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470 lines
15 KiB
470 lines
15 KiB
2 years ago
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*> \brief \b ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download ZTGSY2 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztgsy2.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztgsy2.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztgsy2.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* 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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* .. 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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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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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**H, Im) ] (2)
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*> [ kron(In, D) -kron(E**H, Im) ],
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*>
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*> Ik is the identity matrix of size k and X**H is the conjugate 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**H*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**H * R + D**H * L = scale * C (3)
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*> R * B**H + L * E**H = 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 communication 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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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] TRANS
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*> \verbatim
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*> TRANS is 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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*> \endverbatim
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*>
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*> \param[in] IJOB
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*> \verbatim
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*> IJOB is 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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*> \endverbatim
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*>
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*> \param[in] M
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*> \verbatim
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*> M is 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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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is 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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*> \endverbatim
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*>
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*> \param[in] A
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*> \verbatim
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*> A is COMPLEX*16 array, dimension (LDA, M)
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*> On entry, A contains an upper triangular matrix.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the matrix A. LDA >= max(1, M).
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*> \endverbatim
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*>
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*> \param[in] B
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*> \verbatim
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*> B is COMPLEX*16 array, dimension (LDB, N)
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*> On entry, B contains an upper triangular matrix.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of the matrix B. LDB >= max(1, N).
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*> \endverbatim
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*>
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*> \param[in,out] C
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*> \verbatim
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*> C is 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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*> \endverbatim
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*>
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*> \param[in] LDC
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*> \verbatim
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*> LDC is INTEGER
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*> The leading dimension of the matrix C. LDC >= max(1, M).
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*> \endverbatim
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*>
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*> \param[in] D
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*> \verbatim
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*> D is COMPLEX*16 array, dimension (LDD, M)
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*> On entry, D contains an upper triangular matrix.
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*> \endverbatim
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*>
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*> \param[in] LDD
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*> \verbatim
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*> LDD is INTEGER
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*> The leading dimension of the matrix D. LDD >= max(1, M).
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*> \endverbatim
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*>
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*> \param[in] E
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*> \verbatim
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*> E is COMPLEX*16 array, dimension (LDE, N)
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*> On entry, E contains an upper triangular matrix.
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*> \endverbatim
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*>
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*> \param[in] LDE
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*> \verbatim
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*> LDE is INTEGER
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*> The leading dimension of the matrix E. LDE >= max(1, N).
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*> \endverbatim
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*>
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*> \param[in,out] F
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*> \verbatim
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*> F is 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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*> \endverbatim
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*>
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*> \param[in] LDF
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*> \verbatim
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*> LDF is INTEGER
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*> The leading dimension of the matrix F. LDF >= max(1, M).
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*> \endverbatim
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*>
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*> \param[out] SCALE
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*> \verbatim
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*> SCALE is 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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*> \endverbatim
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*>
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*> \param[in,out] RDSUM
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*> \verbatim
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*> RDSUM is 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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*> \endverbatim
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*>
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*> \param[in,out] RDSCAL
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*> \verbatim
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*> RDSCAL is 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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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is 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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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup complex16SYauxiliary
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*
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*> \par Contributors:
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* ==================
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*>
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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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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 --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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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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* =====================================================================
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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 = -6
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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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|
*
|
||
|
|
Z( 1, 1 ) = A( I, I )
|
||
|
|
Z( 2, 1 ) = D( I, I )
|
||
|
|
Z( 1, 2 ) = -B( J, J )
|
||
|
|
Z( 2, 2 ) = -E( J, J )
|
||
|
|
*
|
||
|
|
* Set up right hand side(s)
|
||
|
|
*
|
||
|
|
RHS( 1 ) = C( I, J )
|
||
|
|
RHS( 2 ) = F( I, J )
|
||
|
|
*
|
||
|
|
* Solve Z * x = RHS
|
||
|
|
*
|
||
|
|
CALL ZGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR )
|
||
|
|
IF( IERR.GT.0 )
|
||
|
|
$ INFO = IERR
|
||
|
|
IF( IJOB.EQ.0 ) THEN
|
||
|
|
CALL ZGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
|
||
|
|
IF( SCALOC.NE.ONE ) THEN
|
||
|
|
DO 10 K = 1, N
|
||
|
|
CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ),
|
||
|
|
$ C( 1, K ), 1 )
|
||
|
|
CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ),
|
||
|
|
$ F( 1, K ), 1 )
|
||
|
|
10 CONTINUE
|
||
|
|
SCALE = SCALE*SCALOC
|
||
|
|
END IF
|
||
|
|
ELSE
|
||
|
|
CALL ZLATDF( IJOB, LDZ, Z, LDZ, RHS, RDSUM, RDSCAL,
|
||
|
|
$ IPIV, JPIV )
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Unpack solution vector(s)
|
||
|
|
*
|
||
|
|
C( I, J ) = RHS( 1 )
|
||
|
|
F( I, J ) = RHS( 2 )
|
||
|
|
*
|
||
|
|
* Substitute R(I, J) and L(I, J) into remaining equation.
