362 lines
12 KiB
FortranFixed
362 lines
12 KiB
FortranFixed
SUBROUTINE ZTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
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$ LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
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$ INFO )
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*
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* -- LAPACK auxiliary routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* November 2006
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*
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* .. Scalar Arguments ..
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CHARACTER TRANS
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INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
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DOUBLE PRECISION RDSCAL, RDSUM, SCALE
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* ..
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* .. Array Arguments ..
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COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
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$ D( LDD, * ), E( LDE, * ), F( LDF, * )
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* ..
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*
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* Purpose
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* =======
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*
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* ZTGSY2 solves the generalized Sylvester equation
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*
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* A * R - L * B = scale * C (1)
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* D * R - L * E = scale * F
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*
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* using Level 1 and 2 BLAS, where R and L are unknown M-by-N matrices,
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* (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
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* N-by-N and M-by-N, respectively. A, B, D and E are upper triangular
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* (i.e., (A,D) and (B,E) in generalized Schur form).
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*
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* The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
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* scaling factor chosen to avoid overflow.
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*
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* In matrix notation solving equation (1) corresponds to solve
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* Zx = scale * b, where Z is defined as
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*
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* Z = [ kron(In, A) -kron(B', Im) ] (2)
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* [ kron(In, D) -kron(E', Im) ],
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*
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* Ik is the identity matrix of size k and X' is the transpose of X.
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* kron(X, Y) is the Kronecker product between the matrices X and Y.
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*
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* If TRANS = 'C', y in the conjugate transposed system Z'y = scale*b
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* is solved for, which is equivalent to solve for R and L in
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*
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* A' * R + D' * L = scale * C (3)
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* R * B' + L * E' = scale * -F
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*
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* This case is used to compute an estimate of Dif[(A, D), (B, E)] =
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* = sigma_min(Z) using reverse communicaton with ZLACON.
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*
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* ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL
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* of an upper bound on the separation between to matrix pairs. Then
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* the input (A, D), (B, E) are sub-pencils of two matrix pairs in
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* ZTGSYL.
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*
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* Arguments
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* =========
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*
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* TRANS (input) CHARACTER*1
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* = 'N', solve the generalized Sylvester equation (1).
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* = 'T': solve the 'transposed' system (3).
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*
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* IJOB (input) INTEGER
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* Specifies what kind of functionality to be performed.
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* =0: solve (1) only.
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* =1: A contribution from this subsystem to a Frobenius
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* norm-based estimate of the separation between two matrix
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* pairs is computed. (look ahead strategy is used).
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* =2: A contribution from this subsystem to a Frobenius
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* norm-based estimate of the separation between two matrix
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* pairs is computed. (DGECON on sub-systems is used.)
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* Not referenced if TRANS = 'T'.
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*
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* M (input) INTEGER
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* On entry, M specifies the order of A and D, and the row
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* dimension of C, F, R and L.
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*
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* N (input) INTEGER
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* On entry, N specifies the order of B and E, and the column
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* dimension of C, F, R and L.
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*
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* A (input) COMPLEX*16 array, dimension (LDA, M)
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* On entry, A contains an upper triangular matrix.
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*
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* LDA (input) INTEGER
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* The leading dimension of the matrix A. LDA >= max(1, M).
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*
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* B (input) COMPLEX*16 array, dimension (LDB, N)
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* On entry, B contains an upper triangular matrix.
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*
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* LDB (input) INTEGER
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* The leading dimension of the matrix B. LDB >= max(1, N).
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*
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* C (input/output) COMPLEX*16 array, dimension (LDC, N)
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* On entry, C contains the right-hand-side of the first matrix
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* equation in (1).
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* On exit, if IJOB = 0, C has been overwritten by the solution
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* R.
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*
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* LDC (input) INTEGER
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* The leading dimension of the matrix C. LDC >= max(1, M).
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*
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* D (input) COMPLEX*16 array, dimension (LDD, M)
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* On entry, D contains an upper triangular matrix.
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*
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* LDD (input) INTEGER
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* The leading dimension of the matrix D. LDD >= max(1, M).
