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realSYauxiliary(3) LAPACK realSYauxiliary(3)

NAME

realSYauxiliary

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.

Date:

December 2016

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.

Date:

December 2016

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.

Date:

June 2016

subroutine ssyswapr (character UPLO, integer N, real, dimension( lda, n ) 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,N)
          On entry, the NB diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by SSYTRF.
          On exit, if INFO = 0, the (symmetric) inverse of the original
          matrix.  If UPLO = 'U', the upper triangular part of the
          inverse is formed and the part of A below the diagonal is not
          referenced; if UPLO = 'L' the lower triangular part of the
          inverse is formed 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.

Date:

December 2016

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 communicaton 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.

Date:

December 2016

Contributors:

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

Author

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