--- rpl/lapack/lapack/ztgsyl.f 2010/04/21 13:45:40 1.2
+++ rpl/lapack/lapack/ztgsyl.f 2023/08/07 08:39:41 1.18
@@ -1,11 +1,301 @@
+*> \brief \b ZTGSYL
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZTGSYL + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
+* LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
+* IWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER TRANS
+* INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
+* $ LWORK, M, N
+* DOUBLE PRECISION DIF, SCALE
+* ..
+* .. Array Arguments ..
+* INTEGER IWORK( * )
+* COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
+* $ D( LDD, * ), E( LDE, * ), F( LDF, * ),
+* $ WORK( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZTGSYL solves the generalized Sylvester equation:
+*>
+*> A * R - L * B = scale * C (1)
+*> D * R - L * E = scale * F
+*>
+*> 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, with complex entries. 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 (1) is equivalent 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) ],
+*>
+*> Here Ix is the identity matrix of size x and X**H is the conjugate
+*> 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 (TRANS = 'C') is used to compute an one-norm-based estimate
+*> of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
+*> and (B,E), using ZLACON.
+*>
+*> If IJOB >= 1, ZTGSYL computes a Frobenius norm-based estimate of
+*> Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
+*> reciprocal of the smallest singular value of Z.
+*>
+*> This is a level-3 BLAS algorithm.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] TRANS
+*> \verbatim
+*> TRANS is CHARACTER*1
+*> = 'N': solve the generalized sylvester equation (1).
+*> = 'C': solve the "conjugate transposed" system (3).
+*> \endverbatim
+*>
+*> \param[in] IJOB
+*> \verbatim
+*> IJOB is INTEGER
+*> Specifies what kind of functionality to be performed.
+*> =0: solve (1) only.
+*> =1: The functionality of 0 and 3.
+*> =2: The functionality of 0 and 4.
+*> =3: Only an estimate of Dif[(A,D), (B,E)] is computed.
+*> (look ahead strategy is used).
+*> =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
+*> (ZGECON on sub-systems is used).
+*> Not referenced if TRANS = 'C'.
+*> \endverbatim
+*>
+*> \param[in] M
+*> \verbatim
+*> M is INTEGER
+*> The order of the matrices A and D, and the row dimension of
+*> the matrices C, F, R and L.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The order of the matrices B and E, and the column dimension
+*> of the matrices C, F, R and L.
+*> \endverbatim
+*>
+*> \param[in] A
+*> \verbatim
+*> A is COMPLEX*16 array, dimension (LDA, M)
+*> The upper triangular matrix A.
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of the array A. LDA >= max(1, M).
+*> \endverbatim
+*>
+*> \param[in] B
+*> \verbatim
+*> B is COMPLEX*16 array, dimension (LDB, N)
+*> The upper triangular matrix B.
+*> \endverbatim
+*>
+*> \param[in] LDB
+*> \verbatim
+*> LDB is INTEGER
+*> The leading dimension of the array 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) or (3).
+*> On exit, if IJOB = 0, 1 or 2, C has been overwritten by
+*> the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
+*> the solution achieved during the computation of the
+*> Dif-estimate.
+*> \endverbatim
+*>
+*> \param[in] LDC
+*> \verbatim
+*> LDC is INTEGER
+*> The leading dimension of the array C. LDC >= max(1, M).
+*> \endverbatim
+*>
+*> \param[in] D
+*> \verbatim
+*> D is COMPLEX*16 array, dimension (LDD, M)
+*> The upper triangular matrix D.
+*> \endverbatim
+*>
+*> \param[in] LDD
+*> \verbatim
+*> LDD is INTEGER
+*> The leading dimension of the array D. LDD >= max(1, M).
+*> \endverbatim
+*>
+*> \param[in] E
+*> \verbatim
+*> E is COMPLEX*16 array, dimension (LDE, N)
+*> The upper triangular matrix E.
+*> \endverbatim
+*>
+*> \param[in] LDE
+*> \verbatim
+*> LDE is INTEGER
+*> The leading dimension of the array 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) or (3).
