realGEauxiliary(3)

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.

Definition at line 116 of file sgesc2.f.

  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.

Definition at line 113 of file sgetc2.f.

  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

Definition at line 116 of file slange.f.

  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

Definition at line 144 of file slaqge.f.

  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.

Definition at line 223 of file stgex2.f.

Author

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