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*
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* E (input) COMPLEX*16 array, dimension (LDE, N)
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* On entry, E contains an upper triangular matrix.
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*
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* LDE (input) INTEGER
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* The leading dimension of the matrix E. LDE >= max(1, N).
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*
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* F (input/output) COMPLEX*16 array, dimension (LDF, N)
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* On entry, F contains the right-hand-side of the second matrix
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* equation in (1).
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* On exit, if IJOB = 0, F has been overwritten by the solution
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* L.
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*
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* LDF (input) INTEGER
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* The leading dimension of the matrix F. LDF >= max(1, M).
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*
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* SCALE (output) DOUBLE PRECISION
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* On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
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* R and L (C and F on entry) will hold the solutions to a
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* slightly perturbed system but the input matrices A, B, D and
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* E have not been changed. If SCALE = 0, R and L will hold the
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* solutions to the homogeneous system with C = F = 0.
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* Normally, SCALE = 1.
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*
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* RDSUM (input/output) DOUBLE PRECISION
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* On entry, the sum of squares of computed contributions to
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* the Dif-estimate under computation by ZTGSYL, where the
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* scaling factor RDSCAL (see below) has been factored out.
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* On exit, the corresponding sum of squares updated with the
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* contributions from the current sub-system.
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* If TRANS = 'T' RDSUM is not touched.
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* NOTE: RDSUM only makes sense when ZTGSY2 is called by
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* ZTGSYL.
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*
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* RDSCAL (input/output) DOUBLE PRECISION
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* On entry, scaling factor used to prevent overflow in RDSUM.
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* On exit, RDSCAL is updated w.r.t. the current contributions
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* in RDSUM.
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* If TRANS = 'T', RDSCAL is not touched.
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* NOTE: RDSCAL only makes sense when ZTGSY2 is called by
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* ZTGSYL.
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*
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* INFO (output) INTEGER
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* On exit, if INFO is set to
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* =0: Successful exit
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* <0: If INFO = -i, input argument number i is illegal.
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* >0: The matrix pairs (A, D) and (B, E) have common or very
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* close eigenvalues.
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*
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* Further Details
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* ===============
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*
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* Based on contributions by
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* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
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* Umea University, S-901 87 Umea, Sweden.
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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INTEGER LDZ
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, LDZ = 2 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOTRAN
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INTEGER I, IERR, J, K
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DOUBLE PRECISION SCALOC
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COMPLEX*16 ALPHA
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* ..
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* .. Local Arrays ..
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INTEGER IPIV( LDZ ), JPIV( LDZ )
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COMPLEX*16 RHS( LDZ ), Z( LDZ, LDZ )
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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* ..
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* .. External Subroutines ..
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EXTERNAL XERBLA, ZAXPY, ZGESC2, ZGETC2, ZLATDF, ZSCAL
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC DCMPLX, DCONJG, MAX
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* ..
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* .. Executable Statements ..
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*
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* Decode and test input parameters
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*
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INFO = 0
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IERR = 0
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NOTRAN = LSAME( TRANS, 'N' )
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IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'C' ) ) THEN
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INFO = -1
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ELSE IF( NOTRAN ) THEN
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IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.2 ) ) THEN
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INFO = -2
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END IF
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END IF
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IF( INFO.EQ.0 ) THEN
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IF( M.LE.0 ) THEN
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INFO = -3
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ELSE IF( N.LE.0 ) THEN
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INFO = -4
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -5
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -8
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ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
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INFO = -10
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ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
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INFO = -12
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ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
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INFO = -14
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ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
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INFO = -16
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END IF
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZTGSY2', -INFO )
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RETURN
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END IF
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*
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IF( NOTRAN ) THEN
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*
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* Solve (I, J) - system
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* A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
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* D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
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* for I = M, M - 1, ..., 1; J = 1, 2, ..., N
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*
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SCALE = ONE
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SCALOC = ONE
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DO 30 J = 1, N
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DO 20 I = M, 1, -1
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*
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* Build 2 by 2 system
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*
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Z( 1, 1 ) = A( I, I )
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Z( 2, 1 ) = D( I, I )
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Z( 1, 2 ) = -B( J, J )
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Z( 2, 2 ) = -E( J, J )
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*
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* Set up right hand side(s)
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*
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RHS( 1 ) = C( I, J )
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RHS( 2 ) = F( I, J )
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*
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* Solve Z * x = RHS
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*
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CALL ZGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR )
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IF( IERR.GT.0 )
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$ INFO = IERR
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IF( IJOB.EQ.0 ) THEN
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CALL ZGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
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IF( SCALOC.NE.ONE ) THEN
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DO 10 K = 1, N
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CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ),
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$ C( 1, K ), 1 )
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CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ),
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$ F( 1, K ), 1 )
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10 CONTINUE
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SCALE = SCALE*SCALOC
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END IF
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ELSE
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CALL ZLATDF( IJOB, LDZ, Z, LDZ, RHS, RDSUM, RDSCAL,
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$ IPIV, JPIV )
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END IF
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*
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* Unpack solution vector(s)
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*
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C( I, J ) = RHS( 1 )
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F( I, J ) = RHS( 2 )
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*
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* Substitute R(I, J) and L(I, J) into remaining equation.