+*> On exit, if IJOB = 0, 1 or 2, F has been overwritten by
+*> the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
+*> the solution achieved during the computation of the
+*> Dif-estimate.
+*> \endverbatim
+*>
+*> \param[in] LDF
+*> \verbatim
+*> LDF is INTEGER
+*> The leading dimension of the array F. LDF >= max(1, M).
+*> \endverbatim
+*>
+*> \param[out] DIF
+*> \verbatim
+*> DIF is DOUBLE PRECISION
+*> On exit DIF is the reciprocal of a lower bound of the
+*> reciprocal of the Dif-function, i.e. DIF is an upper bound of
+*> Dif[(A,D), (B,E)] = sigma-min(Z), where Z as in (2).
+*> IF IJOB = 0 or TRANS = 'C', DIF is not referenced.
+*> \endverbatim
+*>
+*> \param[out] SCALE
+*> \verbatim
+*> SCALE is DOUBLE PRECISION
+*> On exit SCALE is the scaling factor in (1) or (3).
+*> If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
+*> 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.
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
+*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*> LWORK is INTEGER
+*> The dimension of the array WORK. LWORK > = 1.
+*> If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
+*>
+*> If LWORK = -1, then a workspace query is assumed; the routine
+*> only calculates the optimal size of the WORK array, returns
+*> this value as the first entry of the WORK array, and no error
+*> message related to LWORK is issued by XERBLA.
+*> \endverbatim
+*>
+*> \param[out] IWORK
+*> \verbatim
+*> IWORK is INTEGER array, dimension (M+N+2)
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> =0: successful exit
+*> <0: If INFO = -i, the i-th argument had an illegal value.
+*> >0: (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 complex16SYcomputational
+*
+*> \par Contributors:
+* ==================
+*>
+*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
+*> Umea University, S-901 87 Umea, Sweden.
+*
+*> \par References:
+* ================
+*>
+*> [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
+*> for Solving the Generalized Sylvester Equation and Estimating the
+*> Separation between Regular Matrix Pairs, Report UMINF - 93.23,
+*> Department of Computing Science, Umea University, S-901 87 Umea,
+*> Sweden, December 1993, Revised April 1994, Also as LAPACK Working
+*> Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
+*> No 1, 1996.
+*> \n
+*> [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
+*> Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
+*> Appl., 15(4):1045-1060, 1994.
+*> \n
+*> [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
+*> Condition Estimators for Solving the Generalized Sylvester
+*> Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
+*> July 1989, pp 745-751.
+*>
+* =====================================================================
SUBROUTINE ZTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
$ LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
$ IWORK, INFO )
*
-* -- LAPACK routine (version 3.2) --
+* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* January 2007
*
* .. Scalar Arguments ..
CHARACTER TRANS
@@ -20,177 +310,6 @@
$ WORK( * )
* ..
*
-* Purpose
-* =======
-*
-* ZTGSYL solves the generalized Sylvester equation:
-*
-* A * R - L * B = scale * C (1)
-* D * R - L * E = scale * F
-*
-* 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, with complex entries. 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 (1) is equivalent to solve Zx = scale*b, where Z
-* is defined as
-*
-* Z = [ kron(In, A) -kron(B', Im) ] (2)
-* [ kron(In, D) -kron(E', Im) ],
-*
-* Here Ix is the identity matrix of size x and X' is the conjugate
-* 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 (TRANS = 'C') is used to compute an one-norm-based estimate
-* of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
-* and (B,E), using ZLACON.
-*
-* If IJOB >= 1, ZTGSYL computes a Frobenius norm-based estimate of
-* Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
-* reciprocal of the smallest singular value of Z.
-*
-* This is a level-3 BLAS algorithm.
-*
-* Arguments
-* =========
-*
-* TRANS (input) CHARACTER*1
-* = 'N': solve the generalized sylvester equation (1).
-* = 'C': solve the "conjugate transposed" system (3).
-*
-* IJOB (input) INTEGER
-* Specifies what kind of functionality to be performed.
-* =0: solve (1) only.
-* =1: The functionality of 0 and 3.
-* =2: The functionality of 0 and 4.
-* =3: Only an estimate of Dif[(A,D), (B,E)] is computed.
