1076 lines
36 KiB
Fortran
1076 lines
36 KiB
Fortran
*> \brief \b DTGSY2 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 DTGSY2 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsy2.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/dtgsy2.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/dtgsy2.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 DTGSY2( 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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* IWORK, PQ, 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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* $ PQ
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* DOUBLE PRECISION RDSCAL, RDSUM, SCALE
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* ..
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* .. Array Arguments ..
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* INTEGER IWORK( * )
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* DOUBLE PRECISION 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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*> DTGSY2 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, with real entries. (A, D) and (B, E)
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*> must be in generalized Schur canonical form, i.e. A, B are upper
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*> quasi triangular and D, E are upper triangular. The solution (R, L)
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*> overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
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*> 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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*> Z*x = scale*b, where Z is defined as
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*>
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*> Z = [ kron(In, A) -kron(B**T, Im) ] (2)
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*> [ kron(In, D) -kron(E**T, Im) ],
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*>
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*> Ik is the identity matrix of size k and X**T 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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*> In the process of solving (1), we solve a number of such systems
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*> where Dim(In), Dim(In) = 1 or 2.
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*>
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*> If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y,
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*> which is equivalent to solve for R and L in
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*>
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*> A**T * R + D**T * L = scale * C (3)
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*> R * B**T + L * E**T = 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 DLACON.
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*>
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*> DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL
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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 the matrix pair in
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*> DTGSYL. See DTGSYL for details.
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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 DOUBLE PRECISION array, dimension (LDA, M)
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*> On entry, A contains an upper quasi 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 DOUBLE PRECISION array, dimension (LDB, N)
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*> On entry, B contains an upper quasi 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 DOUBLE PRECISION 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
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*> solution 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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
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*> solution 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. Normally,
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*> 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 DTGSYL, 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 DTGSY2 is called by DTGSYL.
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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 DTGSY2 is called by
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*> DTGSYL.
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*> \endverbatim
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension (M+N+2)
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*> \endverbatim
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*>
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*> \param[out] PQ
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*> \verbatim
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*> PQ is INTEGER
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*> On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
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*> 8-by-8) solved by this routine.
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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, the i-th argument had an illegal value.
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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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*> \date September 2012
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*
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*> \ingroup doubleSYauxiliary
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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 DTGSY2( 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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$ IWORK, PQ, INFO )
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*
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* -- LAPACK auxiliary routine (version 3.4.2) --
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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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* September 2012
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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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$ PQ
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DOUBLE PRECISION RDSCAL, RDSUM, SCALE
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* ..
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* .. Array Arguments ..
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INTEGER IWORK( * )
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DOUBLE PRECISION 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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* Replaced various illegal calls to DCOPY by calls to DLASET.
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* Sven Hammarling, 27/5/02.
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*
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* .. Parameters ..
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INTEGER LDZ
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PARAMETER ( LDZ = 8 )
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOTRAN
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INTEGER I, IE, IERR, II, IS, ISP1, J, JE, JJ, JS, JSP1,
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$ K, MB, NB, P, Q, ZDIM
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DOUBLE PRECISION ALPHA, SCALOC
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* ..
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* .. Local Arrays ..
