ZTGSY2(3) LAPACK routine of NEC Numeric Library Collection ZTGSY2(3)
NAME
ZTGSY2
SYNOPSIS
SUBROUTINE ZTGSY2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD,
E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO)
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.
ARGUMENTS
TRANS (input)
TRANS is CHARACTER*1
= 'N', solve the generalized Sylvester equation (1).
= 'T': solve the 'transposed' system (3).
IJOB (input)
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 (input)
M is INTEGER
On entry, M specifies the order of A and D, and the row
dimension of C, F, R and L.
N (input)
N is INTEGER
On entry, N specifies the order of B and E, and the column
dimension of C, F, R and L.
A (input)
A is COMPLEX*16 array, dimension (LDA, M)
On entry, A contains an upper triangular matrix.
LDA (input)
LDA is INTEGER
The leading dimension of the matrix A. LDA >= max(1, M).
B (input)
B is COMPLEX*16 array, dimension (LDB, N)
On entry, B contains an upper triangular matrix.
LDB (input)
LDB is INTEGER
The leading dimension of the matrix B. LDB >= max(1, N).
C (input/output)
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 (input)
LDC is INTEGER
The leading dimension of the matrix C. LDC >= max(1, M).
D (input)
D is COMPLEX*16 array, dimension (LDD, M)
On entry, D contains an upper triangular matrix.
LDD (input)
LDD is INTEGER
The leading dimension of the matrix D. LDD >= max(1, M).
E (input)
E is COMPLEX*16 array, dimension (LDE, N)
On entry, E contains an upper triangular matrix.
LDE (input)
LDE is INTEGER
The leading dimension of the matrix E. LDE >= max(1, N).
F (input/output)
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 (input)
LDF is INTEGER
The leading dimension of the matrix F. LDF >= max(1, M).
SCALE (output)
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 (input/output)
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 (input/output)
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 (output)
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.
LAPACK routine 31 October 2017 ZTGSY2(3)