455 lines
13 KiB
Fortran
455 lines
13 KiB
Fortran
*> \brief \b CTRSYL
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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 CTRSYL + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ctrsyl.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/ctrsyl.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/ctrsyl.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 CTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
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* LDC, SCALE, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER TRANA, TRANB
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* INTEGER INFO, ISGN, LDA, LDB, LDC, M, N
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* REAL SCALE
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* ..
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* .. Array Arguments ..
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* COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * )
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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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*> CTRSYL solves the complex Sylvester matrix equation:
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*>
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*> op(A)*X + X*op(B) = scale*C or
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*> op(A)*X - X*op(B) = scale*C,
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*>
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*> where op(A) = A or A**H, and A and B are both upper triangular. A is
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*> M-by-M and B is N-by-N; the right hand side C and the solution X are
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*> M-by-N; and scale is an output scale factor, set <= 1 to avoid
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*> overflow in X.
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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] TRANA
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*> \verbatim
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*> TRANA is CHARACTER*1
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*> Specifies the option op(A):
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*> = 'N': op(A) = A (No transpose)
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*> = 'C': op(A) = A**H (Conjugate transpose)
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*> \endverbatim
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*>
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*> \param[in] TRANB
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*> \verbatim
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*> TRANB is CHARACTER*1
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*> Specifies the option op(B):
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*> = 'N': op(B) = B (No transpose)
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*> = 'C': op(B) = B**H (Conjugate transpose)
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*> \endverbatim
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*>
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*> \param[in] ISGN
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*> \verbatim
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*> ISGN is INTEGER
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*> Specifies the sign in the equation:
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*> = +1: solve op(A)*X + X*op(B) = scale*C
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*> = -1: solve op(A)*X - X*op(B) = scale*C
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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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*> The order of the matrix A, and the number of rows in the
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*> matrices X and C. M >= 0.
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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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*> The order of the matrix B, and the number of columns in the
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*> matrices X and C. N >= 0.
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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 array, dimension (LDA,M)
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*> The upper triangular matrix A.
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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 array 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 array, dimension (LDB,N)
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*> The upper triangular matrix B.
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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 array 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 array, dimension (LDC,N)
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*> On entry, the M-by-N right hand side matrix C.
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*> On exit, C is overwritten by the solution matrix X.
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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 array C. LDC >= 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 REAL
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*> The scale factor, scale, set <= 1 to avoid overflow in X.
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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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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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*> = 1: A and B have common or very close eigenvalues; perturbed
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*> values were used to solve the equation (but the matrices
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*> A and B are unchanged).
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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 December 2016
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*
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*> \ingroup complexSYcomputational
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*
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* =====================================================================
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SUBROUTINE CTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
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$ LDC, SCALE, INFO )
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*
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* -- LAPACK computational routine (version 3.7.0) --
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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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* December 2016
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*
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* .. Scalar Arguments ..
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CHARACTER TRANA, TRANB
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INTEGER INFO, ISGN, LDA, LDB, LDC, M, N
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REAL SCALE
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* ..
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* .. Array Arguments ..
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COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * )
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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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REAL ONE
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PARAMETER ( ONE = 1.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOTRNA, NOTRNB
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INTEGER J, K, L
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REAL BIGNUM, DA11, DB, EPS, SCALOC, SGN, SMIN,
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$ SMLNUM
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COMPLEX A11, SUML, SUMR, VEC, X11
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* ..
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* .. Local Arrays ..
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REAL DUM( 1 )
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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REAL CLANGE, SLAMCH
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COMPLEX CDOTC, CDOTU, CLADIV
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EXTERNAL LSAME, CLANGE, SLAMCH, CDOTC, CDOTU, CLADIV
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* ..
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* .. External Subroutines ..