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INTEGER IPIV( LDZ ), JPIV( LDZ )
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DOUBLE PRECISION 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 DAXPY, DCOPY, DGEMM, DGEMV, DGER, DGESC2,
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$ DGETC2, DLASET, DLATDF, DSCAL, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC 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, 'T' ) ) 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( 'DTGSY2', -INFO )
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RETURN
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END IF
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*
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* Determine block structure of A
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*
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PQ = 0
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P = 0
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I = 1
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10 CONTINUE
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IF( I.GT.M )
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$ GO TO 20
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P = P + 1
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IWORK( P ) = I
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IF( I.EQ.M )
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$ GO TO 20
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IF( A( I+1, I ).NE.ZERO ) THEN
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I = I + 2
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ELSE
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I = I + 1
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END IF
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GO TO 10
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20 CONTINUE
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IWORK( P+1 ) = M + 1
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*
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* Determine block structure of B
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*
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Q = P + 1
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J = 1
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30 CONTINUE
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IF( J.GT.N )
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$ GO TO 40
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Q = Q + 1
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IWORK( Q ) = J
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IF( J.EQ.N )
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$ GO TO 40
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IF( B( J+1, J ).NE.ZERO ) THEN
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J = J + 2
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ELSE
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J = J + 1
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END IF
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GO TO 30
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40 CONTINUE
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IWORK( Q+1 ) = N + 1
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PQ = P*( Q-P-1 )
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*
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IF( NOTRAN ) THEN
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*
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* Solve (I, J) - subsystem
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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 = P, P - 1, ..., 1; J = 1, 2, ..., Q
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*
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SCALE = ONE
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SCALOC = ONE
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DO 120 J = P + 2, Q
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JS = IWORK( J )
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JSP1 = JS + 1
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JE = IWORK( J+1 ) - 1
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NB = JE - JS + 1
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DO 110 I = P, 1, -1
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*
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IS = IWORK( I )
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ISP1 = IS + 1
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IE = IWORK( I+1 ) - 1
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MB = IE - IS + 1
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ZDIM = MB*NB*2
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*
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IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
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*
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* Build a 2-by-2 system Z * x = RHS
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*
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Z( 1, 1 ) = A( IS, IS )
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Z( 2, 1 ) = D( IS, IS )
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Z( 1, 2 ) = -B( JS, JS )
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Z( 2, 2 ) = -E( JS, JS )
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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( IS, JS )
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RHS( 2 ) = F( IS, JS )
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*
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* Solve Z * x = RHS
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*
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CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
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IF( IERR.GT.0 )
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$ INFO = IERR
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*
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IF( IJOB.EQ.0 ) THEN
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CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
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$ SCALOC )
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IF( SCALOC.NE.ONE ) THEN
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DO 50 K = 1, N
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CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
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CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
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50 CONTINUE
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SCALE = SCALE*SCALOC
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END IF
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ELSE
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CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
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$ RDSCAL, 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( IS, JS ) = RHS( 1 )
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F( IS, JS ) = RHS( 2 )
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*
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* Substitute R(I, J) and L(I, J) into remaining
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* 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 DAXPY( IS-1, ALPHA, A( 1, IS ), 1, C( 1, JS ),
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$ 1 )
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CALL DAXPY( IS-1, ALPHA, D( 1, IS ), 1, F( 1, JS ),
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$ 1 )
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END IF
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IF( J.LT.Q ) THEN
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CALL DAXPY( N-JE, RHS( 2 ), B( JS, JE+1 ), LDB,
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$ C( IS, JE+1 ), LDC )
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CALL DAXPY( N-JE, RHS( 2 ), E( JS, JE+1 ), LDE,
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$ F( IS, JE+1 ), LDF )
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END IF
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*
|
|
ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN
|
|
*
|
|
* Build a 4-by-4 system Z * x = RHS
|
|
*
|
|
Z( 1, 1 ) = A( IS, IS )
|
|
Z( 2, 1 ) = ZERO
|
|
Z( 3, 1 ) = D( IS, IS )
|
|
Z( 4, 1 ) = ZERO
|
|
*
|
|
Z( 1, 2 ) = ZERO
|
|
Z( 2, 2 ) = A( IS, IS )
|
|
Z( 3, 2 ) = ZERO
|
|
Z( 4, 2 ) = D( IS, IS )
|
|
*
|
|
Z( 1, 3 ) = -B( JS, JS )
|
|
Z( 2, 3 ) = -B( JS, JSP1 )
|
|
Z( 3, 3 ) = -E( JS, JS )
|
|
Z( 4, 3 ) = -E( JS, JSP1 )
|
|
*
|
|
Z( 1, 4 ) = -B( JSP1, JS )
|
|
Z( 2, 4 ) = -B( JSP1, JSP1 )
|
|
Z( 3, 4 ) = ZERO
|
|
Z( 4, 4 ) = -E( JSP1, JSP1 )
|
|
*
|
|
* Set up right hand side(s)
|
|
*
|
|
RHS( 1 ) = C( IS, JS )
|
|
RHS( 2 ) = C( IS, JSP1 )
|
|
RHS( 3 ) = F( IS, JS )
|
|
RHS( 4 ) = F( IS, JSP1 )
|
|
*
|
|
* Solve Z * x = RHS
|
|
*
|
|
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
|
|
IF( IERR.GT.0 )
|
|
$ INFO = IERR
|
|
*
|
|
IF( IJOB.EQ.0 ) THEN
|
|
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
|
|
$ SCALOC )
|
|
IF( SCALOC.NE.ONE ) THEN
|
|
DO 60 K = 1, N
|
|
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
|
|
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
|
|
60 CONTINUE
|
|
SCALE = SCALE*SCALOC
|
|
END IF
|
|
ELSE
|
|
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
|
|
$ RDSCAL, IPIV, JPIV )
|
|
END IF
|
|
*
|
|
* Unpack solution vector(s)
|
|
*
|
|
C( IS, JS ) = RHS( 1 )
|
|
C( IS, JSP1 ) = RHS( 2 )
|
|
F( IS, JS ) = RHS( 3 )
|
|
F( IS, JSP1 ) = RHS( 4 )
|
|
*
|
|
* Substitute R(I, J) and L(I, J) into remaining
|
|
* equation.
