realSYauxiliary(3)

NAME
realSYauxiliary - real

SYNOPSIS

  Functions
real function slansy (norm, uplo, n, a, lda, work)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
subroutine slaqsy (uplo, n, a, lda, s, scond, amax, equed)
SLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
subroutine slasy2 (ltranl, ltranr, isgn, n1, n2, tl, ldtl, tr, ldtr, b, ldb, scale, x, ldx, xnorm, info)
SLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
subroutine ssyswapr (uplo, n, a, lda, i1, i2)
SSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
subroutine stgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, iwork, pq, info)
STGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Detailed Description

This is the group of real auxiliary functions for SY matrices

Function Documentation

  real function slansy (character norm, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) work)

SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.

Purpose:

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

Returns
    SLANSY

        SLANSY = ( 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 SLANSY as described
              above.

    UPLO

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

    N

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

    A

              A is REAL array, dimension (LDA,N)
              The symmetric matrix A.  If UPLO = 'U', the leading n by n
              upper triangular part of A contains the upper triangular part
              of the matrix A, and the strictly lower triangular part of A
              is not referenced.  If UPLO = 'L', the leading n by n lower
              triangular part of A contains the lower triangular part of
              the matrix A, and the strictly upper triangular part of A is
              not referenced.

    LDA

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

    WORK

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 121 of file slansy.f.

  subroutine slaqsy (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) s, real scond, real amax, character equed)

SLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.

Purpose:

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

Parameters
    UPLO

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

    N

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

    A

              A is REAL array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = 'U', the leading
              n by n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n by n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, if EQUED = 'Y', the equilibrated matrix:
              diag(S) * A * diag(S).

    LDA

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

    S

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

    SCOND

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

    AMAX

              AMAX is REAL
              Absolute value of largest matrix entry.

    EQUED

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

Internal Parameters:

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 132 of file slaqsy.f.

  subroutine slasy2 (logical ltranl, logical ltranr, integer isgn, integer n1, integer n2, real, dimension( ldtl, * ) tl, integer ldtl, real, dimension( ldtr, * ) tr, integer ldtr, real, dimension( ldb, * ) b, integer ldb, real scale, real, dimension( ldx, * ) x, integer ldx, real xnorm, integer info)

SLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.

Purpose:

     SLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
            op(TL)*X + ISGN*X*op(TR) = SCALE*B,
     where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
     -1.  op(T) = T or T**T, where T**T denotes the transpose of T.

Parameters
    LTRANL

              LTRANL is LOGICAL
              On entry, LTRANL specifies the op(TL):
                 = .FALSE., op(TL) = TL,
                 = .TRUE., op(TL) = TL**T.

    LTRANR

              LTRANR is LOGICAL
              On entry, LTRANR specifies the op(TR):
                = .FALSE., op(TR) = TR,
                = .TRUE., op(TR) = TR**T.

    ISGN

              ISGN is INTEGER
              On entry, ISGN specifies the sign of the equation
              as described before. ISGN may only be 1 or -1.

    N1

              N1 is INTEGER
              On entry, N1 specifies the order of matrix TL.
              N1 may only be 0, 1 or 2.

    N2

              N2 is INTEGER
              On entry, N2 specifies the order of matrix TR.
              N2 may only be 0, 1 or 2.

    TL

              TL is REAL array, dimension (LDTL,2)
              On entry, TL contains an N1 by N1 matrix.

    LDTL

              LDTL is INTEGER
              The leading dimension of the matrix TL. LDTL >= max(1,N1).

    TR

              TR is REAL array, dimension (LDTR,2)
              On entry, TR contains an N2 by N2 matrix.

    LDTR

              LDTR is INTEGER
              The leading dimension of the matrix TR. LDTR >= max(1,N2).

    B

              B is REAL array, dimension (LDB,2)
              On entry, the N1 by N2 matrix B contains the right-hand
              side of the equation.

    LDB

              LDB is INTEGER
              The leading dimension of the matrix B. LDB >= max(1,N1).

    SCALE

              SCALE is REAL
              On exit, SCALE contains the scale factor. SCALE is chosen
              less than or equal to 1 to prevent the solution overflowing.

    X

              X is REAL array, dimension (LDX,2)
              On exit, X contains the N1 by N2 solution.

    LDX

              LDX is INTEGER
              The leading dimension of the matrix X. LDX >= max(1,N1).

    XNORM

              XNORM is REAL
              On exit, XNORM is the infinity-norm of the solution.

    INFO

              INFO is INTEGER
              On exit, INFO is set to
                 0: successful exit.
                 1: TL and TR have too close eigenvalues, so TL or
                    TR is perturbed to get a nonsingular equation.
              NOTE: In the interests of speed, this routine does not
                    check the inputs for errors.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 172 of file slasy2.f.

  subroutine ssyswapr (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer i1, integer i2)

SSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.

Purpose:

     SSYSWAPR applies an elementary permutation on the rows and the columns of
     a symmetric matrix.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the details of the factorization are stored
              as an upper or lower triangular matrix.
              = 'U':  Upper triangular, form is A = U*D*U**T;
              = 'L':  Lower triangular, form is A = L*D*L**T.

    N

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

    A

              A is REAL array, dimension (LDA,*)
              On entry, the N-by-N matrix A. On exit, the permuted matrix
              where the rows I1 and I2 and columns I1 and I2 are interchanged.
              If UPLO = 'U', the interchanges are applied to the upper
              triangular part and the strictly lower triangular part of A is
              not referenced; if UPLO = 'L', the interchanges are applied to
              the lower triangular part and the part of A above the diagonal
              is not referenced.

