'
                     The i-th row must contain the vector which defines the
                     elementary reflector H(i), for i = 1,2,...,k, as returned by
                     DGELQT in the first K rows of its array argument A.

           LDV

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

           T

                     T is DOUBLE PRECISION array, dimension (LDT,K)
                     The upper triangular factors of the block reflectors
                     as returned by DGELQT, stored as a MB-by-K matrix.

           LDT

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

           C

                     C is DOUBLE PRECISION array, dimension (LDC,N)
                     On entry, the M-by-N matrix C.
                     On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q.

           LDC

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

           WORK

                     WORK is DOUBLE PRECISION array. The dimension of
                     WORK is N*MB if SIDE = 'L', or  M*MB if SIDE = 'R'.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

   subroutine dgemqrt (character SIDE, character TRANS, integer M, integer N, integer K, integer
       NB, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension(
       ldt, * ) T, integer LDT, double precision, dimension( ldc, * ) C, integer LDC, double
       precision, dimension( * ) WORK, integer INFO)
       DGEMQRT

       Purpose:

            DGEMQRT overwrites the general real M-by-N matrix C with

                            SIDE = 'L'     SIDE = 'R'
            TRANS = 'N':      Q C            C Q
            TRANS = 'T':   Q**T C            C Q**T

            where Q is a real orthogonal matrix defined as the product of K
            elementary reflectors:

                  Q = H(1) H(2) . . . H(K) = I - V T V**T

            generated using the compact WY representation as returned by DGEQRT.

            Q is of order M if SIDE = 'L' and of order N  if SIDE = 'R'.

       Parameters:
           SIDE

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

           TRANS

                     TRANS is CHARACTER*1
                     = 'N':  No transpose, apply Q;
                     = 'C':  Transpose, apply Q**T.

           M

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

           N

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

           K

                     K is INTEGER
                     The number of elementary reflectors whose product defines
                     the matrix Q.
                     If SIDE = 'L', M >= K >= 0;
                     if SIDE = 'R', N >= K >= 0.

           NB

                     NB is INTEGER
                     The block size used for the storage of T.  K >= NB >= 1.
                     This must be the same value of NB used to generate T
                     in CGEQRT.

           V

                     V is DOUBLE PRECISION array, dimension (LDV,K)
                     The i-th column must contain the vector which defines the
                     elementary reflector H(i), for i = 1,2,...,k, as returned by
                     CGEQRT in the first K columns of its array argument A.

           LDV

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

           T

                     T is DOUBLE PRECISION array, dimension (LDT,K)
                     The upper triangular factors of the block reflectors
                     as returned by CGEQRT, stored as a NB-by-N matrix.

           LDT

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

           C

                     C is DOUBLE PRECISION array, dimension (LDC,N)
                     On entry, the M-by-N matrix C.
                     On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q.

           LDC

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

           WORK

                     WORK is DOUBLE PRECISION array. The dimension of
                     WORK is N*NB if SIDE = 'L', or  M*NB if SIDE = 'R'.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dgeql2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO)
       DGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked
       algorithm.

       Purpose:

            DGEQL2 computes a QL factorization of a real m by n matrix A:
            A = Q * L.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the m by n matrix A.
                     On exit, if m >= n, the lower triangle of the subarray
                     A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
                     if m <= n, the elements on and below the (n-m)-th
                     superdiagonal contain the m by n lower trapezoidal matrix L;
                     the remaining elements, with the array TAU, represent the
                     orthogonal matrix Q as a product of elementary reflectors
                     (see Further Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(k) . . . H(2) H(1), where k = min(m,n).

             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(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
             A(1:m-k+i-1,n-k+i), and tau in TAU(i).

   subroutine dgeqlf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer
       LWORK, integer INFO)
       DGEQLF

       Purpose:

            DGEQLF computes a QL factorization of a real M-by-N matrix A:
            A = Q * L.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit,
                     if m >= n, the lower triangle of the subarray
                     A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
                     if m <= n, the elements on and below the (n-m)-th
                     superdiagonal contain the M-by-N lower trapezoidal matrix L;
                     the remaining elements, with the array TAU, represent the
                     orthogonal matrix Q as a product of elementary reflectors
                     (see Further Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= max(1,N).
                     For optimum performance LWORK >= N*NB, where NB is the
                     optimal blocksize.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(k) . . . H(2) H(1), where k = min(m,n).

             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(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
             A(1:m-k+i-1,n-k+i), and tau in TAU(i).

   subroutine dgeqp3 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) JPVT, double precision, dimension( * ) TAU, double precision,
       dimension( * ) WORK, integer LWORK, integer INFO)
       DGEQP3

       Purpose:

            DGEQP3 computes a QR factorization with column pivoting of a
            matrix A:  A*P = Q*R  using Level 3 BLAS.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, the upper triangle of the array contains the
                     min(M,N)-by-N upper trapezoidal matrix R; the elements below
                     the diagonal, together with the array TAU, represent the
                     orthogonal matrix Q as a product of min(M,N) elementary
                     reflectors.

           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(J).ne.0, the J-th column of A is permuted
                     to the front of A*P (a leading column); if JPVT(J)=0,
                     the J-th column of A is a free column.
                     On exit, if JPVT(J)=K, then the J-th column of A*P was the
                     the K-th column of A.

           TAU

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

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO=0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK. LWORK >= 3*N+1.
                     For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
                     is the optimal blocksize.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0: successful exit.
                     < 0: if INFO = -i, the i-th argument had an illegal value.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(1) H(2) . . . H(k), where k = min(m,n).

             Each H(i) has the form

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

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

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

   subroutine dgeqr2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO)
       DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked
       algorithm.

       Purpose:

            DGEQR2 computes a QR factorization of a real m by n matrix A:
            A = Q * R.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the m by n matrix A.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(m,n) by n upper trapezoidal matrix R (R is
                     upper triangular if m >= n); the elements below the diagonal,
                     with the array TAU, represent the orthogonal matrix Q as a
                     product of elementary reflectors (see Further Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(1) H(2) . . . H(k), where k = min(m,n).

             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-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
             and tau in TAU(i).

   subroutine dgeqr2p (integer M, integer N, double precision, dimension( lda, * ) A, integer
       LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer
       INFO)
       DGEQR2P computes the QR factorization of a general rectangular matrix with non-negative
       diagonal elements using an unblocked algorithm.

       Purpose:

            DGEQR2P computes a QR factorization of a real m by n matrix A:
            A = Q * R. The diagonal entries of R are nonnegative.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the m by n matrix A.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(m,n) by n upper trapezoidal matrix R (R is
                     upper triangular if m >= n). The diagonal entries of R are
                     nonnegative; the elements below the diagonal,
                     with the array TAU, represent the orthogonal matrix Q as a
                     product of elementary reflectors (see Further Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(1) H(2) . . . H(k), where k = min(m,n).

             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-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
             and tau in TAU(i).

            See Lapack Working Note 203 for details

   subroutine dgeqrf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer
       LWORK, integer INFO)
       DGEQRF

       Purpose:

            DGEQRF computes a QR factorization of a real M-by-N matrix A:
            A = Q * R.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(M,N)-by-N upper trapezoidal matrix R (R is
                     upper triangular if m >= n); the elements below the diagonal,
                     with the array TAU, represent the orthogonal matrix Q as a
                     product of min(m,n) elementary reflectors (see Further
                     Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= max(1,N).
                     For optimum performance LWORK >= N*NB, where NB is
                     the optimal blocksize.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(1) H(2) . . . H(k), where k = min(m,n).

             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-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
             and tau in TAU(i).

   subroutine dgeqrfp (integer M, integer N, double precision, dimension( lda, * ) A, integer
       LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer
       LWORK, integer INFO)
       DGEQRFP

       Purpose:

            DGEQRFP computes a QR factorization of a real M-by-N matrix A:
            A = Q * R. The diagonal entries of R are nonnegative.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(M,N)-by-N upper trapezoidal matrix R (R is
                     upper triangular if m >= n). The diagonal entries of R
                     are nonnegative; the elements below the diagonal,
                     with the array TAU, represent the orthogonal matrix Q as a
                     product of min(m,n) elementary reflectors (see Further
                     Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= max(1,N).
                     For optimum performance LWORK >= N*NB, where NB is
                     the optimal blocksize.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(1) H(2) . . . H(k), where k = min(m,n).

             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-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
             and tau in TAU(i).

            See Lapack Working Note 203 for details

   subroutine dgeqrt (integer M, integer N, integer NB, double precision, dimension( lda, * ) A,
       integer LDA, double precision, dimension( ldt, * ) T, integer LDT, double precision,
       dimension( * ) WORK, integer INFO)
       DGEQRT

       Purpose:

            DGEQRT computes a blocked QR factorization of a real M-by-N matrix A
            using the compact WY representation of Q.

       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.

           NB

                     NB is INTEGER
                     The block size to be used in the blocked QR.  MIN(M,N) >= NB >= 1.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(M,N)-by-N upper trapezoidal matrix R (R is
                     upper triangular if M >= N); the elements below the diagonal
                     are the columns of V.

           LDA

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

           T

                     T is DOUBLE PRECISION array, dimension (LDT,MIN(M,N))
                     The upper triangular block reflectors stored in compact form
                     as a sequence of upper triangular blocks.  See below
                     for further details.

           LDT

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

           WORK

                     WORK is DOUBLE PRECISION array, dimension (NB*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

       Further Details:

             The matrix V stores the elementary reflectors H(i) in the i-th column
             below the diagonal. For example, if M=5 and N=3, the matrix V is

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

             where the vi's represent the vectors which define H(i), which are returned
             in the matrix A.  The 1's along the diagonal of V are not stored in A.

             Let K=MIN(M,N).  The number of blocks is B = ceiling(K/NB), where each
             block is of order NB except for the last block, which is of order
             IB = K - (B-1)*NB.  For each of the B blocks, a upper triangular block
             reflector factor is computed: T1, T2, ..., TB.  The NB-by-NB (and IB-by-IB
             for the last block) T's are stored in the NB-by-K matrix T as

                          T = (T1 T2 ... TB).

   subroutine dgeqrt2 (integer M, integer N, double precision, dimension( lda, * ) A, integer
       LDA, double precision, dimension( ldt, * ) T, integer LDT, integer INFO)
       DGEQRT2 computes a QR factorization of a general real or complex matrix using the compact
       WY representation of Q.

       Purpose:

            DGEQRT2 computes a QR factorization of a real M-by-N matrix A,
            using the compact WY representation of Q.

