Provided by: liblapack-doc_3.8.0-2_all bug

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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