|
|
*
|
|
IF( I.GT.1 ) THEN
|
|
CALL DGER( IS-1, NB, -ONE, A( 1, IS ), 1, RHS( 1 ),
|
|
$ 1, C( 1, JS ), LDC )
|
|
CALL DGER( IS-1, NB, -ONE, D( 1, IS ), 1, RHS( 1 ),
|
|
$ 1, F( 1, JS ), LDF )
|
|
END IF
|
|
IF( J.LT.Q ) THEN
|
|
CALL DAXPY( N-JE, RHS( 3 ), B( JS, JE+1 ), LDB,
|
|
$ C( IS, JE+1 ), LDC )
|
|
CALL DAXPY( N-JE, RHS( 3 ), E( JS, JE+1 ), LDE,
|
|
$ F( IS, JE+1 ), LDF )
|
|
CALL DAXPY( N-JE, RHS( 4 ), B( JSP1, JE+1 ), LDB,
|
|
$ C( IS, JE+1 ), LDC )
|
|
CALL DAXPY( N-JE, RHS( 4 ), E( JSP1, JE+1 ), LDE,
|
|
$ F( IS, JE+1 ), LDF )
|
|
END IF
|
|
*
|
|
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN
|
|
*
|
|
* Build a 4-by-4 system Z * x = RHS
|
|
*
|
|
Z( 1, 1 ) = A( IS, IS )
|
|
Z( 2, 1 ) = A( ISP1, IS )
|
|
Z( 3, 1 ) = D( IS, IS )
|
|
Z( 4, 1 ) = ZERO
|
|
*
|
|
Z( 1, 2 ) = A( IS, ISP1 )
|
|
Z( 2, 2 ) = A( ISP1, ISP1 )
|
|
Z( 3, 2 ) = D( IS, ISP1 )
|
|
Z( 4, 2 ) = D( ISP1, ISP1 )
|
|
*
|
|
Z( 1, 3 ) = -B( JS, JS )
|
|
Z( 2, 3 ) = ZERO
|
|
Z( 3, 3 ) = -E( JS, JS )
|
|
Z( 4, 3 ) = ZERO
|
|
*
|
|
Z( 1, 4 ) = ZERO
|
|
Z( 2, 4 ) = -B( JS, JS )
|
|
Z( 3, 4 ) = ZERO
|
|
Z( 4, 4 ) = -E( JS, JS )
|
|
*
|
|
* Set up right hand side(s)
|
|
*
|
|
RHS( 1 ) = C( IS, JS )
|
|
RHS( 2 ) = C( ISP1, JS )
|
|
RHS( 3 ) = F( IS, JS )
|
|
RHS( 4 ) = F( ISP1, JS )
|
|
*
|
|
* Solve Z * x = RHS
|
|
*
|
|
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
|
|
IF( IERR.GT.0 )
|
|
$ INFO = IERR
|
|
IF( IJOB.EQ.0 ) THEN
|
|
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
|
|
$ SCALOC )
|
|
IF( SCALOC.NE.ONE ) THEN
|
|
DO 70 K = 1, N
|
|
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
|
|
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
|
|
70 CONTINUE
|
|
SCALE = SCALE*SCALOC
|
|
END IF
|
|
ELSE
|
|
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
|
|
$ RDSCAL, IPIV, JPIV )
|
|
END IF
|
|
*
|
|
* Unpack solution vector(s)
|
|
*
|
|
C( IS, JS ) = RHS( 1 )
|
|
C( ISP1, JS ) = RHS( 2 )
|
|
F( IS, JS ) = RHS( 3 )
|
|
F( ISP1, JS ) = RHS( 4 )
|
|
*
|
|
* Substitute R(I, J) and L(I, J) into remaining
|
|
* equation.