       Parameters:
           M

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

           N

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the real M-by-N matrix A.  On exit, the elements on and
                     above the diagonal contain the N-by-N upper triangular matrix R; the
                     elements below the diagonal are the columns of V.  See below for
                     further details.

           LDA

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

           T

                     T is DOUBLE PRECISION array, dimension (LDT,N)
                     The N-by-N upper triangular factor of the block reflector.
                     The elements on and above the diagonal contain the block
                     reflector T; the elements below the diagonal are not used.
                     See below for further details.

           LDT

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

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix V stores the elementary reflectors H(i) in the i-th column
             below the diagonal. For example, if M=5 and N=3, the matrix V is

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

             where the vi's represent the vectors which define H(i), which are returned
             in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
             block reflector H is then given by

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

             where V**T is the transpose of V.

   recursive subroutine dgeqrt3 (integer M, integer N, double precision, dimension( lda, * ) A,
       integer LDA, double precision, dimension( ldt, * ) T, integer LDT, integer INFO)
       DGEQRT3 recursively computes a QR factorization of a general real or complex matrix using
       the compact WY representation of Q.

       Purpose:

            DGEQRT3 recursively computes a QR factorization of a real M-by-N
            matrix A, using the compact WY representation of Q.

            Based on the algorithm of Elmroth and Gustavson,
            IBM J. Res. Develop. Vol 44 No. 4 July 2000.

       Parameters:
           M

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

           N

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the real M-by-N matrix A.  On exit, the elements on and
                     above the diagonal contain the N-by-N upper triangular matrix R; the
                     elements below the diagonal are the columns of V.  See below for
                     further details.

           LDA

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

           T

                     T is DOUBLE PRECISION array, dimension (LDT,N)
                     The N-by-N upper triangular factor of the block reflector.
                     The elements on and above the diagonal contain the block
                     reflector T; the elements below the diagonal are not used.
                     See below for further details.

           LDT

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

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2016

       Further Details:

             The matrix V stores the elementary reflectors H(i) in the i-th column
             below the diagonal. For example, if M=5 and N=3, the matrix V is

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

             where the vi's represent the vectors which define H(i), which are returned
             in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
             block reflector H is then given by

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

             where V**T is the transpose of V.

             For details of the algorithm, see Elmroth and Gustavson (cited above).

   subroutine dgerfs (character TRANS, integer N, integer NRHS, double precision, dimension( lda,
       * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer,
       dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double
       precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR,
       double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer,
       dimension( * ) IWORK, integer INFO)
       DGERFS

       Purpose:

            DGERFS improves the computed solution to a system of linear
            equations and provides error bounds and backward error estimates for
            the solution.

       Parameters:
           TRANS

                     TRANS is CHARACTER*1
                     Specifies the form of the system of equations:
                     = 'N':  A * X = B     (No transpose)
                     = 'T':  A**T * X = B  (Transpose)
                     = 'C':  A**H * X = B  (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 B and X.  NRHS >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     The original N-by-N matrix A.

           LDA

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

           AF

                     AF is DOUBLE PRECISION array, dimension (LDAF,N)
                     The factors L and U from the factorization A = P*L*U
                     as computed by DGETRF.

           LDAF

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

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     The pivot indices from DGETRF; for 1<=i<=N, row i of the
                     matrix was interchanged with row IPIV(i).

           B

                     B is DOUBLE PRECISION array, dimension (LDB,NRHS)
                     The right hand side matrix B.

           LDB

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

           X

                     X is DOUBLE PRECISION array, dimension (LDX,NRHS)
                     On entry, the solution matrix X, as computed by DGETRS.
                     On exit, the improved solution matrix X.

           LDX

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

           FERR

                     FERR is DOUBLE PRECISION array, dimension (NRHS)
                     The estimated forward error bound for each solution vector
                     X(j) (the j-th column of the solution matrix X).
                     If XTRUE is the true solution corresponding to X(j), FERR(j)
                     is an estimated upper bound for the magnitude of the largest
                     element in (X(j) - XTRUE) divided by the magnitude of the
                     largest element in X(j).  The estimate is as reliable as
                     the estimate for RCOND, and is almost always a slight
                     overestimate of the true error.

           BERR

                     BERR is DOUBLE PRECISION array, dimension (NRHS)
                     The componentwise relative backward error of each solution
                     vector X(j) (i.e., the smallest relative change in
                     any element of A or B that makes X(j) an exact solution).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (3*N)

           IWORK

                     IWORK is INTEGER array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Internal Parameters:

             ITMAX is the maximum number of steps of iterative refinement.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dgerfsx (character TRANS, character EQUED, integer N, integer NRHS, double
       precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF,
       integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) R, double
       precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double
       precision, dimension( ldx , * ) X, integer LDX, double precision RCOND, double precision,
       dimension( * ) BERR, integer N_ERR_BNDS, double precision, dimension( nrhs, * )
       ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS,
       double precision, dimension( * ) PARAMS, double precision, dimension( * ) WORK, integer,
       dimension( * ) IWORK, integer INFO)
       DGERFSX

       Purpose:

               DGERFSX improves the computed solution to a system of linear
               equations and provides error bounds and backward error estimates
               for the solution.  In addition to normwise error bound, the code
               provides maximum componentwise error bound if possible.  See
               comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
               error bounds.

               The original system of linear equations may have been equilibrated
               before calling this routine, as described by arguments EQUED, R
               and C below. In this case, the solution and error bounds returned
               are for the original unequilibrated system.

                Some optional parameters are bundled in the PARAMS array.  These
                settings determine how refinement is performed, but often the
                defaults are acceptable.  If the defaults are acceptable, users
                can pass NPARAMS = 0 which prevents the source code from accessing
                the PARAMS argument.

       Parameters:
           TRANS

                     TRANS is CHARACTER*1
                Specifies the form of the system of equations:
                  = 'N':  A * X = B     (No transpose)
                  = 'T':  A**T * X = B  (Transpose)
                  = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

           EQUED

                     EQUED is CHARACTER*1
                Specifies the form of equilibration that was done to A
                before calling this routine. This is needed to compute
                the solution and error bounds correctly.
                  = 'N':  No equilibration
                  = 'R':  Row equilibration, i.e., A has been premultiplied by
                          diag(R).
                  = 'C':  Column equilibration, i.e., A has been postmultiplied
                          by diag(C).
                  = 'B':  Both row and column equilibration, i.e., A has been
                          replaced by diag(R) * A * diag(C).
                          The right hand side B has been changed accordingly.

           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 B and X.  NRHS >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                The original N-by-N matrix A.

           LDA

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

           AF

                     AF is DOUBLE PRECISION array, dimension (LDAF,N)
                The factors L and U from the factorization A = P*L*U
                as computed by DGETRF.

           LDAF

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

           IPIV

                     IPIV is INTEGER array, dimension (N)
                The pivot indices from DGETRF; for 1<=i<=N, row i of the
                matrix was interchanged with row IPIV(i).

           R

                     R is DOUBLE PRECISION array, dimension (N)
                The row scale factors for A.  If EQUED = 'R' or 'B', A is
                multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
                is not accessed.
                If R is accessed, each element of R should be a power of the radix
                to ensure a reliable solution and error estimates. Scaling by
                powers of the radix does not cause rounding errors unless the
                result underflows or overflows. Rounding errors during scaling
                lead to refining with a matrix that is not equivalent to the
                input matrix, producing error estimates that may not be
                reliable.

           C

                     C is DOUBLE PRECISION array, dimension (N)
                The column scale factors for A.  If EQUED = 'C' or 'B', A is
                multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
                is not accessed.
                If C is accessed, each element of C should be a power of the radix
                to ensure a reliable solution and error estimates. Scaling by
                powers of the radix does not cause rounding errors unless the
                result underflows or overflows. Rounding errors during scaling
                lead to refining with a matrix that is not equivalent to the
                input matrix, producing error estimates that may not be
                reliable.

           B

                     B is DOUBLE PRECISION array, dimension (LDB,NRHS)
                The right hand side matrix B.

           LDB

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

           X

                     X is DOUBLE PRECISION array, dimension (LDX,NRHS)
                On entry, the solution matrix X, as computed by DGETRS.
                On exit, the improved solution matrix X.

           LDX

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

           RCOND

                     RCOND is DOUBLE PRECISION
                Reciprocal scaled condition number.  This is an estimate of the
                reciprocal Skeel condition number of the matrix A after
                equilibration (if done).  If this is less than the machine
                precision (in particular, if it is zero), the matrix is singular
                to working precision.  Note that the error may still be small even
                if this number is very small and the matrix appears ill-
                conditioned.

           BERR

                     BERR is DOUBLE PRECISION array, dimension (NRHS)
                Componentwise relative backward error.  This is the
                componentwise relative backward error of each solution vector X(j)
                (i.e., the smallest relative change in any element of A or B that
                makes X(j) an exact solution).

           N_ERR_BNDS

                     N_ERR_BNDS is INTEGER
                Number of error bounds to return for each right hand side
                and each type (normwise or componentwise).  See ERR_BNDS_NORM and
                ERR_BNDS_COMP below.

           ERR_BNDS_NORM

                     ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                normwise relative error, which is defined as follows:

                Normwise relative error in the ith solution vector:
                        max_j (abs(XTRUE(j,i) - X(j,i)))
                       ------------------------------
                             max_j abs(X(j,i))

                The array is indexed by the type of error information as described
                below. There currently are up to three pieces of information
                returned.

                The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
                right-hand side.

                The second index in ERR_BNDS_NORM(:,err) contains the following
                three fields:
                err = 1 "Trust/don't trust" boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * dlamch('Epsilon').

                err = 2 "Guaranteed" error bound: The estimated forward error,
                         almost certainly within a factor of 10 of the true error
                         so long as the next entry is greater than the threshold
                         sqrt(n) * dlamch('Epsilon'). This error bound should only
                         be trusted if the previous boolean is true.

                err = 3  Reciprocal condition number: Estimated normwise
                         reciprocal condition number.  Compared with the threshold
                         sqrt(n) * dlamch('Epsilon') to determine if the error
                         estimate is "guaranteed". These reciprocal condition
                         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                         appropriately scaled matrix Z.
                         Let Z = S*A, where S scales each row by a power of the
                         radix so all absolute row sums of Z are approximately 1.

                See Lapack Working Note 165 for further details and extra
                cautions.