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*
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IF( I.GT.1 ) THEN
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ALPHA = -RHS( 1 )
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CALL ZAXPY( I-1, ALPHA, A( 1, I ), 1, C( 1, J ), 1 )
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CALL ZAXPY( I-1, ALPHA, D( 1, I ), 1, F( 1, J ), 1 )
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END IF
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IF( J.LT.N ) THEN
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CALL ZAXPY( N-J, RHS( 2 ), B( J, J+1 ), LDB,
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$ C( I, J+1 ), LDC )
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CALL ZAXPY( N-J, RHS( 2 ), E( J, J+1 ), LDE,
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$ F( I, J+1 ), LDF )
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END IF
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*
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20 CONTINUE
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30 CONTINUE
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ELSE
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*
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* Solve transposed (I, J) - system:
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* A(I, I)' * R(I, J) + D(I, I)' * L(J, J) = C(I, J)
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* R(I, I) * B(J, J) + L(I, J) * E(J, J) = -F(I, J)
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* for I = 1, 2, ..., M, J = N, N - 1, ..., 1
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*
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SCALE = ONE
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SCALOC = ONE
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DO 80 I = 1, M
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DO 70 J = N, 1, -1
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*
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* Build 2 by 2 system Z'
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*
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Z( 1, 1 ) = DCONJG( A( I, I ) )
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Z( 2, 1 ) = -DCONJG( B( J, J ) )
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Z( 1, 2 ) = DCONJG( D( I, I ) )
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Z( 2, 2 ) = -DCONJG( E( J, J ) )
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*
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*
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* Set up right hand side(s)
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*
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RHS( 1 ) = C( I, J )
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RHS( 2 ) = F( I, J )
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*
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* Solve Z' * x = RHS
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*
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CALL ZGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR )
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IF( IERR.GT.0 )
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$ INFO = IERR
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CALL ZGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
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IF( SCALOC.NE.ONE ) THEN
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DO 40 K = 1, N
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CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), C( 1, K ),
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$ 1 )
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CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), F( 1, K ),
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$ 1 )
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40 CONTINUE
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SCALE = SCALE*SCALOC
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END IF
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*
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* Unpack solution vector(s)
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*
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C( I, J ) = RHS( 1 )
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F( I, J ) = RHS( 2 )
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*
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* Substitute R(I, J) and L(I, J) into remaining equation.
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*
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DO 50 K = 1, J - 1
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F( I, K ) = F( I, K ) + RHS( 1 )*DCONJG( B( K, J ) ) +
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$ RHS( 2 )*DCONJG( E( K, J ) )
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50 CONTINUE
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DO 60 K = I + 1, M
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C( K, J ) = C( K, J ) - DCONJG( A( I, K ) )*RHS( 1 ) -
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$ DCONJG( D( I, K ) )*RHS( 2 )
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60 CONTINUE
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*
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70 CONTINUE
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80 CONTINUE
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END IF
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RETURN
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*
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* End of ZTGSY2
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*
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END
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