    LDA

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

    I1

              I1 is INTEGER
              Index of the first row to swap

    I2

              I2 is INTEGER
              Index of the second row to swap

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 99 of file ssyswapr.f.

  subroutine stgsy2 (character trans, integer ijob, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldc, * ) c, integer ldc, real, dimension( ldd, * ) d, integer ldd, real, dimension( lde, * ) e, integer lde, real, dimension( ldf, * ) f, integer ldf, real scale, real rdsum, real rdscal, integer, dimension( * ) iwork, integer pq, integer info)

STGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:

     STGSY2 solves the generalized Sylvester equation:
                 A * R - L * B = scale * C                (1)
                 D * R - L * E = scale * F,
     using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,
     (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
     N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)
     must be in generalized Schur canonical form, i.e. A, B are upper
     quasi triangular and D, E are upper triangular. The solution (R, L)
     overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
     chosen to avoid overflow.
     In matrix notation solving equation (1) corresponds to solve
     Z*x = scale*b, where Z is defined as
            Z = [ kron(In, A)  -kron(B**T, Im) ]             (2)
                [ kron(In, D)  -kron(E**T, Im) ],
     Ik is the identity matrix of size k and X**T is the transpose of X.
     kron(X, Y) is the Kronecker product between the matrices X and Y.
     In the process of solving (1), we solve a number of such systems
     where Dim(In), Dim(In) = 1 or 2.
     If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y,
     which is equivalent to solve for R and L in
                 A**T * R  + D**T * L   = scale * C           (3)
                 R  * B**T + L  * E**T  = scale * -F
     This case is used to compute an estimate of Dif[(A, D), (B, E)] =
     sigma_min(Z) using reverse communication with SLACON.
     STGSY2 also (IJOB >= 1) contributes to the computation in STGSYL
     of an upper bound on the separation between to matrix pairs. Then
     the input (A, D), (B, E) are sub-pencils of the matrix pair in
     STGSYL. See STGSYL for details.

Parameters
    TRANS

              TRANS is CHARACTER*1
              = 'N': solve the generalized Sylvester equation (1).
              = 'T': solve the 'transposed' system (3).

    IJOB

              IJOB is INTEGER
              Specifies what kind of functionality to be performed.
              = 0: solve (1) only.
              = 1: A contribution from this subsystem to a Frobenius
                   norm-based estimate of the separation between two matrix
                   pairs is computed. (look ahead strategy is used).
              = 2: A contribution from this subsystem to a Frobenius
                   norm-based estimate of the separation between two matrix
                   pairs is computed. (SGECON on sub-systems is used.)
              Not referenced if TRANS = 'T'.

    M

              M is INTEGER
              On entry, M specifies the order of A and D, and the row
              dimension of C, F, R and L.

    N

              N is INTEGER
              On entry, N specifies the order of B and E, and the column
              dimension of C, F, R and L.

    A

              A is REAL array, dimension (LDA, M)
              On entry, A contains an upper quasi triangular matrix.

    LDA

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

    B

              B is REAL array, dimension (LDB, N)
              On entry, B contains an upper quasi triangular matrix.

    LDB

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

    C

              C is REAL array, dimension (LDC, N)
              On entry, C contains the right-hand-side of the first matrix
              equation in (1).
              On exit, if IJOB = 0, C has been overwritten by the
              solution R.

    LDC

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

    D

              D is REAL array, dimension (LDD, M)
              On entry, D contains an upper triangular matrix.

    LDD

              LDD is INTEGER
              The leading dimension of the matrix D. LDD >= max(1, M).

    E

              E is REAL array, dimension (LDE, N)
              On entry, E contains an upper triangular matrix.

    LDE

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

    F

              F is REAL array, dimension (LDF, N)
              On entry, F contains the right-hand-side of the second matrix
              equation in (1).
              On exit, if IJOB = 0, F has been overwritten by the
              solution L.

    LDF

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

    SCALE

              SCALE is REAL
              On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
              R and L (C and F on entry) will hold the solutions to a
              slightly perturbed system but the input matrices A, B, D and
              E have not been changed. If SCALE = 0, R and L will hold the
              solutions to the homogeneous system with C = F = 0. Normally,
              SCALE = 1.

    RDSUM

              RDSUM is REAL
              On entry, the sum of squares of computed contributions to
              the Dif-estimate under computation by STGSYL, where the
              scaling factor RDSCAL (see below) has been factored out.
              On exit, the corresponding sum of squares updated with the
              contributions from the current sub-system.
              If TRANS = 'T' RDSUM is not touched.
              NOTE: RDSUM only makes sense when STGSY2 is called by STGSYL.

    RDSCAL

              RDSCAL is REAL
              On entry, scaling factor used to prevent overflow in RDSUM.
              On exit, RDSCAL is updated w.r.t. the current contributions
              in RDSUM.
              If TRANS = 'T', RDSCAL is not touched.
              NOTE: RDSCAL only makes sense when STGSY2 is called by
                    STGSYL.

    IWORK

              IWORK is INTEGER array, dimension (M+N+2)

    PQ

              PQ is INTEGER
              On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
              8-by-8) solved by this routine.

    INFO

              INFO is INTEGER
              On exit, if INFO is set to
                =0: Successful exit
                <0: If INFO = -i, the i-th argument had an illegal value.
                >0: The matrix pairs (A, D) and (B, E) have common or very
                    close eigenvalues.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

Definition at line 271 of file stgsy2.f.

Author

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