|
||
|
|
*
|
||
|
|
IF( I.GT.1 ) THEN
|
||
|
|
ALPHA = -RHS( 1 )
|
||
|
|
CALL ZAXPY( I-1, ALPHA, A( 1, I ), 1, C( 1, J ), 1 )
|
||
|
|
CALL ZAXPY( I-1, ALPHA, D( 1, I ), 1, F( 1, J ), 1 )
|
||
|
|
END IF
|
||
|
|
IF( J.LT.N ) THEN
|
||
|
|
CALL ZAXPY( N-J, RHS( 2 ), B( J, J+1 ), LDB,
|
||
|
|
$ C( I, J+1 ), LDC )
|
||
|
|
CALL ZAXPY( N-J, RHS( 2 ), E( J, J+1 ), LDE,
|
||
|
|
$ F( I, J+1 ), LDF )
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
20 CONTINUE
|
||
|
|
30 CONTINUE
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* Solve transposed (I, J) - system:
|
||
|
|
* 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
|
||
|
|
*
|
||
|
|
SCALE = ONE
|
||
|
|
SCALOC = ONE
|
||
|
|
DO 80 I = 1, M
|
||
|
|
DO 70 J = N, 1, -1
|
||
|
|
*
|
||
|
|
* Build 2 by 2 system Z**H
|
||
|
|
*
|
||
|
|
Z( 1, 1 ) = DCONJG( A( I, I ) )
|
||
|
|
Z( 2, 1 ) = -DCONJG( B( J, J ) )
|
||
|
|
Z( 1, 2 ) = DCONJG( D( I, I ) )
|
||
|
|
Z( 2, 2 ) = -DCONJG( E( J, J ) )
|
||
|
|
*
|
||
|
|
*
|
||
|
|
* Set up right hand side(s)
|
||
|
|
*
|
||
|
|
RHS( 1 ) = C( I, J )
|
||
|
|
RHS( 2 ) = F( I, J )
|
||
|
|
*
|
||
|
|
* Solve Z**H * x = RHS
|
||
|
|
*
|
||
|
|
CALL ZGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR )
|
||
|
|
IF( IERR.GT.0 )
|
||
|
|
$ INFO = IERR
|
||
|
|
CALL ZGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
|
||
|
|
IF( SCALOC.NE.ONE ) THEN
|
||
|
|
DO 40 K = 1, N
|
||
|
|
CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), C( 1, K ),
|
||
|
|
$ 1 )
|
||
|
|
CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), F( 1, K ),
|
||
|
|
$ 1 )
|
||
|
|
40 CONTINUE
|
||
|
|
SCALE = SCALE*SCALOC
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Unpack solution vector(s)
|
||
|
|
*
|
||
|
|
C( I, J ) = RHS( 1 )
|
||
|
|
F( I, J ) = RHS( 2 )
|
||
|
|
*
|
||
|
|
* Substitute R(I, J) and L(I, J) into remaining equation.
|
||
|
|
*
|
||
|
|
DO 50 K = 1, J - 1
|
||
|
|
F( I, K ) = F( I, K ) + RHS( 1 )*DCONJG( B( K, J ) ) +
|
||
|
|
$ RHS( 2 )*DCONJG( E( K, J ) )
|
||
|
|
50 CONTINUE
|
||
|
|
DO 60 K = I + 1, M
|
||
|
|
C( K, J ) = C( K, J ) - DCONJG( A( I, K ) )*RHS( 1 ) -
|
||
|
|
$ DCONJG( D( I, K ) )*RHS( 2 )
|
||
|
|
60 CONTINUE
|
||
|
|
*
|
||
|
|
70 CONTINUE
|
||
|
|
80 CONTINUE
|
||
|
|
END IF
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of ZTGSY2
|
||
|
|
*
|
||
|
|
END
|