|
|
*
|
|
IF( I.GT.1 ) THEN
|
|
CALL DGEMV( 'N', IS-1, MB, -ONE, A( 1, IS ), LDA,
|
|
$ RHS( 1 ), 1, ONE, C( 1, JS ), 1 )
|
|
CALL DGEMV( 'N', IS-1, MB, -ONE, D( 1, IS ), LDD,
|
|
$ RHS( 1 ), 1, ONE, F( 1, JS ), 1 )
|
|
END IF
|
|
IF( J.LT.Q ) THEN
|
|
CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1,
|
|
$ B( JS, JE+1 ), LDB, C( IS, JE+1 ), LDC )
|
|
CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1,
|
|
$ E( JS, JE+1 ), LDE, F( IS, JE+1 ), LDF )
|
|
END IF
|
|
*
|
|
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN
|
|
*
|
|
* Build an 8-by-8 system Z * x = RHS
|
|
*
|
|
CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ )
|
|
*
|
|
Z( 1, 1 ) = A( IS, IS )
|
|
Z( 2, 1 ) = A( ISP1, IS )
|
|
Z( 5, 1 ) = D( IS, IS )
|
|
*
|
|
Z( 1, 2 ) = A( IS, ISP1 )
|
|
Z( 2, 2 ) = A( ISP1, ISP1 )
|
|
Z( 5, 2 ) = D( IS, ISP1 )
|
|
Z( 6, 2 ) = D( ISP1, ISP1 )
|
|
*
|
|
Z( 3, 3 ) = A( IS, IS )
|
|
Z( 4, 3 ) = A( ISP1, IS )
|
|
Z( 7, 3 ) = D( IS, IS )
|
|
*
|
|
Z( 3, 4 ) = A( IS, ISP1 )
|
|
Z( 4, 4 ) = A( ISP1, ISP1 )
|
|
Z( 7, 4 ) = D( IS, ISP1 )
|
|
Z( 8, 4 ) = D( ISP1, ISP1 )
|
|
*
|
|
Z( 1, 5 ) = -B( JS, JS )
|
|
Z( 3, 5 ) = -B( JS, JSP1 )
|
|
Z( 5, 5 ) = -E( JS, JS )
|
|
Z( 7, 5 ) = -E( JS, JSP1 )
|
|
*
|
|
Z( 2, 6 ) = -B( JS, JS )
|
|
Z( 4, 6 ) = -B( JS, JSP1 )
|
|
Z( 6, 6 ) = -E( JS, JS )
|
|
Z( 8, 6 ) = -E( JS, JSP1 )
|
|
*
|
|
Z( 1, 7 ) = -B( JSP1, JS )
|
|
Z( 3, 7 ) = -B( JSP1, JSP1 )
|
|
Z( 7, 7 ) = -E( JSP1, JSP1 )
|
|
*
|
|
Z( 2, 8 ) = -B( JSP1, JS )
|
|
Z( 4, 8 ) = -B( JSP1, JSP1 )
|
|
Z( 8, 8 ) = -E( JSP1, JSP1 )
|
|
*
|
|
* Set up right hand side(s)
|
|
*
|
|
K = 1
|
|
II = MB*NB + 1
|
|
DO 80 JJ = 0, NB - 1
|
|
CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 )
|
|
CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ), 1 )
|
|
K = K + MB
|
|
II = II + MB
|
|
80 CONTINUE
|
|
*
|
|
* Solve Z * x = RHS
|
|
*
|
|
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
|
|
IF( IERR.GT.0 )
|
|
$ INFO = IERR
|
|
IF( IJOB.EQ.0 ) THEN
|
|
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
|
|
$ SCALOC )
|
|
IF( SCALOC.NE.ONE ) THEN
|
|
DO 90 K = 1, N
|
|
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
|
|
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
|
|
90 CONTINUE
|
|
SCALE = SCALE*SCALOC
|
|
END IF
|
|
ELSE
|
|
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
|
|
$ RDSCAL, IPIV, JPIV )
|
|
END IF
|
|
*
|
|
* Unpack solution vector(s)
|
|
*
|
|
K = 1
|
|
II = MB*NB + 1
|
|
DO 100 JJ = 0, NB - 1
|
|
CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 )
|
|
CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ), 1 )
|
|
K = K + MB
|
|
II = II + MB
|
|
100 CONTINUE
|
|
*
|
|
* Substitute R(I, J) and L(I, J) into remaining
|
|
* equation.