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EXTERNAL CSSCAL, SLABAD, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, MIN, REAL
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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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NOTRNA = LSAME( TRANA, 'N' )
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NOTRNB = LSAME( TRANB, 'N' )
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*
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INFO = 0
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IF( .NOT.NOTRNA .AND. .NOT.LSAME( TRANA, 'C' ) ) THEN
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INFO = -1
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ELSE IF( .NOT.NOTRNB .AND. .NOT.LSAME( TRANB, 'C' ) ) THEN
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INFO = -2
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ELSE IF( ISGN.NE.1 .AND. ISGN.NE.-1 ) THEN
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INFO = -3
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ELSE IF( M.LT.0 ) THEN
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INFO = -4
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ELSE IF( N.LT.0 ) THEN
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INFO = -5
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -7
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -9
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ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
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INFO = -11
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CTRSYL', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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SCALE = ONE
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IF( M.EQ.0 .OR. N.EQ.0 )
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$ RETURN
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*
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* Set constants to control overflow
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*
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EPS = SLAMCH( 'P' )
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SMLNUM = SLAMCH( 'S' )
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BIGNUM = ONE / SMLNUM
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CALL SLABAD( SMLNUM, BIGNUM )
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SMLNUM = SMLNUM*REAL( M*N ) / EPS
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BIGNUM = ONE / SMLNUM
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SMIN = MAX( SMLNUM, EPS*CLANGE( 'M', M, M, A, LDA, DUM ),
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$ EPS*CLANGE( 'M', N, N, B, LDB, DUM ) )
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SGN = ISGN
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*
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IF( NOTRNA .AND. NOTRNB ) THEN
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*
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* Solve A*X + ISGN*X*B = scale*C.
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*
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* The (K,L)th block of X is determined starting from
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* bottom-left corner column by column by
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*
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* A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
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*
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* Where
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* M L-1
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* R(K,L) = SUM [A(K,I)*X(I,L)] +ISGN*SUM [X(K,J)*B(J,L)].
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* I=K+1 J=1
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*
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DO 30 L = 1, N
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DO 20 K = M, 1, -1
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*
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SUML = CDOTU( M-K, A( K, MIN( K+1, M ) ), LDA,
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$ C( MIN( K+1, M ), L ), 1 )
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SUMR = CDOTU( L-1, C( K, 1 ), LDC, B( 1, L ), 1 )
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VEC = C( K, L ) - ( SUML+SGN*SUMR )
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*
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SCALOC = ONE
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A11 = A( K, K ) + SGN*B( L, L )
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DA11 = ABS( REAL( A11 ) ) + ABS( AIMAG( A11 ) )
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IF( DA11.LE.SMIN ) THEN
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A11 = SMIN
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DA11 = SMIN
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INFO = 1
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END IF
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DB = ABS( REAL( VEC ) ) + ABS( AIMAG( VEC ) )
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IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
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IF( DB.GT.BIGNUM*DA11 )
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$ SCALOC = ONE / DB
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END IF
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X11 = CLADIV( VEC*CMPLX( SCALOC ), A11 )
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*
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IF( SCALOC.NE.ONE ) THEN
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DO 10 J = 1, N
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CALL CSSCAL( M, SCALOC, C( 1, J ), 1 )
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10 CONTINUE
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SCALE = SCALE*SCALOC
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END IF
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C( K, L ) = X11
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*
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20 CONTINUE
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30 CONTINUE
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*
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ELSE IF( .NOT.NOTRNA .AND. NOTRNB ) THEN
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*
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* Solve A**H *X + ISGN*X*B = scale*C.
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*
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* The (K,L)th block of X is determined starting from
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* upper-left corner column by column by
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*
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* A**H(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
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*
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* Where
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* K-1 L-1
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* R(K,L) = SUM [A**H(I,K)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)]
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* I=1 J=1
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*
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DO 60 L = 1, N
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DO 50 K = 1, M
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*
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SUML = CDOTC( K-1, A( 1, K ), 1, C( 1, L ), 1 )
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SUMR = CDOTU( L-1, C( K, 1 ), LDC, B( 1, L ), 1 )
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VEC = C( K, L ) - ( SUML+SGN*SUMR )
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*
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SCALOC = ONE
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A11 = CONJG( A( K, K ) ) + SGN*B( L, L )
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DA11 = ABS( REAL( A11 ) ) + ABS( AIMAG( A11 ) )
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IF( DA11.LE.SMIN ) THEN
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A11 = SMIN
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DA11 = SMIN
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INFO = 1
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END IF
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DB = ABS( REAL( VEC ) ) + ABS( AIMAG( VEC ) )
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IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
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IF( DB.GT.BIGNUM*DA11 )
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$ SCALOC = ONE / DB
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END IF
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*
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X11 = CLADIV( VEC*CMPLX( SCALOC ), A11 )
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*
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IF( SCALOC.NE.ONE ) THEN
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DO 40 J = 1, N
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CALL CSSCAL( M, SCALOC, C( 1, J ), 1 )
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40 CONTINUE
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SCALE = SCALE*SCALOC
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END IF
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C( K, L ) = X11
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*
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50 CONTINUE
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60 CONTINUE
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*
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ELSE IF( .NOT.NOTRNA .AND. .NOT.NOTRNB ) THEN
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*
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* Solve A**H*X + ISGN*X*B**H = C.