           ERR_BNDS_COMP

                     ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                componentwise relative error, which is defined as follows:

                Componentwise relative error in the ith solution vector:
                               abs(XTRUE(j,i) - X(j,i))
                        max_j ----------------------
                                    abs(X(j,i))

                The array is indexed by the right-hand side i (on which the
                componentwise relative error depends), and the type of error
                information as described below. There currently are up to three
                pieces of information returned for each right-hand side. If
                componentwise accuracy is not requested (PARAMS(3) = 0.0), then
                ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
                the first (:,N_ERR_BNDS) entries are returned.

                The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
                right-hand side.

                The second index in ERR_BNDS_COMP(:,err) contains the following
                three fields:
                err = 1 "Trust/don't trust" boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * dlamch('Epsilon').

                err = 2 "Guaranteed" error bound: The estimated forward error,
                         almost certainly within a factor of 10 of the true error
                         so long as the next entry is greater than the threshold
                         sqrt(n) * dlamch('Epsilon'). This error bound should only
                         be trusted if the previous boolean is true.

                err = 3  Reciprocal condition number: Estimated componentwise
                         reciprocal condition number.  Compared with the threshold
                         sqrt(n) * dlamch('Epsilon') to determine if the error
                         estimate is "guaranteed". These reciprocal condition
                         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                         appropriately scaled matrix Z.
                         Let Z = S*(A*diag(x)), where x is the solution for the
                         current right-hand side and S scales each row of
                         A*diag(x) by a power of the radix so all absolute row
                         sums of Z are approximately 1.

                See Lapack Working Note 165 for further details and extra
                cautions.

           NPARAMS

                     NPARAMS is INTEGER
                Specifies the number of parameters set in PARAMS.  If .LE. 0, the
                PARAMS array is never referenced and default values are used.

           PARAMS

                     PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
                Specifies algorithm parameters.  If an entry is .LT. 0.0, then
                that entry will be filled with default value used for that
                parameter.  Only positions up to NPARAMS are accessed; defaults
                are used for higher-numbered parameters.

                  PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
                       refinement or not.
                    Default: 1.0D+0
                       = 0.0 : No refinement is performed, and no error bounds are
                               computed.
                       = 1.0 : Use the double-precision refinement algorithm,
                               possibly with doubled-single computations if the
                               compilation environment does not support DOUBLE
                               PRECISION.
                         (other values are reserved for future use)

                  PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
                       computations allowed for refinement.
                    Default: 10
                    Aggressive: Set to 100 to permit convergence using approximate
                                factorizations or factorizations other than LU. If
                                the factorization uses a technique other than
                                Gaussian elimination, the guarantees in
                                err_bnds_norm and err_bnds_comp may no longer be
                                trustworthy.

                  PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
                       will attempt to find a solution with small componentwise
                       relative error in the double-precision algorithm.  Positive
                       is true, 0.0 is false.
                    Default: 1.0 (attempt componentwise convergence)

           WORK

                     WORK is DOUBLE PRECISION array, dimension (4*N)

           IWORK

                     IWORK is INTEGER array, dimension (N)

           INFO

                     INFO is INTEGER
                  = 0:  Successful exit. The solution to every right-hand side is
                    guaranteed.
                  < 0:  If INFO = -i, the i-th argument had an illegal value
                  > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
                    has been completed, but the factor U is exactly singular, so
                    the solution and error bounds could not be computed. RCOND = 0
                    is returned.
                  = N+J: The solution corresponding to the Jth right-hand side is
                    not guaranteed. The solutions corresponding to other right-
                    hand sides K with K > J may not be guaranteed as well, but
                    only the first such right-hand side is reported. If a small
                    componentwise error is not requested (PARAMS(3) = 0.0) then
                    the Jth right-hand side is the first with a normwise error
                    bound that is not guaranteed (the smallest J such
                    that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
                    the Jth right-hand side is the first with either a normwise or
                    componentwise error bound that is not guaranteed (the smallest
                    J such that either ERR_BNDS_NORM(J,1) = 0.0 or
                    ERR_BNDS_COMP(J,1) = 0.0). See the definition of
                    ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
                    about all of the right-hand sides check ERR_BNDS_NORM or
                    ERR_BNDS_COMP.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dgerq2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO)
       DGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked
       algorithm.

       Purpose:

            DGERQ2 computes an RQ factorization of a real m by n matrix A:
            A = R * Q.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the m by n matrix A.
                     On exit, if m <= n, the upper triangle of the subarray
                     A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
                     if m >= n, the elements on and above the (m-n)-th subdiagonal
                     contain the m by n upper trapezoidal matrix R; the remaining
                     elements, with the array TAU, represent the orthogonal matrix
                     Q as a product of elementary reflectors (see Further
                     Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (M)

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(1) H(2) . . . H(k), where k = min(m,n).

             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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
             A(m-k+i,1:n-k+i-1), and tau in TAU(i).

   subroutine dgerqf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer
       LWORK, integer INFO)
       DGERQF

       Purpose:

            DGERQF computes an RQ factorization of a real M-by-N matrix A:
            A = R * Q.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit,
                     if m <= n, the upper triangle of the subarray
                     A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
                     if m >= n, the elements on and above the (m-n)-th subdiagonal
                     contain the M-by-N upper trapezoidal matrix R;
                     the remaining elements, with the array TAU, represent the
                     orthogonal matrix Q as a product of min(m,n) elementary
                     reflectors (see Further Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= max(1,M).
                     For optimum performance LWORK >= M*NB, where NB is
                     the optimal blocksize.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The matrix Q is represented as a product of elementary reflectors

                Q = H(1) H(2) . . . H(k), where k = min(m,n).

             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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
             A(m-k+i,1:n-k+i-1), and tau in TAU(i).

   subroutine dgesvj (character*1 JOBA, character*1 JOBU, character*1 JOBV, integer M, integer N,
       double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( n )
       SVA, integer MV, double precision, dimension( ldv, * ) V, integer LDV, double precision,
       dimension( lwork ) WORK, integer LWORK, integer INFO)
       DGESVJ

       Purpose:

            DGESVJ computes the singular value decomposition (SVD) of a real
            M-by-N matrix A, where M >= N. The SVD of A is written as
                                               [++]   [xx]   [x0]   [xx]
                         A = U * SIGMA * V^t,  [++] = [xx] * [ox] * [xx]
                                               [++]   [xx]
            where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
            matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
            of SIGMA are the singular values of A. The columns of U and V are the
            left and the right singular vectors of A, respectively.
            DGESVJ can sometimes compute tiny singular values and their singular vectors much
            more accurately than other SVD routines, see below under Further Details.

       Parameters:
           JOBA

                     JOBA is CHARACTER*1
                     Specifies the structure of A.
                     = 'L': The input matrix A is lower triangular;
                     = 'U': The input matrix A is upper triangular;
                     = 'G': The input matrix A is general M-by-N matrix, M >= N.

           JOBU

                     JOBU is CHARACTER*1
                     Specifies whether to compute the left singular vectors
                     (columns of U):
                     = 'U': The left singular vectors corresponding to the nonzero
                            singular values are computed and returned in the leading
                            columns of A. See more details in the description of A.
                            The default numerical orthogonality threshold is set to
                            approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E').
                     = 'C': Analogous to JOBU='U', except that user can control the
                            level of numerical orthogonality of the computed left
                            singular vectors. TOL can be set to TOL = CTOL*EPS, where
                            CTOL is given on input in the array WORK.
                            No CTOL smaller than ONE is allowed. CTOL greater
                            than 1 / EPS is meaningless. The option 'C'
                            can be used if M*EPS is satisfactory orthogonality
                            of the computed left singular vectors, so CTOL=M could
                            save few sweeps of Jacobi rotations.
                            See the descriptions of A and WORK(1).
                     = 'N': The matrix U is not computed. However, see the
                            description of A.

           JOBV

                     JOBV is CHARACTER*1
                     Specifies whether to compute the right singular vectors, that
                     is, the matrix V:
                     = 'V' : the matrix V is computed and returned in the array V
                     = 'A' : the Jacobi rotations are applied to the MV-by-N
                             array V. In other words, the right singular vector
                             matrix V is not computed explicitly, instead it is
                             applied to an MV-by-N matrix initially stored in the
                             first MV rows of V.
                     = 'N' : the matrix V is not computed and the array V is not
                             referenced

           M

                     M is INTEGER
                     The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.

           N

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit :
                     If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C' :
                            If INFO .EQ. 0 :
                            RANKA orthonormal columns of U are returned in the
                            leading RANKA columns of the array A. Here RANKA <= N
                            is the number of computed singular values of A that are
                            above the underflow threshold DLAMCH('S'). The singular
                            vectors corresponding to underflowed or zero singular
                            values are not computed. The value of RANKA is returned
                            in the array WORK as RANKA=NINT(WORK(2)). Also see the
                            descriptions of SVA and WORK. The computed columns of U
                            are mutually numerically orthogonal up to approximately
                            TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),
                            see the description of JOBU.
                            If INFO .GT. 0 :
                            the procedure DGESVJ did not converge in the given number
                            of iterations (sweeps). In that case, the computed
                            columns of U may not be orthogonal up to TOL. The output
                            U (stored in A), SIGMA (given by the computed singular
                            values in SVA(1:N)) and V is still a decomposition of the
                            input matrix A in the sense that the residual
                            ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.

                     If JOBU .EQ. 'N' :
                            If INFO .EQ. 0 :
                            Note that the left singular vectors are 'for free' in the
                            one-sided Jacobi SVD algorithm. However, if only the
                            singular values are needed, the level of numerical
                            orthogonality of U is not an issue and iterations are
                            stopped when the columns of the iterated matrix are
                            numerically orthogonal up to approximately M*EPS. Thus,
                            on exit, A contains the columns of U scaled with the
                            corresponding singular values.
                            If INFO .GT. 0 :
                            the procedure DGESVJ did not converge in the given number
                            of iterations (sweeps).

           LDA

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

           SVA

                     SVA is DOUBLE PRECISION array, dimension (N)
                     On exit :
                     If INFO .EQ. 0 :
                     depending on the value SCALE = WORK(1), we have:
                            If SCALE .EQ. ONE :
                            SVA(1:N) contains the computed singular values of A.
                            During the computation SVA contains the Euclidean column
                            norms of the iterated matrices in the array A.
                            If SCALE .NE. ONE :
                            The singular values of A are SCALE*SVA(1:N), and this
                            factored representation is due to the fact that some of the
                            singular values of A might underflow or overflow.
                     If INFO .GT. 0 :
                     the procedure DGESVJ did not converge in the given number of
                     iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.