|
|
*
|
|
IF( I.GT.1 ) THEN
|
|
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
|
|
$ A( 1, IS ), LDA, RHS( 1 ), MB, ONE,
|
|
$ C( 1, JS ), LDC )
|
|
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
|
|
$ D( 1, IS ), LDD, RHS( 1 ), MB, ONE,
|
|
$ F( 1, JS ), LDF )
|
|
END IF
|
|
IF( J.LT.Q ) THEN
|
|
K = MB*NB + 1
|
|
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE, RHS( K ),
|
|
$ MB, B( JS, JE+1 ), LDB, ONE,
|
|
$ C( IS, JE+1 ), LDC )
|
|
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE, RHS( K ),
|
|
$ MB, E( JS, JE+1 ), LDE, ONE,
|
|
$ F( IS, JE+1 ), LDF )
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
110 CONTINUE
|
|
120 CONTINUE
|
|
ELSE
|
|
*
|
|
* Solve (I, J) - subsystem
|
|
* A(I, I)**T * R(I, J) + D(I, I)**T * 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, ..., P, J = Q, Q - 1, ..., 1
|
|
*
|
|
SCALE = ONE
|
|
SCALOC = ONE
|
|
DO 200 I = 1, P
|
|
*
|
|
IS = IWORK( I )
|
|
ISP1 = IS + 1
|
|
IE = IWORK ( I+1 ) - 1
|
|
MB = IE - IS + 1
|
|
DO 190 J = Q, P + 2, -1
|
|
*
|
|
JS = IWORK( J )
|
|
JSP1 = JS + 1
|
|
JE = IWORK( J+1 ) - 1
|
|
NB = JE - JS + 1
|
|
ZDIM = MB*NB*2
|
|
IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
|
|
*
|
|
* Build a 2-by-2 system Z**T * x = RHS
|
|
*
|
|
Z( 1, 1 ) = A( IS, IS )
|
|
Z( 2, 1 ) = -B( JS, JS )
|
|
Z( 1, 2 ) = D( IS, IS )
|
|
Z( 2, 2 ) = -E( JS, JS )
|
|
*
|
|
* Set up right hand side(s)
|
|
*
|
|
RHS( 1 ) = C( IS, JS )
|
|
RHS( 2 ) = F( IS, JS )
|
|
*
|
|
* Solve Z**T * x = RHS
|
|
*
|
|
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
|
|
IF( IERR.GT.0 )
|
|
$ INFO = IERR
|
|
*
|
|
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
|
|
IF( SCALOC.NE.ONE ) THEN
|
|
DO 130 K = 1, N
|
|
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
|
|
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
|
|
130 CONTINUE
|
|
SCALE = SCALE*SCALOC
|
|
END IF
|
|
*
|
|
* Unpack solution vector(s)
|
|
*
|
|
C( IS, JS ) = RHS( 1 )
|
|
F( IS, JS ) = RHS( 2 )
|
|
*
|
|
* Substitute R(I, J) and L(I, J) into remaining
|
|
* equation.
