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ztgsy2.f(3) LAPACK ztgsy2.f(3)

ztgsy2.f -


subroutine ztgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO)
 
ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
Purpose:
 ZTGSY2 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 conjuguate 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 ZLACON.
ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL 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 ZTGSYL.
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 COMPLEX*16 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*16 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*16 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*16 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*16 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*16 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 ZTGSYL, 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 ZTGSY2 is called by
          ZTGSYL.
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 ZTGSY2 is called by
          ZTGSYL.
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:
September 2012
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.
Definition at line 258 of file ztgsy2.f.

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