           MV

                     MV is INTEGER
                     If JOBV .EQ. 'A', then the product of Jacobi rotations in DGESVJ
                     is applied to the first MV rows of V. See the description of JOBV.

           V

                     V is DOUBLE PRECISION array, dimension (LDV,N)
                     If JOBV = 'V', then V contains on exit the N-by-N matrix of
                                    the right singular vectors;
                     If JOBV = 'A', then V contains the product of the computed right
                                    singular vector matrix and the initial matrix in
                                    the array V.
                     If JOBV = 'N', then V is not referenced.

           LDV

                     LDV is INTEGER
                     The leading dimension of the array V, LDV .GE. 1.
                     If JOBV .EQ. 'V', then LDV .GE. max(1,N).
                     If JOBV .EQ. 'A', then LDV .GE. max(1,MV) .

           WORK

                     WORK is DOUBLE PRECISION array, dimension (LWORK)
                     On entry :
                     If JOBU .EQ. 'C' :
                     WORK(1) = CTOL, where CTOL defines the threshold for convergence.
                               The process stops if all columns of A are mutually
                               orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
                               It is required that CTOL >= ONE, i.e. it is not
                               allowed to force the routine to obtain orthogonality
                               below EPS.
                     On exit :
                     WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
                               are the computed singular values of A.
                               (See description of SVA().)
                     WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
                               singular values.
                     WORK(3) = NINT(WORK(3)) is the number of the computed singular
                               values that are larger than the underflow threshold.
                     WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
                               rotations needed for numerical convergence.
                     WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
                               This is useful information in cases when DGESVJ did
                               not converge, as it can be used to estimate whether
                               the output is stil useful and for post festum analysis.
                     WORK(6) = the largest absolute value over all sines of the
                               Jacobi rotation angles in the last sweep. It can be
                               useful for a post festum analysis.

           LWORK

                     LWORK is INTEGER
                     length of WORK, WORK >= MAX(6,M+N)

           INFO

                     INFO is INTEGER
                     = 0 : successful exit.
                     < 0 : if INFO = -i, then the i-th argument had an illegal value
                     > 0 : DGESVJ did not converge in the maximal allowed number (30)
                           of sweeps. The output may still be useful. See the
                           description of WORK.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

       Further Details:

             The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
             rotations. The rotations are implemented as fast scaled rotations of
             Anda and Park [1]. In the case of underflow of the Jacobi angle, a
             modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
             column interchanges of de Rijk [2]. The relative accuracy of the computed
             singular values and the accuracy of the computed singular vectors (in
             angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
             The condition number that determines the accuracy in the full rank case
             is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
             spectral condition number. The best performance of this Jacobi SVD
             procedure is achieved if used in an  accelerated version of Drmac and
             Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
             Some tunning parameters (marked with [TP]) are available for the
             implementer.
             The computational range for the nonzero singular values is the  machine
             number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
             denormalized singular values can be computed with the corresponding
             gradual loss of accurate digits.

       Contributors:

             ============

             Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

       References:

            [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
                SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
            [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
                singular value decomposition on a vector computer.
                SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
            [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
            [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
                value computation in floating point arithmetic.
                SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
            [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
                SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
                LAPACK Working note 169.
            [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
                SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
                LAPACK Working note 170.
            [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
                QSVD, (H,K)-SVD computations.
                Department of Mathematics, University of Zagreb, 2008.

       Bugs, examples and comments:

             ===========================
             Please report all bugs and send interesting test examples and comments to
             drmac@math.hr. Thank you.

   subroutine dgetf2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) IPIV, integer INFO)
       DGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting
       with row interchanges (unblocked algorithm).

       Purpose:

            DGETF2 computes an LU factorization of a general m-by-n matrix A
            using partial pivoting with row interchanges.

            The factorization has the form
               A = P * L * U
            where P is a permutation matrix, L is lower triangular with unit
            diagonal elements (lower trapezoidal if m > n), and U is upper
            triangular (upper trapezoidal if m < n).

            This is the right-looking Level 2 BLAS version of the algorithm.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the m by n matrix to be factored.
                     On exit, the factors L and U from the factorization
                     A = P*L*U; the unit diagonal elements of L are not stored.

           LDA

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

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -k, the k-th argument had an illegal value
                     > 0: if INFO = k, U(k,k) is exactly zero. The factorization
                          has been completed, but the factor U is exactly
                          singular, and division by zero will occur if it is used
                          to solve a system of equations.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dgetrf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) IPIV, integer INFO)
       DGETRF DGETRF VARIANT: iterative version of Sivan Toledo's recursive LU algorithm

       DGETRF VARIANT: left-looking Level 3 BLAS version of the algorithm.

       Purpose:

            DGETRF computes an LU factorization of a general M-by-N matrix A
            using partial pivoting with row interchanges.

            The factorization has the form
               A = P * L * U
            where P is a permutation matrix, L is lower triangular with unit
            diagonal elements (lower trapezoidal if m > n), and U is upper
            triangular (upper trapezoidal if m < n).

            This is the right-looking Level 3 BLAS version of the algorithm.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix to be factored.
                     On exit, the factors L and U from the factorization
                     A = P*L*U; the unit diagonal elements of L are not stored.

           LDA

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

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Purpose:

        DGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This is the left-looking Level 3 BLAS version of the algorithm.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix to be factored.
                     On exit, the factors L and U from the factorization
                     A = P*L*U; the unit diagonal elements of L are not stored.

           LDA

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

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Purpose:

        DGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This code implements an iterative version of Sivan Toledo's recursive
        LU algorithm[1].  For square matrices, this iterative versions should
        be within a factor of two of the optimum number of memory transfers.

        The pattern is as follows, with the large blocks of U being updated
        in one call to DTRSM, and the dotted lines denoting sections that
        have had all pending permutations applied:

         1 2 3 4 5 6 7 8
        +-+-+---+-------+------
        | |1|   |       |
        |.+-+ 2 |       |
        | | |   |       |
        |.|.+-+-+   4   |
        | | | |1|       |
        | | |.+-+       |
        | | | | |       |
        |.|.|.|.+-+-+---+  8
        | | | | | |1|   |
        | | | | |.+-+ 2 |
        | | | | | | |   |
        | | | | |.|.+-+-+
        | | | | | | | |1|
        | | | | | | |.+-+
        | | | | | | | | |
        |.|.|.|.|.|.|.|.+-----
        | | | | | | | | |

        The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
        the binary expansion of the current column.  Each Schur update is
        applied as soon as the necessary portion of U is available.

        [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
        Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
        1065-1081. http://dx.doi.org/10.1137/S0895479896297744

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix to be factored.
                     On exit, the factors L and U from the factorization
                     A = P*L*U; the unit diagonal elements of L are not stored.

           LDA

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

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   recursive subroutine dgetrf2 (integer M, integer N, double precision, dimension( lda, * ) A,
       integer LDA, integer, dimension( * ) IPIV, integer INFO)
       DGETRF2

       Purpose:

            DGETRF2 computes an LU factorization of a general M-by-N matrix A
            using partial pivoting with row interchanges.

            The factorization has the form
               A = P * L * U
            where P is a permutation matrix, L is lower triangular with unit
            diagonal elements (lower trapezoidal if m > n), and U is upper
            triangular (upper trapezoidal if m < n).

            This is the recursive version of the algorithm. It divides
            the matrix into four submatrices:

                   [  A11 | A12  ]  where A11 is n1 by n1 and A22 is n2 by n2
               A = [ -----|----- ]  with n1 = min(m,n)/2
                   [  A21 | A22  ]       n2 = n-n1

                                                  [ A11 ]
            The subroutine calls itself to factor [ --- ],
                                                  [ A12 ]
                            [ A12 ]
            do the swaps on [ --- ], solve A12, update A22,
                            [ A22 ]

            then calls itself to factor A22 and do the swaps on A21.

       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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix to be factored.
                     On exit, the factors L and U from the factorization
                     A = P*L*U; the unit diagonal elements of L are not stored.

           LDA

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

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2016

   subroutine dgetri (integer N, double precision, dimension( lda, * ) A, integer LDA, integer,
       dimension( * ) IPIV, double precision, dimension( * ) WORK, integer LWORK, integer INFO)
       DGETRI

       Purpose:

            DGETRI computes the inverse of a matrix using the LU factorization
            computed by DGETRF.

            This method inverts U and then computes inv(A) by solving the system
            inv(A)*L = inv(U) for inv(A).

       Parameters:
           N

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the factors L and U from the factorization
                     A = P*L*U as computed by DGETRF.
                     On exit, if INFO = 0, the inverse of the original matrix A.

           LDA

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

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     The pivot indices from DGETRF; for 1<=i<=N, row i of the
                     matrix was interchanged with row IPIV(i).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO=0, then WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= max(1,N).
                     For optimal performance LWORK >= N*NB, where NB is
                     the optimal blocksize returned by ILAENV.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero; the matrix is
                           singular and its inverse could not be computed.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dgetrs (character TRANS, integer N, integer NRHS, double precision, dimension( lda,
       * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B,
       integer LDB, integer INFO)
       DGETRS

       Purpose:

            DGETRS solves a system of linear equations
               A * X = B  or  A**T * X = B
            with a general N-by-N matrix A using the LU factorization computed
            by DGETRF.

       Parameters:
           TRANS

                     TRANS is CHARACTER*1
                     Specifies the form of the system of equations:
                     = 'N':  A * X = B  (No transpose)
                     = 'T':  A**T* X = B  (Transpose)
                     = 'C':  A**T* X = B  (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 matrix B.  NRHS >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     The factors L and U from the factorization A = P*L*U
                     as computed by DGETRF.

           LDA

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

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     The pivot indices from DGETRF; for 1<=i<=N, row i of the
                     matrix was interchanged with row IPIV(i).

           B

                     B is DOUBLE PRECISION array, dimension (LDB,NRHS)
                     On entry, the right hand side matrix B.
                     On exit, the solution matrix X.

           LDB

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

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dhgeqz (character JOB, character COMPQ, character COMPZ, integer N, integer ILO,
       integer IHI, double precision, dimension( ldh, * ) H, integer LDH, double precision,
       dimension( ldt, * ) T, integer LDT, double precision, dimension( * ) ALPHAR, double
       precision, dimension( * ) ALPHAI, double precision, dimension( * ) BETA, double precision,
       dimension( ldq, * ) Q, integer LDQ, double precision, dimension( ldz, * ) Z, integer LDZ,
       double precision, dimension( * ) WORK, integer LWORK, integer INFO)
       DHGEQZ

       Purpose:

            DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
            where H is an upper Hessenberg matrix and T is upper triangular,
            using the double-shift QZ method.
            Matrix pairs of this type are produced by the reduction to
            generalized upper Hessenberg form of a real matrix pair (A,B):

               A = Q1*H*Z1**T,  B = Q1*T*Z1**T,

            as computed by DGGHRD.

