--- rpl/lapack/lapack/ztgsy2.f 2010/12/21 13:53:57 1.7
+++ rpl/lapack/lapack/ztgsy2.f 2023/08/07 08:39:41 1.21
@@ -1,11 +1,265 @@
+*> \brief \b ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZTGSY2 + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
+* LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
+* INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER TRANS
+* INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
+* DOUBLE PRECISION RDSCAL, RDSUM, SCALE
+* ..
+* .. Array Arguments ..
+* COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
+* $ D( LDD, * ), E( LDE, * ), F( LDF, * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> 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 communication 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.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] TRANS
+*> \verbatim
+*> TRANS is CHARACTER*1
+*> = 'N': solve the generalized Sylvester equation (1).
+*> = 'T': solve the 'transposed' system (3).
+*> \endverbatim
+*>
+*> \param[in] IJOB
+*> \verbatim
+*> 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'.
+*> \endverbatim
+*>
+*> \param[in] M
+*> \verbatim
+*> M is INTEGER
+*> On entry, M specifies the order of A and D, and the row
+*> dimension of C, F, R and L.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> On entry, N specifies the order of B and E, and the column
+*> dimension of C, F, R and L.
+*> \endverbatim
+*>
+*> \param[in] A
+*> \verbatim
+*> A is COMPLEX*16 array, dimension (LDA, M)
+*> On entry, A contains an upper triangular matrix.
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of the matrix A. LDA >= max(1, M).
+*> \endverbatim
+*>
+*> \param[in] B
+*> \verbatim
+*> B is COMPLEX*16 array, dimension (LDB, N)
+*> On entry, B contains an upper triangular matrix.
+*> \endverbatim
+*>
+*> \param[in] LDB
+*> \verbatim
+*> LDB is INTEGER
+*> The leading dimension of the matrix B. LDB >= max(1, N).
+*> \endverbatim
+*>
+*> \param[in,out] C
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[in] LDC
+*> \verbatim
+*> LDC is INTEGER
+*> The leading dimension of the matrix C. LDC >= max(1, M).
+*> \endverbatim
+*>
+*> \param[in] D
+*> \verbatim
+*> D is COMPLEX*16 array, dimension (LDD, M)
+*> On entry, D contains an upper triangular matrix.
+*> \endverbatim
+*>
+*> \param[in] LDD
+*> \verbatim
+*> LDD is INTEGER
+*> The leading dimension of the matrix D. LDD >= max(1, M).
+*> \endverbatim
+*>
+*> \param[in] E
+*> \verbatim
+*> E is COMPLEX*16 array, dimension (LDE, N)
+*> On entry, E contains an upper triangular matrix.
+*> \endverbatim
+*>
+*> \param[in] LDE
+*> \verbatim
+*> LDE is INTEGER
+*> The leading dimension of the matrix E. LDE >= max(1, N).
+*> \endverbatim
+*>
+*> \param[in,out] F
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[in] LDF
+*> \verbatim
+*> LDF is INTEGER
+*> The leading dimension of the matrix F. LDF >= max(1, M).
+*> \endverbatim
+*>
+*> \param[out] SCALE
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[in,out] RDSUM
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[in,out] RDSCAL
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> 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.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \ingroup complex16SYauxiliary
+*
+*> \par Contributors:
+* ==================
+*>
+*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
+*> Umea University, S-901 87 Umea, Sweden.
+*
+* =====================================================================
SUBROUTINE ZTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
$ LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
$ INFO )
*
-* -- LAPACK auxiliary routine (version 3.2) --
+* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2006
*
* .. Scalar Arguments ..
CHARACTER TRANS
@@ -17,153 +271,6 @@
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
* ..
*
-* 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', Im) ] (2)
-* [ kron(In, D) -kron(E', Im) ],
-*
-* Ik is the identity matrix of size k and X' 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'y = scale*b
-* is solved for, which is equivalent to solve for R and L in
-*
-* A' * R + D' * L = scale * C (3)
-* R * B' + L * E' = 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) CHARACTER*1
-* = 'N', solve the generalized Sylvester equation (1).
-* = 'T': solve the 'transposed' system (3).
-*
-* IJOB (input) 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) INTEGER
-* On entry, M specifies the order of A and D, and the row
-* dimension of C, F, R and L.
-*
-* N (input) INTEGER
-* On entry, N specifies the order of B and E, and the column
-* dimension of C, F, R and L.
-*
-* A (input) COMPLEX*16 array, dimension (LDA, M)
-* On entry, A contains an upper triangular matrix.
-*
-* LDA (input) INTEGER
-* The leading dimension of the matrix A. LDA >= max(1, M).
-*
-* B (input) COMPLEX*16 array, dimension (LDB, N)
-* On entry, B contains an upper triangular matrix.
-*
-* LDB (input) INTEGER
-* The leading dimension of the matrix B. LDB >= max(1, N).
-*
-* C (input/output) 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) INTEGER
-* The leading dimension of the matrix C. LDC >= max(1, M).
-*
-* D (input) COMPLEX*16 array, dimension (LDD, M)
-* On entry, D contains an upper triangular matrix.
-*
-* LDD (input) INTEGER
-* The leading dimension of the matrix D. LDD >= max(1, M).
-*
-* E (input) COMPLEX*16 array, dimension (LDE, N)
-* On entry, E contains an upper triangular matrix.
-*
-* LDE (input) INTEGER
-* The leading dimension of the matrix E. LDE >= max(1, N).
-*
-* F (input/output) 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) INTEGER
-* The leading dimension of the matrix F. LDF >= max(1, M).
-*
-* SCALE (output) 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) 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) 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) 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.
-*
-* Further Details
-* ===============
-*
-* Based on contributions by
-* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
-* Umea University, S-901 87 Umea, Sweden.
-*
* =====================================================================
*
* .. Parameters ..
@@ -211,7 +318,7 @@
ELSE IF( N.LE.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
- INFO = -5
+ INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
@@ -298,7 +405,7 @@
ELSE
*
* Solve transposed (I, J) - system:
-* A(I, I)' * R(I, J) + D(I, I)' * L(J, J) = C(I, J)
+* A(I, I)**H * R(I, J) + D(I, I)**H * L(J, J) = C(I, J)
* R(I, I) * B(J, J) + L(I, J) * E(J, J) = -F(I, J)
* for I = 1, 2, ..., M, J = N, N - 1, ..., 1
*
@@ -307,7 +414,7 @@
DO 80 I = 1, M
DO 70 J = N, 1, -1
*
-* Build 2 by 2 system Z'
+* Build 2 by 2 system Z**H
*
Z( 1, 1 ) = DCONJG( A( I, I ) )
Z( 2, 1 ) = -DCONJG( B( J, J ) )
@@ -320,7 +427,7 @@
RHS( 1 ) = C( I, J )
RHS( 2 ) = F( I, J )
*
-* Solve Z' * x = RHS
+* Solve Z**H * x = RHS
*
CALL ZGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )