ctgsy2.f - Man Page

subroutine ctgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO)
CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).  

Purpose:

 CTGSY2 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. A, B, D and E are upper triangular
 (i.e., (A,D) and (B,E) in generalized Schur form).

 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
 Zx = scale * b, where Z is defined as

        Z = [ kron(In, A)  -kron(B**H, Im) ]             (2)
            [ kron(In, D)  -kron(E**H, Im) ],

 Ik is the identity matrix of size k and X**H is the transpose of X.
 kron(X, Y) is the Kronecker product between the matrices X and Y.

 If TRANS = 'C', y in the conjugate transposed system Z**H*y = scale*b
 is solved for, which is equivalent to solve for R and L in

             A**H * R  + D**H * L   = scale * C           (3)
             R  * B**H + L  * E**H  = scale * -F

 This case is used to compute an estimate of Dif[(A, D), (B, E)] =
 = sigma_min(Z) using reverse communicaton with CLACON.

 CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL
 of an upper bound on the separation between to matrix pairs. Then
 the input (A, D), (B, E) are sub-pencils of two matrix pairs in
 CTGSYL.

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

LDA

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

B

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

LDB

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

C

          C is COMPLEX 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 COMPLEX 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 COMPLEX 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 COMPLEX 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 CTGSYL, 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 CTGSY2 is called by
          CTGSYL.

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 CTGSY2 is called by
          CTGSYL.

INFO

          INFO is INTEGER
          On exit, if INFO is set to
            =0: Successful exit
            <0: If INFO = -i, input argument number i is illegal.
            >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.

Definition at line 261 of file ctgsy2.f.