            If JOB='S', then the Hessenberg-triangular pair (H,T) is
            also reduced to generalized Schur form,

               H = Q*S*Z**T,  T = Q*P*Z**T,

            where Q and Z are orthogonal matrices, P is an upper triangular
            matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
            diagonal blocks.

            The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
            (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
            eigenvalues.

            Additionally, the 2-by-2 upper triangular diagonal blocks of P
            corresponding to 2-by-2 blocks of S are reduced to positive diagonal
            form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
            P(j,j) > 0, and P(j+1,j+1) > 0.

            Optionally, the orthogonal matrix Q from the generalized Schur
            factorization may be postmultiplied into an input matrix Q1, and the
            orthogonal matrix Z may be postmultiplied into an input matrix Z1.
            If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
            the matrix pair (A,B) to generalized upper Hessenberg form, then the
            output matrices Q1*Q and Z1*Z are the orthogonal factors from the
            generalized Schur factorization of (A,B):

               A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.

            To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
            of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
            complex and beta real.
            If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
            generalized nonsymmetric eigenvalue problem (GNEP)
               A*x = lambda*B*x
            and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
            alternate form of the GNEP
               mu*A*y = B*y.
            Real eigenvalues can be read directly from the generalized Schur
            form:
              alpha = S(i,i), beta = P(i,i).

            Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
                 Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
                 pp. 241--256.

       Parameters:
           JOB

                     JOB is CHARACTER*1
                     = 'E': Compute eigenvalues only;
                     = 'S': Compute eigenvalues and the Schur form.

           COMPQ

                     COMPQ is CHARACTER*1
                     = 'N': Left Schur vectors (Q) are not computed;
                     = 'I': Q is initialized to the unit matrix and the matrix Q
                            of left Schur vectors of (H,T) is returned;
                     = 'V': Q must contain an orthogonal matrix Q1 on entry and
                            the product Q1*Q is returned.

           COMPZ

                     COMPZ is CHARACTER*1
                     = 'N': Right Schur vectors (Z) are not computed;
                     = 'I': Z is initialized to the unit matrix and the matrix Z
                            of right Schur vectors of (H,T) is returned;
                     = 'V': Z must contain an orthogonal matrix Z1 on entry and
                            the product Z1*Z is returned.

           N

                     N is INTEGER
                     The order of the matrices H, T, Q, and Z.  N >= 0.

           ILO

                     ILO is INTEGER

           IHI

                     IHI is INTEGER
                     ILO and IHI mark the rows and columns of H which are in
                     Hessenberg form.  It is assumed that A is already upper
                     triangular in rows and columns 1:ILO-1 and IHI+1:N.
                     If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

           H

                     H is DOUBLE PRECISION array, dimension (LDH, N)
                     On entry, the N-by-N upper Hessenberg matrix H.
                     On exit, if JOB = 'S', H contains the upper quasi-triangular
                     matrix S from the generalized Schur factorization.
                     If JOB = 'E', the diagonal blocks of H match those of S, but
                     the rest of H is unspecified.

           LDH

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

           T

                     T is DOUBLE PRECISION array, dimension (LDT, N)
                     On entry, the N-by-N upper triangular matrix T.
                     On exit, if JOB = 'S', T contains the upper triangular
                     matrix P from the generalized Schur factorization;
                     2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
                     are reduced to positive diagonal form, i.e., if H(j+1,j) is
                     non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
                     T(j+1,j+1) > 0.
                     If JOB = 'E', the diagonal blocks of T match those of P, but
                     the rest of T is unspecified.

           LDT

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

           ALPHAR

                     ALPHAR is DOUBLE PRECISION array, dimension (N)
                     The real parts of each scalar alpha defining an eigenvalue
                     of GNEP.

           ALPHAI

                     ALPHAI is DOUBLE PRECISION array, dimension (N)
                     The imaginary parts of each scalar alpha defining an
                     eigenvalue of GNEP.
                     If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
                     positive, then the j-th and (j+1)-st eigenvalues are a
                     complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

           BETA

                     BETA is DOUBLE PRECISION array, dimension (N)
                     The scalars beta that define the eigenvalues of GNEP.
                     Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
                     beta = BETA(j) represent the j-th eigenvalue of the matrix
                     pair (A,B), in one of the forms lambda = alpha/beta or
                     mu = beta/alpha.  Since either lambda or mu may overflow,
                     they should not, in general, be computed.

           Q

                     Q is DOUBLE PRECISION array, dimension (LDQ, N)
                     On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
                     the reduction of (A,B) to generalized Hessenberg form.
                     On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
                     vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix
                     of left Schur vectors of (A,B).
                     Not referenced if COMPQ = 'N'.

           LDQ

                     LDQ is INTEGER
                     The leading dimension of the array Q.  LDQ >= 1.
                     If COMPQ='V' or 'I', then LDQ >= N.

           Z

                     Z is DOUBLE PRECISION array, dimension (LDZ, N)
                     On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
                     the reduction of (A,B) to generalized Hessenberg form.
                     On exit, if COMPZ = 'I', the orthogonal matrix of
                     right Schur vectors of (H,T), and if COMPZ = 'V', the
                     orthogonal matrix of right Schur vectors of (A,B).
                     Not referenced if COMPZ = 'N'.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1.
                     If COMPZ='V' or 'I', then LDZ >= N.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= max(1,N).

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value
                     = 1,...,N: the QZ iteration did not converge.  (H,T) is not
                                in Schur form, but ALPHAR(i), ALPHAI(i), and
                                BETA(i), i=INFO+1,...,N should be correct.
                     = N+1,...,2*N: the shift calculation failed.  (H,T) is not
                                in Schur form, but ALPHAR(i), ALPHAI(i), and
                                BETA(i), i=INFO-N+1,...,N should be correct.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2016

       Further Details:

             Iteration counters:

             JITER  -- counts iterations.
             IITER  -- counts iterations run since ILAST was last
                       changed.  This is therefore reset only when a 1-by-1 or
                       2-by-2 block deflates off the bottom.

   subroutine dla_geamv (integer TRANS, integer M, integer N, double precision ALPHA, double
       precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) X, integer
       INCX, double precision BETA, double precision, dimension( * ) Y, integer INCY)
       DLA_GEAMV computes a matrix-vector product using a general matrix to calculate error
       bounds.

       Purpose:

            DLA_GEAMV  performs one of the matrix-vector operations

                    y := alpha*abs(A)*abs(x) + beta*abs(y),
               or   y := alpha*abs(A)**T*abs(x) + beta*abs(y),

            where alpha and beta are scalars, x and y are vectors and A is an
            m by n matrix.

            This function is primarily used in calculating error bounds.
            To protect against underflow during evaluation, components in
            the resulting vector are perturbed away from zero by (N+1)
            times the underflow threshold.  To prevent unnecessarily large
            errors for block-structure embedded in general matrices,
            "symbolically" zero components are not perturbed.  A zero
            entry is considered "symbolic" if all multiplications involved
            in computing that entry have at least one zero multiplicand.

       Parameters:
           TRANS

                     TRANS is INTEGER
                      On entry, TRANS specifies the operation to be performed as
                      follows:

                        BLAS_NO_TRANS      y := alpha*abs(A)*abs(x) + beta*abs(y)
                        BLAS_TRANS         y := alpha*abs(A**T)*abs(x) + beta*abs(y)
                        BLAS_CONJ_TRANS    y := alpha*abs(A**T)*abs(x) + beta*abs(y)

                      Unchanged on exit.

           M

                     M is INTEGER
                      On entry, M specifies the number of rows of the matrix A.
                      M must be at least zero.
                      Unchanged on exit.

           N

                     N is INTEGER
                      On entry, N specifies the number of columns of the matrix A.
                      N must be at least zero.
                      Unchanged on exit.

           ALPHA

                     ALPHA is DOUBLE PRECISION
                      On entry, ALPHA specifies the scalar alpha.
                      Unchanged on exit.

           A

                     A is DOUBLE PRECISION array, dimension ( LDA, n )
                      Before entry, the leading m by n part of the array A must
                      contain the matrix of coefficients.
                      Unchanged on exit.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      max( 1, m ).
                      Unchanged on exit.

           X

                     X is DOUBLE PRECISION array, dimension
                      ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
                      and at least
                      ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
                      Before entry, the incremented array X must contain the
                      vector x.
                      Unchanged on exit.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.
                      Unchanged on exit.

           BETA

                     BETA is DOUBLE PRECISION
                      On entry, BETA specifies the scalar beta. When BETA is
                      supplied as zero then Y need not be set on input.
                      Unchanged on exit.

           Y

                     Y is DOUBLE PRECISION array,
                      dimension at least
                      ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
                      and at least
                      ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
                      Before entry with BETA non-zero, the incremented array Y
                      must contain the vector y. On exit, Y is overwritten by the
                      updated vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.
                      Unchanged on exit.

             Level 2 Blas routine.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

   double precision function dla_gercond (character TRANS, integer N, double precision,
       dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer
       LDAF, integer, dimension( * ) IPIV, integer CMODE, double precision, dimension( * ) C,
       integer INFO, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK)
       DLA_GERCOND estimates the Skeel condition number for a general matrix.

       Purpose:

               DLA_GERCOND estimates the Skeel condition number of op(A) * op2(C)
               where op2 is determined by CMODE as follows
               CMODE =  1    op2(C) = C
               CMODE =  0    op2(C) = I
               CMODE = -1    op2(C) = inv(C)
               The Skeel condition number cond(A) = norminf( |inv(A)||A| )
               is computed by computing scaling factors R such that
               diag(R)*A*op2(C) is row equilibrated and computing the standard
               infinity-norm condition number.

       Parameters:
           TRANS

                     TRANS is CHARACTER*1
                Specifies the form of the system of equations:
                  = 'N':  A * X = B     (No transpose)
                  = 'T':  A**T * X = B  (Transpose)
                  = 'C':  A**H * X = B  (Conjugate Transpose = Transpose)

           N

                     N is INTEGER
                The number of linear equations, i.e., the order of the
                matrix A.  N >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                On entry, the N-by-N matrix A.

           LDA

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

           AF

                     AF is DOUBLE PRECISION array, dimension (LDAF,N)
                The factors L and U from the factorization
                A = P*L*U as computed by DGETRF.