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*
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* The (K,L)th block of X is determined starting from
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* upper-right corner column by column by
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*
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* A**H(K,K)*X(K,L) + ISGN*X(K,L)*B**H(L,L) = C(K,L) - R(K,L)
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*
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* Where
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* K-1
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* R(K,L) = SUM [A**H(I,K)*X(I,L)] +
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* I=1
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* N
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* ISGN*SUM [X(K,J)*B**H(L,J)].
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* J=L+1
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*
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DO 90 L = N, 1, -1
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DO 80 K = 1, M
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*
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SUML = CDOTC( K-1, A( 1, K ), 1, C( 1, L ), 1 )
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SUMR = CDOTC( N-L, C( K, MIN( L+1, N ) ), LDC,
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$ B( L, MIN( L+1, N ) ), LDB )
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VEC = C( K, L ) - ( SUML+SGN*CONJG( SUMR ) )
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*
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SCALOC = ONE
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A11 = CONJG( A( K, K )+SGN*B( L, L ) )
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DA11 = ABS( REAL( A11 ) ) + ABS( AIMAG( A11 ) )
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IF( DA11.LE.SMIN ) THEN
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A11 = SMIN
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DA11 = SMIN
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INFO = 1
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END IF
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DB = ABS( REAL( VEC ) ) + ABS( AIMAG( VEC ) )
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IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
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IF( DB.GT.BIGNUM*DA11 )
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$ SCALOC = ONE / DB
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END IF
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*
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X11 = CLADIV( VEC*CMPLX( SCALOC ), A11 )
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*
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IF( SCALOC.NE.ONE ) THEN
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DO 70 J = 1, N
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CALL CSSCAL( M, SCALOC, C( 1, J ), 1 )
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70 CONTINUE
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SCALE = SCALE*SCALOC
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END IF
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C( K, L ) = X11
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*
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80 CONTINUE
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90 CONTINUE
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*
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ELSE IF( NOTRNA .AND. .NOT.NOTRNB ) THEN
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*
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* Solve A*X + ISGN*X*B**H = C.
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*
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* The (K,L)th block of X is determined starting from
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* bottom-left corner column by column by
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*
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* A(K,K)*X(K,L) + ISGN*X(K,L)*B**H(L,L) = C(K,L) - R(K,L)
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*
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* Where
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* M N
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* R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B**H(L,J)]
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* I=K+1 J=L+1
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*
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DO 120 L = N, 1, -1
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DO 110 K = M, 1, -1
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*
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SUML = CDOTU( M-K, A( K, MIN( K+1, M ) ), LDA,
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$ C( MIN( K+1, M ), L ), 1 )
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SUMR = CDOTC( N-L, C( K, MIN( L+1, N ) ), LDC,
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$ B( L, MIN( L+1, N ) ), LDB )
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VEC = C( K, L ) - ( SUML+SGN*CONJG( SUMR ) )
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*
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SCALOC = ONE
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A11 = A( K, K ) + SGN*CONJG( B( L, L ) )
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DA11 = ABS( REAL( A11 ) ) + ABS( AIMAG( A11 ) )
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IF( DA11.LE.SMIN ) THEN
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A11 = SMIN
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DA11 = SMIN
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INFO = 1
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END IF
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DB = ABS( REAL( VEC ) ) + ABS( AIMAG( VEC ) )
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IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
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IF( DB.GT.BIGNUM*DA11 )
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$ SCALOC = ONE / DB
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END IF
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*
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X11 = CLADIV( VEC*CMPLX( SCALOC ), A11 )
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*
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IF( SCALOC.NE.ONE ) THEN
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DO 100 J = 1, N
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CALL CSSCAL( M, SCALOC, C( 1, J ), 1 )
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100 CONTINUE
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SCALE = SCALE*SCALOC
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END IF
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C( K, L ) = X11
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*
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110 CONTINUE
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120 CONTINUE
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*
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END IF
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*
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RETURN
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*
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* End of CTRSYL
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*
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END
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