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NAME

       realGEauxiliary

SYNOPSIS

   Functions
       subroutine sgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE)
           SGESC2 solves a system of linear equations using the LU factorization with complete
           pivoting computed by sgetc2.
       subroutine sgetc2 (N, A, LDA, IPIV, JPIV, INFO)
           SGETC2 computes the LU factorization with complete pivoting of the general n-by-n
           matrix.
       real function slange (NORM, M, N, A, LDA, WORK)
           SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
           absolute value of any element of a general rectangular matrix.
       subroutine slaqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
           SLAQGE scales a general rectangular matrix, using row and column scaling factors
           computed by sgeequ.
       subroutine stgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, N1, N2, WORK,
           LWORK, INFO)
           STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an
           orthogonal equivalence transformation.

Detailed Description

       This is the group of real auxiliary functions for GE matrices

Function Documentation

   subroutine sgesc2 (integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * )
       RHS, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, real SCALE)
       SGESC2 solves a system of linear equations using the LU factorization with complete
       pivoting computed by sgetc2.

       Purpose:

            SGESC2 solves a system of linear equations

                      A * X = scale* RHS

            with a general N-by-N matrix A using the LU factorization with
            complete pivoting computed by SGETC2.

       Parameters:
           N

                     N is INTEGER
                     The order of the matrix A.

           A

                     A is REAL array, dimension (LDA,N)
                     On entry, the  LU part of the factorization of the n-by-n
                     matrix A computed by SGETC2:  A = P * L * U * Q

           LDA

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

           RHS

                     RHS is REAL array, dimension (N).
                     On entry, the right hand side vector b.
                     On exit, the solution vector X.

           IPIV

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

           JPIV

                     JPIV is INTEGER array, dimension (N).
                     The pivot indices; for 1 <= j <= N, column j of the
                     matrix has been interchanged with column JPIV(j).

           SCALE

                     SCALE is REAL
                      On exit, SCALE contains the scale factor. SCALE is chosen
                      0 <= SCALE <= 1 to prevent owerflow in the solution.

       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.

   subroutine sgetc2 (integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( *
       ) IPIV, integer, dimension( * ) JPIV, integer INFO)
       SGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.

       Purpose:

            SGETC2 computes an LU factorization with complete pivoting of the
            n-by-n matrix A. The factorization has the form A = P * L * U * Q,
            where P and Q are permutation matrices, L is lower triangular with
            unit diagonal elements and U is upper triangular.

            This is the Level 2 BLAS algorithm.

       Parameters:
           N

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

           A

                     A is REAL array, dimension (LDA, N)
                     On entry, the n-by-n matrix A to be factored.
                     On exit, the factors L and U from the factorization
                     A = P*L*U*Q; the unit diagonal elements of L are not stored.
                     If U(k, k) appears to be less than SMIN, U(k, k) is given the
                     value of SMIN, i.e., giving a nonsingular perturbed system.

           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; for 1 <= i <= N, row i of the
                     matrix has been interchanged with row IPIV(i).

           JPIV

                     JPIV is INTEGER array, dimension(N).
                     The pivot indices; for 1 <= j <= N, column j of the
                     matrix has been interchanged with column JPIV(j).

           INFO

                     INFO is INTEGER
                      = 0: successful exit
                      > 0: if INFO = k, U(k, k) is likely to produce owerflow if
                           we try to solve for x in Ax = b. So U is perturbed to
                           avoid the overflow.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2016

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

   real function slange (character NORM, integer M, integer N, real, dimension( lda, * ) A,
       integer LDA, real, dimension( * ) WORK)
       SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
       absolute value of any element of a general rectangular matrix.

       Purpose:

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

       Returns:
           SLANGE

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

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

       Parameters:
           NORM

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

           M

                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.  When M = 0,
                     SLANGE is set to zero.

           N

                     N is INTEGER
                     The number of columns of the matrix A.  N >= 0.  When N = 0,
                     SLANGE is set to zero.

           A

                     A is REAL array, dimension (LDA,N)
                     The m by n matrix A.

           LDA

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

           WORK

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine slaqge (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real,
       dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, character
       EQUED)
       SLAQGE scales a general rectangular matrix, using row and column scaling factors computed
       by sgeequ.

       Purpose:

            SLAQGE equilibrates a general M by N matrix A using the row and
            column scaling factors in the vectors R and C.

       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 REAL array, dimension (LDA,N)
                     On entry, the M by N matrix A.
                     On exit, the equilibrated matrix.  See EQUED for the form of
                     the equilibrated matrix.

           LDA

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

           R

                     R is REAL array, dimension (M)
                     The row scale factors for A.

           C

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

           ROWCND

                     ROWCND is REAL
                     Ratio of the smallest R(i) to the largest R(i).

           COLCND

                     COLCND is REAL
                     Ratio of the smallest C(i) to the largest C(i).

           AMAX

                     AMAX is REAL
                     Absolute value of largest matrix entry.

           EQUED

                     EQUED is CHARACTER*1
                     Specifies the form of equilibration that was done.
                     = '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).

       Internal Parameters:

             THRESH is a threshold value used to decide if row or column scaling
             should be done based on the ratio of the row or column scaling
             factors.  If ROWCND < THRESH, row scaling is done, and if
             COLCND < THRESH, column scaling is done.

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine stgex2 (logical WANTQ, logical WANTZ, integer N, real, dimension( lda, * ) A,
       integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldq, * ) Q,
       integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, integer J1, integer N1, integer N2,
       real, dimension( * ) WORK, integer LWORK, integer INFO)
       STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an
       orthogonal equivalence transformation.

       Purpose:

            STGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
            of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
            (A, B) by an orthogonal equivalence transformation.

            (A, B) must be in generalized real Schur canonical form (as returned
            by SGGES), 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 REAL array, dimension (LDA,N)
                     On entry, the matrix A in the pair (A, B).
                     On exit, the updated matrix A.

           LDA

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

           B

                     B is REAL array, dimension (LDB,N)
                     On entry, the matrix B in the pair (A, B).
                     On exit, the updated matrix B.

           LDB

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

           Q

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

           LDQ

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

           Z

                     Z is REAL array, dimension (LDZ,N)
                     On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
                     On exit, the updated matrix Z.
                     Not referenced if WANTZ = .FALSE..

           LDZ

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

           J1

                     J1 is INTEGER
                     The index to the first block (A11, B11). 1 <= J1 <= N.

           N1

                     N1 is INTEGER
                     The order of the first block (A11, B11). N1 = 0, 1 or 2.

           N2

                     N2 is INTEGER
                     The order of the second block (A22, B22). N2 = 0, 1 or 2.

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)).

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.
                     LWORK >=  MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 )

           INFO

                     INFO is INTEGER
                       =0: Successful exit
                       >0: If INFO = 1, the transformed matrix (A, B) would be
                           too far from generalized Schur form; the blocks are
                           not swapped and (A, B) and (Q, Z) are unchanged.
                           The problem of swapping is too ill-conditioned.
                       <0: If INFO = -16: LWORK is too small. Appropriate value
                           for LWORK is returned in WORK(1).

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

       Further Details:
           In the current code both weak and strong stability tests are performed. The user can
           omit the strong stability test by changing the internal logical parameter WANDS to
           .FALSE.. See ref. [2] for details.

       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.

             [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
                 Eigenvalues of a Regular Matrix Pair (A, B) and Condition
                 Estimation: Theory, Algorithms and Software,
                 Report UMINF - 94.04, Department of Computing Science, Umea
                 University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
                 Note 87. To appear in Numerical Algorithms, 1996.

Author

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