           LDAF

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

           IPIV

                     IPIV is INTEGER array, dimension (N)
                The pivot indices from the factorization A = P*L*U
                as computed by DGETRF; row i of the matrix was interchanged
                with row IPIV(i).

           CMODE

                     CMODE is INTEGER
                Determines op2(C) in the formula op(A) * op2(C) as follows:
                CMODE =  1    op2(C) = C
                CMODE =  0    op2(C) = I
                CMODE = -1    op2(C) = inv(C)

           C

                     C is DOUBLE PRECISION array, dimension (N)
                The vector C in the formula op(A) * op2(C).

           INFO

                     INFO is INTEGER
                  = 0:  Successful exit.
                i > 0:  The ith argument is invalid.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (3*N).
                Workspace.

           IWORK

                     IWORK is INTEGER array, dimension (N).
                Workspace.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dla_gerfsx_extended (integer PREC_TYPE, integer TRANS_TYPE, integer N, integer
       NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension(
       ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, double
       precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double
       precision, dimension( ldy, * ) Y, integer LDY, double precision, dimension( * ) BERR_OUT,
       integer N_NORMS, double precision, dimension( nrhs, * ) ERRS_N, double precision,
       dimension( nrhs, * ) ERRS_C, double precision, dimension( * ) RES, double precision,
       dimension( * ) AYB, double precision, dimension( * ) DY, double precision, dimension( * )
       Y_TAIL, double precision RCOND, integer ITHRESH, double precision RTHRESH, double
       precision DZ_UB, logical IGNORE_CWISE, integer INFO)
       DLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for
       general matrices by performing extra-precise iterative refinement and provides error
       bounds and backward error estimates for the solution.

       Purpose:

            DLA_GERFSX_EXTENDED improves the computed solution to a system of
            linear equations by performing extra-precise iterative refinement
            and provides error bounds and backward error estimates for the solution.
            This subroutine is called by DGERFSX to perform iterative refinement.
            In addition to normwise error bound, the code provides maximum
            componentwise error bound if possible. See comments for ERRS_N
            and ERRS_C for details of the error bounds. Note that this
            subroutine is only resonsible for setting the second fields of
            ERRS_N and ERRS_C.

       Parameters:
           PREC_TYPE

                     PREC_TYPE is INTEGER
                Specifies the intermediate precision to be used in refinement.
                The value is defined by ILAPREC(P) where P is a CHARACTER and
                P    = 'S':  Single
                     = 'D':  Double
                     = 'I':  Indigenous
                     = 'X', 'E':  Extra

           TRANS_TYPE

                     TRANS_TYPE is INTEGER
                Specifies the transposition operation on A.
                The value is defined by ILATRANS(T) where T is a CHARACTER and
                T    = 'N':  No transpose
                     = 'T':  Transpose
                     = 'C':  Conjugate transpose

           N

                     N is INTEGER
                The number of linear equations, i.e., 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
                matrix B.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                On entry, the N-by-N matrix A.

           LDA

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

           AF

                     AF is DOUBLE PRECISION array, dimension (LDAF,N)
                The factors L and U from the factorization
                A = P*L*U as computed by DGETRF.

           LDAF

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

           IPIV

                     IPIV is INTEGER array, dimension (N)
                The pivot indices from the factorization A = P*L*U
                as computed by DGETRF; row i of the matrix was interchanged
                with row IPIV(i).

           COLEQU

                     COLEQU is LOGICAL
                If .TRUE. then column equilibration was done to A before calling
                this routine. This is needed to compute the solution and error
                bounds correctly.

           C

                     C is DOUBLE PRECISION array, dimension (N)
                The column scale factors for A. If COLEQU = .FALSE., C
                is not accessed. If C is input, each element of C should be a power
                of the radix to ensure a reliable solution and error estimates.
                Scaling by powers of the radix does not cause rounding errors unless
                the result underflows or overflows. Rounding errors during scaling
                lead to refining with a matrix that is not equivalent to the
                input matrix, producing error estimates that may not be
                reliable.

           B

                     B is DOUBLE PRECISION array, dimension (LDB,NRHS)
                The right-hand-side matrix B.

           LDB

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

           Y

                     Y is DOUBLE PRECISION array, dimension (LDY,NRHS)
                On entry, the solution matrix X, as computed by DGETRS.
                On exit, the improved solution matrix Y.

           LDY

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

           BERR_OUT

                     BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
                On exit, BERR_OUT(j) contains the componentwise relative backward
                error for right-hand-side j from the formula
                    max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
                where abs(Z) is the componentwise absolute value of the matrix
                or vector Z. This is computed by DLA_LIN_BERR.

           N_NORMS

                     N_NORMS is INTEGER
                Determines which error bounds to return (see ERRS_N
                and ERRS_C).
                If N_NORMS >= 1 return normwise error bounds.
                If N_NORMS >= 2 return componentwise error bounds.

           ERRS_N

                     ERRS_N is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                normwise relative error, which is defined as follows:

                Normwise relative error in the ith solution vector:
                        max_j (abs(XTRUE(j,i) - X(j,i)))
                       ------------------------------
                             max_j abs(X(j,i))

                The array is indexed by the type of error information as described
                below. There currently are up to three pieces of information
                returned.

                The first index in ERRS_N(i,:) corresponds to the ith
                right-hand side.

                The second index in ERRS_N(:,err) contains the following
                three fields:
                err = 1 "Trust/don't trust" boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * slamch('Epsilon').

                err = 2 "Guaranteed" error bound: The estimated forward error,
                         almost certainly within a factor of 10 of the true error
                         so long as the next entry is greater than the threshold
                         sqrt(n) * slamch('Epsilon'). This error bound should only
                         be trusted if the previous boolean is true.

                err = 3  Reciprocal condition number: Estimated normwise
                         reciprocal condition number.  Compared with the threshold
                         sqrt(n) * slamch('Epsilon') to determine if the error
                         estimate is "guaranteed". These reciprocal condition
                         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                         appropriately scaled matrix Z.
                         Let Z = S*A, where S scales each row by a power of the
                         radix so all absolute row sums of Z are approximately 1.

                This subroutine is only responsible for setting the second field
                above.
                See Lapack Working Note 165 for further details and extra
                cautions.

           ERRS_C

                     ERRS_C is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                componentwise relative error, which is defined as follows:

                Componentwise relative error in the ith solution vector:
                               abs(XTRUE(j,i) - X(j,i))
                        max_j ----------------------
                                    abs(X(j,i))

                The array is indexed by the right-hand side i (on which the
                componentwise relative error depends), and the type of error
                information as described below. There currently are up to three
                pieces of information returned for each right-hand side. If
                componentwise accuracy is not requested (PARAMS(3) = 0.0), then
                ERRS_C is not accessed.  If N_ERR_BNDS .LT. 3, then at most
                the first (:,N_ERR_BNDS) entries are returned.

                The first index in ERRS_C(i,:) corresponds to the ith
                right-hand side.

                The second index in ERRS_C(:,err) contains the following
                three fields:
                err = 1 "Trust/don't trust" boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * slamch('Epsilon').

                err = 2 "Guaranteed" error bound: The estimated forward error,
                         almost certainly within a factor of 10 of the true error
                         so long as the next entry is greater than the threshold
                         sqrt(n) * slamch('Epsilon'). This error bound should only
                         be trusted if the previous boolean is true.

                err = 3  Reciprocal condition number: Estimated componentwise
                         reciprocal condition number.  Compared with the threshold
                         sqrt(n) * slamch('Epsilon') to determine if the error
                         estimate is "guaranteed". These reciprocal condition
                         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                         appropriately scaled matrix Z.
                         Let Z = S*(A*diag(x)), where x is the solution for the
                         current right-hand side and S scales each row of
                         A*diag(x) by a power of the radix so all absolute row
                         sums of Z are approximately 1.

                This subroutine is only responsible for setting the second field
                above.
                See Lapack Working Note 165 for further details and extra
                cautions.

           RES

                     RES is DOUBLE PRECISION array, dimension (N)
                Workspace to hold the intermediate residual.

           AYB

                     AYB is DOUBLE PRECISION array, dimension (N)
                Workspace. This can be the same workspace passed for Y_TAIL.

           DY

                     DY is DOUBLE PRECISION array, dimension (N)
                Workspace to hold the intermediate solution.

           Y_TAIL

                     Y_TAIL is DOUBLE PRECISION array, dimension (N)
                Workspace to hold the trailing bits of the intermediate solution.

           RCOND

                     RCOND is DOUBLE PRECISION
                Reciprocal scaled condition number.  This is an estimate of the
                reciprocal Skeel condition number of the matrix A after
                equilibration (if done).  If this is less than the machine
                precision (in particular, if it is zero), the matrix is singular
                to working precision.  Note that the error may still be small even
                if this number is very small and the matrix appears ill-
                conditioned.

           ITHRESH

                     ITHRESH is INTEGER
                The maximum number of residual computations allowed for
                refinement. The default is 10. For 'aggressive' set to 100 to
                permit convergence using approximate factorizations or
                factorizations other than LU. If the factorization uses a
                technique other than Gaussian elimination, the guarantees in
                ERRS_N and ERRS_C may no longer be trustworthy.

           RTHRESH

                     RTHRESH is DOUBLE PRECISION
                Determines when to stop refinement if the error estimate stops
                decreasing. Refinement will stop when the next solution no longer
                satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
                the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
                default value is 0.5. For 'aggressive' set to 0.9 to permit
                convergence on extremely ill-conditioned matrices. See LAWN 165
                for more details.

           DZ_UB

                     DZ_UB is DOUBLE PRECISION
                Determines when to start considering componentwise convergence.
                Componentwise convergence is only considered after each component
                of the solution Y is stable, which we definte as the relative
                change in each component being less than DZ_UB. The default value
                is 0.25, requiring the first bit to be stable. See LAWN 165 for
                more details.

           IGNORE_CWISE

                     IGNORE_CWISE is LOGICAL
                If .TRUE. then ignore componentwise convergence. Default value
                is .FALSE..

           INFO

                     INFO is INTEGER
                  = 0:  Successful exit.
                  < 0:  if INFO = -i, the ith argument to DGETRS had an illegal
                        value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

   double precision function dla_gerpvgrw (integer N, integer NCOLS, double precision, dimension(
       lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF)
       DLA_GERPVGRW

       Purpose:

            DLA_GERPVGRW computes the reciprocal pivot growth factor
            norm(A)/norm(U). The "max absolute element" norm is used. If this is
            much less than 1, the stability of the LU factorization of the
            (equilibrated) matrix A could be poor. This also means that the
            solution X, estimated condition numbers, and error bounds could be
            unreliable.

