doubleSYauxiliary(3)

NAME
doubleSYauxiliary - double

SYNOPSIS

  Functions
double precision function dlansy (norm, uplo, n, a, lda, work)
DLANSY 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 dlaqsy (uplo, n, a, lda, s, scond, amax, equed)
DLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
subroutine dlasy2 (ltranl, ltranr, isgn, n1, n2, tl, ldtl, tr, ldtr, b, ldb, scale, x, ldx, xnorm, info)
DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
subroutine dsyswapr (uplo, n, a, lda, i1, i2)
DSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
subroutine dtgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, iwork, pq, info)
DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Detailed Description

This is the group of double auxiliary functions for SY matrices

Function Documentation

  double precision function dlansy (character norm, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) work)

DLANSY 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:

     DLANSY  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
    DLANSY

        DLANSY = ( 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 DLANSY 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, DLANSY is
              set to zero.

    A

              A is DOUBLE PRECISION 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 DOUBLE PRECISION 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 dlansy.f.

  subroutine dlaqsy (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) s, double precision scond, double precision amax, character equed)

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

Purpose:

     DLAQSY 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)
              The scale factors for A.

    SCOND

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

    AMAX

              AMAX is DOUBLE PRECISION
              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 dlaqsy.f.

  subroutine dlasy2 (logical ltranl, logical ltranr, integer isgn, integer n1, integer n2, double precision, dimension( ldtl, * ) tl, integer ldtl, double precision, dimension( ldtr, * ) tr, integer ldtr, double precision, dimension( ldb, * ) b, integer ldb, double precision scale, double precision, dimension( ldx, * ) x, integer ldx, double precision xnorm, integer info)

DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.

Purpose:

     DLASY2 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
              On exit, SCALE contains the scale factor. SCALE is chosen
              less than or equal to 1 to prevent the solution overflowing.

    X

              X is DOUBLE PRECISION 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 DOUBLE PRECISION
              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 dlasy2.f.

  subroutine dsyswapr (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, integer i1, integer i2)

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

Purpose:

     DSYSWAPR 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 DOUBLE PRECISION 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 dsyswapr.f.

  subroutine dtgsy2 (character trans, integer ijob, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( ldd, * ) d, integer ldd, double precision, dimension( lde, * ) e, integer lde, double precision, dimension( ldf, * ) f, integer ldf, double precision scale, double precision rdsum, double precision rdscal, integer, dimension( * ) iwork, integer pq, integer info)

DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:

     DTGSY2 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 DLACON.
     DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL
     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
     DTGSYL. See DTGSYL 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. (DGECON 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
              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 DOUBLE PRECISION
              On entry, the sum of squares of computed contributions to
              the Dif-estimate under computation by DTGSYL, 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 DTGSY2 is called by DTGSYL.

    RDSCAL

              RDSCAL is DOUBLE PRECISION
              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 DTGSY2 is called by
                    DTGSYL.

    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 dtgsy2.f.

Author

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