|
|
*
|
|
IF( J.GT.P+2 ) THEN
|
|
ALPHA = RHS( 1 )
|
|
CALL DAXPY( JS-1, ALPHA, B( 1, JS ), 1, F( IS, 1 ),
|
|
$ LDF )
|
|
ALPHA = RHS( 2 )
|
|
CALL DAXPY( JS-1, ALPHA, E( 1, JS ), 1, F( IS, 1 ),
|
|
$ LDF )
|
|
END IF
|
|
IF( I.LT.P ) THEN
|
|
ALPHA = -RHS( 1 )
|
|
CALL DAXPY( M-IE, ALPHA, A( IS, IE+1 ), LDA,
|
|
$ C( IE+1, JS ), 1 )
|
|
ALPHA = -RHS( 2 )
|
|
CALL DAXPY( M-IE, ALPHA, D( IS, IE+1 ), LDD,
|
|
$ C( IE+1, JS ), 1 )
|
|
END IF
|
|
*
|
|
ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN
|
|
*
|
|
* Build a 4-by-4 system Z**T * x = RHS
|
|
*
|
|
Z( 1, 1 ) = A( IS, IS )
|
|
Z( 2, 1 ) = ZERO
|
|
Z( 3, 1 ) = -B( JS, JS )
|
|
Z( 4, 1 ) = -B( JSP1, JS )
|
|
*
|
|
Z( 1, 2 ) = ZERO
|
|
Z( 2, 2 ) = A( IS, IS )
|
|
Z( 3, 2 ) = -B( JS, JSP1 )
|
|
Z( 4, 2 ) = -B( JSP1, JSP1 )
|
|
*
|
|
Z( 1, 3 ) = D( IS, IS )
|
|
Z( 2, 3 ) = ZERO
|
|
Z( 3, 3 ) = -E( JS, JS )
|
|
Z( 4, 3 ) = ZERO
|
|
*
|
|
Z( 1, 4 ) = ZERO
|
|
Z( 2, 4 ) = D( IS, IS )
|
|
Z( 3, 4 ) = -E( JS, JSP1 )
|
|
Z( 4, 4 ) = -E( JSP1, JSP1 )
|
|
*
|
|
* Set up right hand side(s)
|
|
*
|
|
RHS( 1 ) = C( IS, JS )
|
|
RHS( 2 ) = C( IS, JSP1 )
|
|
RHS( 3 ) = F( IS, JS )
|
|
RHS( 4 ) = F( IS, JSP1 )
|
|
*
|
|
* Solve Z**T * x = RHS
|
|
*
|
|
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
|
|
IF( IERR.GT.0 )
|
|
$ INFO = IERR
|
|
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
|
|
IF( SCALOC.NE.ONE ) THEN
|
|
DO 140 K = 1, N
|
|
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
|
|
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
|
|
140 CONTINUE
|
|
SCALE = SCALE*SCALOC
|
|
END IF
|
|
*
|
|
* Unpack solution vector(s)
|
|
*
|
|
C( IS, JS ) = RHS( 1 )
|
|
C( IS, JSP1 ) = RHS( 2 )
|
|
F( IS, JS ) = RHS( 3 )
|
|
F( IS, JSP1 ) = RHS( 4 )
|
|
*
|
|
* Substitute R(I, J) and L(I, J) into remaining
|
|
* equation.