       Parameters:
           N

                     N is INTEGER
                The number of linear equations, i.e., the order of the
                matrix A.  N >= 0.

           NCOLS

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                On entry, the N-by-N matrix A.

           LDA

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

           AF

                     AF is DOUBLE PRECISION array, dimension (LDAF,N)
                The factors L and U from the factorization
                A = P*L*U as computed by DGETRF.

           LDAF

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dtgevc (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer
       N, double precision, dimension( lds, * ) S, integer LDS, double precision, dimension( ldp,
       * ) P, integer LDP, double precision, dimension( ldvl, * ) VL, integer LDVL, double
       precision, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, double precision,
       dimension( * ) WORK, integer INFO)
       DTGEVC

       Purpose:

            DTGEVC computes some or all of the right and/or left eigenvectors of
            a pair of real matrices (S,P), where S is a quasi-triangular matrix
            and P is upper triangular.  Matrix pairs of this type are produced by
            the generalized Schur factorization of a matrix pair (A,B):

               A = Q*S*Z**T,  B = Q*P*Z**T

            as computed by DGGHRD + DHGEQZ.

            The right eigenvector x and the left eigenvector y of (S,P)
            corresponding to an eigenvalue w are defined by:

               S*x = w*P*x,  (y**H)*S = w*(y**H)*P,

            where y**H denotes the conjugate tranpose of y.
            The eigenvalues are not input to this routine, but are computed
            directly from the diagonal blocks of S and P.

            This routine returns the matrices X and/or Y of right and left
            eigenvectors of (S,P), or the products Z*X and/or Q*Y,
            where Z and Q are input matrices.
            If Q and Z are the orthogonal factors from the generalized Schur
            factorization of a matrix pair (A,B), then Z*X and Q*Y
            are the matrices of right and left eigenvectors of (A,B).

       Parameters:
           SIDE

                     SIDE is CHARACTER*1
                     = 'R': compute right eigenvectors only;
                     = 'L': compute left eigenvectors only;
                     = 'B': compute both right and left eigenvectors.

           HOWMNY

                     HOWMNY is CHARACTER*1
                     = 'A': compute all right and/or left eigenvectors;
                     = 'B': compute all right and/or left eigenvectors,
                            backtransformed by the matrices in VR and/or VL;
                     = 'S': compute selected right and/or left eigenvectors,
                            specified by the logical array SELECT.

           SELECT

                     SELECT is LOGICAL array, dimension (N)
                     If HOWMNY='S', SELECT specifies the eigenvectors to be
                     computed.  If w(j) is a real eigenvalue, the corresponding
                     real eigenvector is computed if SELECT(j) is .TRUE..
                     If w(j) and w(j+1) are the real and imaginary parts of a
                     complex eigenvalue, the corresponding complex eigenvector
                     is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
                     and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
                     set to .FALSE..
                     Not referenced if HOWMNY = 'A' or 'B'.

           N

                     N is INTEGER
                     The order of the matrices S and P.  N >= 0.

           S

                     S is DOUBLE PRECISION array, dimension (LDS,N)
                     The upper quasi-triangular matrix S from a generalized Schur
                     factorization, as computed by DHGEQZ.

           LDS

                     LDS is INTEGER
                     The leading dimension of array S.  LDS >= max(1,N).

           P

                     P is DOUBLE PRECISION array, dimension (LDP,N)
                     The upper triangular matrix P from a generalized Schur
                     factorization, as computed by DHGEQZ.
                     2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
                     of S must be in positive diagonal form.

           LDP

                     LDP is INTEGER
                     The leading dimension of array P.  LDP >= max(1,N).

           VL

                     VL is DOUBLE PRECISION array, dimension (LDVL,MM)
                     On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
                     contain an N-by-N matrix Q (usually the orthogonal matrix Q
                     of left Schur vectors returned by DHGEQZ).
                     On exit, if SIDE = 'L' or 'B', VL contains:
                     if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
                     if HOWMNY = 'B', the matrix Q*Y;
                     if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
                                 SELECT, stored consecutively in the columns of
                                 VL, in the same order as their eigenvalues.

                     A complex eigenvector corresponding to a complex eigenvalue
                     is stored in two consecutive columns, the first holding the
                     real part, and the second the imaginary part.

                     Not referenced if SIDE = 'R'.

           LDVL

                     LDVL is INTEGER
                     The leading dimension of array VL.  LDVL >= 1, and if
                     SIDE = 'L' or 'B', LDVL >= N.

           VR

                     VR is DOUBLE PRECISION array, dimension (LDVR,MM)
                     On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
                     contain an N-by-N matrix Z (usually the orthogonal matrix Z
                     of right Schur vectors returned by DHGEQZ).

                     On exit, if SIDE = 'R' or 'B', VR contains:
                     if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
                     if HOWMNY = 'B' or 'b', the matrix Z*X;
                     if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
                                 specified by SELECT, stored consecutively in the
                                 columns of VR, in the same order as their
                                 eigenvalues.

                     A complex eigenvector corresponding to a complex eigenvalue
                     is stored in two consecutive columns, the first holding the
                     real part and the second the imaginary part.

                     Not referenced if SIDE = 'L'.

           LDVR

                     LDVR is INTEGER
                     The leading dimension of the array VR.  LDVR >= 1, and if
                     SIDE = 'R' or 'B', LDVR >= N.

           MM

                     MM is INTEGER
                     The number of columns in the arrays VL and/or VR. MM >= M.

           M

                     M is INTEGER
                     The number of columns in the arrays VL and/or VR actually
                     used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
                     is set to N.  Each selected real eigenvector occupies one
                     column and each selected complex eigenvector occupies two
                     columns.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (6*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex
                           eigenvalue.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             Allocation of workspace:
             ---------- -- ---------

                WORK( j ) = 1-norm of j-th column of A, above the diagonal
                WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
                WORK( 2*N+1:3*N ) = real part of eigenvector
                WORK( 3*N+1:4*N ) = imaginary part of eigenvector
                WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
                WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector

             Rowwise vs. columnwise solution methods:
             ------- --  ---------- -------- -------

             Finding a generalized eigenvector consists basically of solving the
             singular triangular system

              (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)

             Consider finding the i-th right eigenvector (assume all eigenvalues
             are real). The equation to be solved is:
                  n                   i
             0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
                 k=j                 k=j

             where  C = (A - w B)  (The components v(i+1:n) are 0.)

             The "rowwise" method is:

             (1)  v(i) := 1
             for j = i-1,. . .,1:
                                     i
                 (2) compute  s = - sum C(j,k) v(k)   and
                                   k=j+1

                 (3) v(j) := s / C(j,j)

             Step 2 is sometimes called the "dot product" step, since it is an
             inner product between the j-th row and the portion of the eigenvector
             that has been computed so far.

             The "columnwise" method consists basically in doing the sums
             for all the rows in parallel.  As each v(j) is computed, the
             contribution of v(j) times the j-th column of C is added to the
             partial sums.  Since FORTRAN arrays are stored columnwise, this has
             the advantage that at each step, the elements of C that are accessed
             are adjacent to one another, whereas with the rowwise method, the
             elements accessed at a step are spaced LDS (and LDP) words apart.

             When finding left eigenvectors, the matrix in question is the
             transpose of the one in storage, so the rowwise method then
             actually accesses columns of A and B at each step, and so is the
             preferred method.

   subroutine dtgexc (logical WANTQ, logical WANTZ, integer N, double precision, dimension( lda,
       * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double
       precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( ldz, * ) Z,
       integer LDZ, integer IFST, integer ILST, double precision, dimension( * ) WORK, integer
       LWORK, integer INFO)
       DTGEXC

       Purpose:

            DTGEXC reorders the generalized real Schur decomposition of a real
            matrix pair (A,B) using an orthogonal equivalence transformation

                           (A, B) = Q * (A, B) * Z**T,

            so that the diagonal block of (A, B) with row index IFST is moved
            to row ILST.

            (A, B) must be in generalized real Schur canonical form (as returned
            by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
            diagonal blocks. B is upper triangular.

            Optionally, the matrices Q and Z of generalized Schur vectors are
            updated.

                   Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
                   Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T

       Parameters:
           WANTQ

                     WANTQ is LOGICAL
                     .TRUE. : update the left transformation matrix Q;
                     .FALSE.: do not update Q.

           WANTZ

                     WANTZ is LOGICAL
                     .TRUE. : update the right transformation matrix Z;
                     .FALSE.: do not update Z.

           N

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

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the matrix A in generalized real Schur canonical
                     form.
                     On exit, the updated matrix A, again in generalized
                     real Schur canonical form.

           LDA

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

           B

                     B is DOUBLE PRECISION array, dimension (LDB,N)
                     On entry, the matrix B in generalized real Schur canonical
                     form (A,B).
                     On exit, the updated matrix B, again in generalized
                     real Schur canonical form (A,B).

           LDB

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

           Q

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

           LDQ

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

           Z

                     Z is DOUBLE PRECISION array, dimension (LDZ,N)
                     On entry, if WANTZ = .TRUE., the orthogonal matrix Z.
                     On exit, the updated matrix Z.
                     If WANTZ = .FALSE., Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z. LDZ >= 1.
                     If WANTZ = .TRUE., LDZ >= N.

           IFST

                     IFST is INTEGER

           ILST

                     ILST is INTEGER
                     Specify the reordering of the diagonal blocks of (A, B).
                     The block with row index IFST is moved to row ILST, by a
                     sequence of swapping between adjacent blocks.
                     On exit, if IFST pointed on entry to the second row of
                     a 2-by-2 block, it is changed to point to the first row;
                     ILST always points to the first row of the block in its
                     final position (which may differ from its input value by
                     +1 or -1). 1 <= IFST, ILST <= N.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.
                     LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                      =0:  successful exit.
                      <0:  if INFO = -i, the i-th argument had an illegal value.
                      =1:  The transformed matrix pair (A, B) would be too far
                           from generalized Schur form; the problem is ill-
                           conditioned. (A, B) may have been partially reordered,
                           and ILST points to the first row of the current
                           position of the block being moved.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Contributors:
           Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901
           87 Umea, Sweden.