-* (look ahead strategy is used).
-* =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
-* (ZGECON on sub-systems is used).
-* Not referenced if TRANS = 'C'.
-*
-* M (input) INTEGER
-* The order of the matrices A and D, and the row dimension of
-* the matrices C, F, R and L.
-*
-* N (input) INTEGER
-* The order of the matrices B and E, and the column dimension
-* of the matrices C, F, R and L.
-*
-* A (input) COMPLEX*16 array, dimension (LDA, M)
-* The upper triangular matrix A.
-*
-* LDA (input) INTEGER
-* The leading dimension of the array A. LDA >= max(1, M).
-*
-* B (input) COMPLEX*16 array, dimension (LDB, N)
-* The upper triangular matrix B.
-*
-* LDB (input) INTEGER
-* The leading dimension of the array 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) or (3).
-* On exit, if IJOB = 0, 1 or 2, C has been overwritten by
-* the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
-* the solution achieved during the computation of the
-* Dif-estimate.
-*
-* LDC (input) INTEGER
-* The leading dimension of the array C. LDC >= max(1, M).
-*
-* D (input) COMPLEX*16 array, dimension (LDD, M)
-* The upper triangular matrix D.
-*
-* LDD (input) INTEGER
-* The leading dimension of the array D. LDD >= max(1, M).
-*
-* E (input) COMPLEX*16 array, dimension (LDE, N)
-* The upper triangular matrix E.
-*
-* LDE (input) INTEGER
-* The leading dimension of the array 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) or (3).
-* On exit, if IJOB = 0, 1 or 2, F has been overwritten by
-* the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
-* the solution achieved during the computation of the
-* Dif-estimate.
-*
-* LDF (input) INTEGER
-* The leading dimension of the array F. LDF >= max(1, M).
-*
-* DIF (output) DOUBLE PRECISION
-* On exit DIF is the reciprocal of a lower bound of the
-* reciprocal of the Dif-function, i.e. DIF is an upper bound of
-* Dif[(A,D), (B,E)] = sigma-min(Z), where Z as in (2).
-* IF IJOB = 0 or TRANS = 'C', DIF is not referenced.
-*
-* SCALE (output) DOUBLE PRECISION
-* On exit SCALE is the scaling factor in (1) or (3).
-* If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
-* 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 homogenious system with C = F = 0.
-*
-* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
-* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
-*
-* LWORK (input) INTEGER
-* The dimension of the array WORK. LWORK > = 1.
-* If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
-*
-* If LWORK = -1, then a workspace query is assumed; the routine
-* only calculates the optimal size of the WORK array, returns
-* this value as the first entry of the WORK array, and no error
-* message related to LWORK is issued by XERBLA.
-*
-* IWORK (workspace) INTEGER array, dimension (M+N+2)
-*
-* INFO (output) INTEGER
-* =0: successful exit
-* <0: If INFO = -i, the i-th argument had an illegal value.
-* >0: (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.
-*
-* [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
-* for Solving the Generalized Sylvester Equation and Estimating the
-* Separation between Regular Matrix Pairs, Report UMINF - 93.23,
-* Department of Computing Science, Umea University, S-901 87 Umea,
-* Sweden, December 1993, Revised April 1994, Also as LAPACK Working
-* Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
-* No 1, 1996.
-*
-* [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
-* Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
-* Appl., 15(4):1045-1060, 1994.
-*
-* [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
-* Condition Estimators for Solving the Generalized Sylvester
-* Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
-* July 1989, pp 745-751.
-*
* =====================================================================
* Replaced various illegal calls to CCOPY by calls to CLASET.
* Sven Hammarling, 1/5/02.
@@ -492,7 +611,7 @@
ELSE
*
* Solve transposed (I, J)-subsystem
-* A(I, I)' * R(I, J) + D(I, I)' * L(I, J) = C(I, J)
+* A(I, I)**H * R(I, J) + D(I, I)**H * L(I, J) = C(I, J)
* R(I, J) * B(J, J) + L(I, J) * E(J, J) = -F(I, J)
* for I = 1,2,..., P; J = Q, Q-1,..., 1
*