|
|
*
|
|
IF( J.GT.P+2 ) THEN
|
|
CALL DAXPY( JS-1, RHS( 1 ), B( 1, JS ), 1,
|
|
$ F( IS, 1 ), LDF )
|
|
CALL DAXPY( JS-1, RHS( 2 ), B( 1, JSP1 ), 1,
|
|
$ F( IS, 1 ), LDF )
|
|
CALL DAXPY( JS-1, RHS( 3 ), E( 1, JS ), 1,
|
|
$ F( IS, 1 ), LDF )
|
|
CALL DAXPY( JS-1, RHS( 4 ), E( 1, JSP1 ), 1,
|
|
$ F( IS, 1 ), LDF )
|
|
END IF
|
|
IF( I.LT.P ) THEN
|
|
CALL DGER( M-IE, NB, -ONE, A( IS, IE+1 ), LDA,
|
|
$ RHS( 1 ), 1, C( IE+1, JS ), LDC )
|
|
CALL DGER( M-IE, NB, -ONE, D( IS, IE+1 ), LDD,
|
|
$ RHS( 3 ), 1, C( IE+1, JS ), LDC )
|
|
END IF
|
|
*
|
|
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN
|
|
*
|
|
* Build a 4-by-4 system Z**T * x = RHS
|
|
*
|
|
Z( 1, 1 ) = A( IS, IS )
|
|
Z( 2, 1 ) = A( IS, ISP1 )
|
|
Z( 3, 1 ) = -B( JS, JS )
|
|
Z( 4, 1 ) = ZERO
|
|
*
|
|
Z( 1, 2 ) = A( ISP1, IS )
|
|
Z( 2, 2 ) = A( ISP1, ISP1 )
|
|
Z( 3, 2 ) = ZERO
|
|
Z( 4, 2 ) = -B( JS, JS )
|
|
*
|
|
Z( 1, 3 ) = D( IS, IS )
|
|
Z( 2, 3 ) = D( IS, ISP1 )
|
|
Z( 3, 3 ) = -E( JS, JS )
|
|
Z( 4, 3 ) = ZERO
|
|
*
|
|
Z( 1, 4 ) = ZERO
|
|
Z( 2, 4 ) = D( ISP1, ISP1 )
|
|
Z( 3, 4 ) = ZERO
|
|
Z( 4, 4 ) = -E( JS, JS )
|
|
*
|
|
* Set up right hand side(s)
|
|
*
|
|
RHS( 1 ) = C( IS, JS )
|
|
RHS( 2 ) = C( ISP1, JS )
|
|
RHS( 3 ) = F( IS, JS )
|
|
RHS( 4 ) = F( ISP1, JS )
|
|
*
|
|
* Solve Z**T * x = RHS
|
|
*
|
|
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
|
|
IF( IERR.GT.0 )
|
|
$ INFO = IERR
|
|
*
|
|
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
|
|
IF( SCALOC.NE.ONE ) THEN
|
|
DO 150 K = 1, N
|
|
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
|
|
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
|
|
150 CONTINUE
|
|
SCALE = SCALE*SCALOC
|
|
END IF
|
|
*
|
|
* Unpack solution vector(s)
|
|
*
|
|
C( IS, JS ) = RHS( 1 )
|
|
C( ISP1, JS ) = RHS( 2 )
|
|
F( IS, JS ) = RHS( 3 )
|
|
F( ISP1, JS ) = RHS( 4 )
|
|
*
|
|
* Substitute R(I, J) and L(I, J) into remaining
|
|
* equation.
|
|
*
|
|
IF( J.GT.P+2 ) THEN
|
|
CALL DGER( MB, JS-1, ONE, RHS( 1 ), 1, B( 1, JS ),
|
|
$ 1, F( IS, 1 ), LDF )
|
|
CALL DGER( MB, JS-1, ONE, RHS( 3 ), 1, E( 1, JS ),
|
|
$ 1, F( IS, 1 ), LDF )
|
|
END IF
|
|
IF( I.LT.P ) THEN
|
|
CALL DGEMV( 'T', MB, M-IE, -ONE, A( IS, IE+1 ),
|
|
$ LDA, RHS( 1 ), 1, ONE, C( IE+1, JS ),
|
|
$ 1 )
|
|
CALL DGEMV( 'T', MB, M-IE, -ONE, D( IS, IE+1 ),
|
|
$ LDD, RHS( 3 ), 1, ONE, C( IE+1, JS ),
|
|
$ 1 )
|
|
END IF
|
|
*
|
|
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN
|
|
*
|
|
* Build an 8-by-8 system Z**T * x = RHS
|
|
*
|
|
CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ )
|
|
*
|
|
Z( 1, 1 ) = A( IS, IS )
|
|
Z( 2, 1 ) = A( IS, ISP1 )
|
|