       References:

             [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
                 Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
                 M.S. Moonen et al (eds), Linear Algebra for Large Scale and
                 Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

   subroutine sgelqt (integer M, integer N, integer MB, real, dimension( lda, * ) A, integer LDA,
       real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) WORK, integer INFO)
       Purpose:

       DGELQT computes a blocked LQ factorization of a real M-by-N matrix A using the compact WY
       representation of Q.

       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.

           MB

                     MB is INTEGER
                     The block size to be used in the blocked QR.  MIN(M,N) >= MB >= 1.

           A

                     A is REAL array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, the elements on and below the diagonal of the array
                     contain the M-by-MIN(M,N) lower trapezoidal matrix L (L is
                     lower triangular if M <= N); the elements above the diagonal
                     are the rows of V.

           LDA

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

           T

                     T is REAL array, dimension (LDT,MIN(M,N))
                     The upper triangular block reflectors stored in compact form
                     as a sequence of upper triangular blocks.  See below
                     for further details.

           LDT

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

           WORK

                     WORK is REAL array, dimension (MB*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

       Further Details:

       The matrix V stores the elementary reflectors H(i) in the i-th row above the diagonal. For
       example, if M=5 and N=3, the matrix V is

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

       where the vi's represent the vectors which define H(i), which are returned in the matrix
       A. The 1's along the diagonal of V are not stored in A. Let K=MIN(M,N). The number of
       blocks is B = ceiling(K/MB), where each block is of order MB except for the last block,
       which is of order IB = K - (B-1)*MB. For each of the B blocks, a upper triangular block
       reflector factor is computed: T1, T2, ..., TB. The MB-by-MB (and IB-by-IB for the last
       block) T's are stored in the MB-by-K matrix T as

       T = (T1 T2 ... TB).

   recursive subroutine sgelqt3 (integer M, integer N, real, dimension( lda, * ) A, integer LDA,
       real, dimension( ldt, * ) T, integer LDT, integer INFO)
       Purpose:

       DGELQT3 recursively computes a LQ factorization of a real M-by-N matrix A, using the
       compact WY representation of Q.

       Based on the algorithm of Elmroth and Gustavson, IBM J. Res. Develop. Vol 44 No. 4 July
       2000.

       Parameters:
           M

                     M is INTEGER
                     The number of rows of the matrix A.  M =< N.

           N

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

           A

                     A is REAL array, dimension (LDA,N)
                     On entry, the real M-by-N matrix A.  On exit, the elements on and
                     below the diagonal contain the N-by-N lower triangular matrix L; the
                     elements above the diagonal are the rows of V.  See below for
                     further details.

           LDA

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

           T

                     T is REAL array, dimension (LDT,N)
                     The N-by-N upper triangular factor of the block reflector.
                     The elements on and above the diagonal contain the block
                     reflector T; the elements below the diagonal are not used.
                     See below for further details.

           LDT

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

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

       Further Details:

       The matrix V stores the elementary reflectors H(i) in the i-th row above the diagonal. For
       example, if M=5 and N=3, the matrix V is

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

       where the vi's represent the vectors which define H(i), which are returned in the matrix
       A. The 1's along the diagonal of V are not stored in A. The block reflector H is then
       given by

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

       where V**T is the transpose of V.

       For details of the algorithm, see Elmroth and Gustavson (cited above).

   subroutine sgemlqt (character SIDE, character TRANS, integer M, integer N, integer K, integer
       MB, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * ) T, integer LDT,
       real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK, integer INFO)
       Purpose:

       DGEMLQT overwrites the general real M-by-N matrix C with

       SIDE = 'L' SIDE = 'R' TRANS = 'N': Q C C Q TRANS = 'T': Q**T C C Q**T

       where Q is a real orthogonal matrix defined as the product of K elementary reflectors:

       Q = H(1) H(2) . . . H(K) = I - V T V**T

       generated using the compact WY representation as returned by DGELQT.

       Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

       Parameters:
           SIDE

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

           TRANS

                     TRANS is CHARACTER*1
                     = 'N':  No transpose, apply Q;
                     = 'C':  Transpose, apply Q**T.

           M

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

           N

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

           K

                     K is INTEGER
                     The number of elementary reflectors whose product defines
                     the matrix Q.
                     If SIDE = 'L', M >= K >= 0;
                     if SIDE = 'R', N >= K >= 0.

           MB

                     MB is INTEGER
                     The block size used for the storage of T.  K >= MB >= 1.
                     This must be the same value of MB used to generate T
                     in DGELQT.

           V

                     V is REAL array, dimension
                                          (LDV,M) if SIDE = 'L',
                                          (LDV,N) if SIDE = 'R'
                     The i-th row must contain the vector which defines the
                     elementary reflector H(i), for i = 1,2,...,k, as returned by
                     DGELQT in the first K rows of its array argument A.

           LDV

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

           T

                     T is REAL array, dimension (LDT,K)
                     The upper triangular factors of the block reflectors
                     as returned by DGELQT, stored as a MB-by-K matrix.

           LDT

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

           C

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

           LDC

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

           WORK

                     WORK is REAL array. The dimension of
                     WORK is N*MB if SIDE = 'L', or  M*MB if SIDE = 'R'.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

   subroutine zgelqt (integer M, integer N, integer MB, complex*16, dimension( lda, * ) A,
       integer LDA, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( * )
       WORK, integer INFO)
       ZGELQT

       Purpose:

            ZGELQT computes a blocked LQ factorization of a complex M-by-N matrix A
            using the compact WY representation of Q.

       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.

           MB

                     MB is INTEGER
                     The block size to be used in the blocked QR.  MIN(M,N) >= MB >= 1.

           A

                     A is COMPLEX*16 array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, the elements on and below the diagonal of the array
                     contain the M-by-MIN(M,N) lower trapezoidal matrix L (L is
                     lower triangular if M <= N); the elements above the diagonal
                     are the rows of V.

           LDA

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

           T

                     T is COMPLEX*16 array, dimension (LDT,MIN(M,N))
                     The upper triangular block reflectors stored in compact form
                     as a sequence of upper triangular blocks.  See below
                     for further details.

           LDT

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

           WORK

                     WORK is COMPLEX*16 array, dimension (MB*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

       Further Details:

             The matrix V stores the elementary reflectors H(i) in the i-th row
             above the diagonal. For example, if M=5 and N=3, the matrix V is

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

             where the vi's represent the vectors which define H(i), which are returned
             in the matrix A.  The 1's along the diagonal of V are not stored in A.
             Let K=MIN(M,N).  The number of blocks is B = ceiling(K/MB), where each
             block is of order MB except for the last block, which is of order
             IB = K - (B-1)*MB.  For each of the B blocks, a upper triangular block
             reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
             for the last block) T's are stored in the MB-by-K matrix T as

                          T = (T1 T2 ... TB).

   recursive subroutine zgelqt3 (integer M, integer N, complex*16, dimension( lda, * ) A, integer
       LDA, complex*16, dimension( ldt, * ) T, integer LDT, integer INFO)
       ZGELQT3 recursively computes a LQ factorization of a general real or complex matrix using
       the compact WY representation of Q.

       Purpose:

            DGELQT3 recursively computes a LQ factorization of a complex M-by-N
            matrix A, using the compact WY representation of Q.

            Based on the algorithm of Elmroth and Gustavson,
            IBM J. Res. Develop. Vol 44 No. 4 July 2000.

       Parameters:
           M

                     M is INTEGER
                     The number of rows of the matrix A.  M =< N.

           N

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

           A

                     A is COMPLEX*16 array, dimension (LDA,N)
                     On entry, the real M-by-N matrix A.  On exit, the elements on and
                     below the diagonal contain the N-by-N lower triangular matrix L; the
                     elements above the diagonal are the rows of V.  See below for
                     further details.

           LDA

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

           T

                     T is COMPLEX*16 array, dimension (LDT,N)
                     The N-by-N upper triangular factor of the block reflector.
                     The elements on and above the diagonal contain the block
                     reflector T; the elements below the diagonal are not used.
                     See below for further details.

           LDT

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

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

       Further Details:

             The matrix V stores the elementary reflectors H(i) in the i-th row
             above the diagonal. For example, if M=5 and N=3, the matrix V is

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

             where the vi's represent the vectors which define H(i), which are returned
             in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
             block reflector H is then given by

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

             where V**T is the transpose of V.

             For details of the algorithm, see Elmroth and Gustavson (cited above).

   subroutine zgemlqt (character SIDE, character TRANS, integer M, integer N, integer K, integer
       MB, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldt, * ) T,
       integer LDT, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * )
       WORK, integer INFO)
       ZGEMLQT

       Purpose:

            ZGEMLQT overwrites the general real M-by-N matrix C with

                            SIDE = 'L'     SIDE = 'R'
            TRANS = 'N':      Q C            C Q
            TRANS = 'C':   Q**H C            C Q**H

            where Q is a complex orthogonal matrix defined as the product of K
            elementary reflectors:

                  Q = H(1) H(2) . . . H(K) = I - V T V**H

            generated using the compact WY representation as returned by ZGELQT.

            Q is of order M if SIDE = 'L' and of order N  if SIDE = 'R'.

       Parameters:
           SIDE

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

           TRANS

                     TRANS is CHARACTER*1
                     = 'N':  No transpose, apply Q;
                     = 'C':  Transpose, apply Q**H.

           M

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

           N

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

           K

                     K is INTEGER
                     The number of elementary reflectors whose product defines
                     the matrix Q.
                     If SIDE = 'L', M >= K >= 0;
                     if SIDE = 'R', N >= K >= 0.

           MB

                     MB is INTEGER
                     The block size used for the storage of T.  K >= MB >= 1.
                     This must be the same value of MB used to generate T
                     in DGELQT.

           V

                     V is COMPLEX*16 array, dimension
                                          (LDV,M) if SIDE = 'L',
                                          (LDV,N) if SIDE = 'R'
                     The i-th row must contain the vector which defines the
                     elementary reflector H(i), for i = 1,2,...,k, as returned by
                     DGELQT in the first K rows of its array argument A.

           LDV

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

           T

                     T is COMPLEX*16 array, dimension (LDT,K)
                     The upper triangular factors of the block reflectors
                     as returned by DGELQT, stored as a MB-by-K matrix.

           LDT

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

           C

                     C is COMPLEX*16 array, dimension (LDC,N)
                     On entry, the M-by-N matrix C.
                     On exit, C is overwritten by Q C, Q**H C, C Q**H or C Q.

           LDC

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

           WORK

                     WORK is COMPLEX*16 array. The dimension of
                     WORK is N*MB if SIDE = 'L', or  M*MB if SIDE = 'R'.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2017

Author

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