Z( 5, 1 ) = -B( JS, JS )
|
|
Z( 7, 1 ) = -B( JSP1, JS )
|
|
*
|
|
Z( 1, 2 ) = A( ISP1, IS )
|
|
Z( 2, 2 ) = A( ISP1, ISP1 )
|
|
Z( 6, 2 ) = -B( JS, JS )
|
|
Z( 8, 2 ) = -B( JSP1, JS )
|
|
*
|
|
Z( 3, 3 ) = A( IS, IS )
|
|
Z( 4, 3 ) = A( IS, ISP1 )
|
|
Z( 5, 3 ) = -B( JS, JSP1 )
|
|
Z( 7, 3 ) = -B( JSP1, JSP1 )
|
|
*
|
|
Z( 3, 4 ) = A( ISP1, IS )
|
|
Z( 4, 4 ) = A( ISP1, ISP1 )
|
|
Z( 6, 4 ) = -B( JS, JSP1 )
|
|
Z( 8, 4 ) = -B( JSP1, JSP1 )
|
|
*
|
|
Z( 1, 5 ) = D( IS, IS )
|
|
Z( 2, 5 ) = D( IS, ISP1 )
|
|
Z( 5, 5 ) = -E( JS, JS )
|
|
*
|
|
Z( 2, 6 ) = D( ISP1, ISP1 )
|
|
Z( 6, 6 ) = -E( JS, JS )
|
|
*
|
|
Z( 3, 7 ) = D( IS, IS )
|
|
Z( 4, 7 ) = D( IS, ISP1 )
|
|
Z( 5, 7 ) = -E( JS, JSP1 )
|
|
Z( 7, 7 ) = -E( JSP1, JSP1 )
|
|
*
|
|
Z( 4, 8 ) = D( ISP1, ISP1 )
|
|
Z( 6, 8 ) = -E( JS, JSP1 )
|
|
Z( 8, 8 ) = -E( JSP1, JSP1 )
|
|
*
|
|
* Set up right hand side(s)
|
|
*
|
|
K = 1
|
|
II = MB*NB + 1
|
|
DO 160 JJ = 0, NB - 1
|
|
CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 )
|
|
CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ), 1 )
|
|
K = K + MB
|
|
II = II + MB
|
|
160 CONTINUE
|
|
*
|
|
*
|
|
* Solve Z**T * x = RHS
|
|
*
|
|
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
|
|
IF( IERR.GT.0 )
|
|
$ INFO = IERR
|
|
*
|
|
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
|
|
IF( SCALOC.NE.ONE ) THEN
|
|
DO 170 K = 1, N
|
|
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
|
|
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
|
|
170 CONTINUE
|
|
SCALE = SCALE*SCALOC
|
|
END IF
|
|
*
|
|
* Unpack solution vector(s)
|
|
*
|
|
K = 1
|
|
II = MB*NB + 1
|
|
DO 180 JJ = 0, NB - 1
|
|
CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 )
|
|
CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ), 1 )
|
|
K = K + MB
|
|
II = II + MB
|
|
180 CONTINUE
|
|
*
|
|
* Substitute R(I, J) and L(I, J) into remaining
|
|
* equation.
|
|
*
|
|
IF( J.GT.P+2 ) THEN
|
|
CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE,
|
|
$ C( IS, JS ), LDC, B( 1, JS ), LDB, ONE,
|
|
$ F( IS, 1 ), LDF )
|
|
CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE,
|
|
$ F( IS, JS ), LDF, E( 1, JS ), LDE, ONE,
|
|
$ F( IS, 1 ), LDF )
|
|
END IF
|
|
IF( I.LT.P ) THEN
|
|
CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
|
|
$ A( IS, IE+1 ), LDA, C( IS, JS ), LDC,
|
|
$ ONE, C( IE+1, JS ), LDC )
|
|
CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
|
|
$ D( IS, IE+1 ), LDD, F( IS, JS ), LDF,
|
|
$ ONE, C( IE+1, JS ), LDC )
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
190 CONTINUE
|
|
200 CONTINUE
|
|
*
|
|
END IF
|
|
RETURN
|
|
*
|
|
* End of DTGSY2
|
|
*
|
|
END
|