517 lines
16 KiB
Fortran
517 lines
16 KiB
Fortran
*> \brief \b SGBTRF
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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 SGBTRF + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgbtrf.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/sgbtrf.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/sgbtrf.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 SGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER INFO, KL, KU, LDAB, M, N
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* ..
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* .. Array Arguments ..
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* INTEGER IPIV( * )
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* REAL AB( LDAB, * )
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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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*> SGBTRF computes an LU factorization of a real m-by-n band matrix A
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*> using partial pivoting with row interchanges.
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*>
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*> This is the blocked version of the algorithm, calling Level 3 BLAS.
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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] M
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*> \verbatim
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*> M is INTEGER
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*> The number of rows of the matrix A. 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 number of columns of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] KL
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*> \verbatim
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*> KL is INTEGER
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*> The number of subdiagonals within the band of A. KL >= 0.
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*> \endverbatim
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*>
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*> \param[in] KU
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*> \verbatim
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*> KU is INTEGER
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*> The number of superdiagonals within the band of A. KU >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] AB
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*> \verbatim
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*> AB is REAL array, dimension (LDAB,N)
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*> On entry, the matrix A in band storage, in rows KL+1 to
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*> 2*KL+KU+1; rows 1 to KL of the array need not be set.
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*> The j-th column of A is stored in the j-th column of the
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*> array AB as follows:
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*> AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
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*>
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*> On exit, details of the factorization: U is stored as an
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*> upper triangular band matrix with KL+KU superdiagonals in
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*> rows 1 to KL+KU+1, and the multipliers used during the
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*> factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
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*> See below for further details.
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*> \endverbatim
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*>
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*> \param[in] LDAB
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*> \verbatim
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*> LDAB is INTEGER
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*> The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
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*> \endverbatim
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*>
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*> \param[out] IPIV
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*> \verbatim
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*> IPIV is INTEGER array, dimension (min(M,N))
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*> The pivot indices; for 1 <= i <= min(M,N), row i of the
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*> matrix was interchanged with row IPIV(i).
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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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*> > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
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*> has been completed, but the factor U is exactly
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*> singular, and division by zero will occur if it is used
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*> to solve a system of equations.
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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 November 2011
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*
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*> \ingroup realGBcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> The band storage scheme is illustrated by the following example, when
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*> M = N = 6, KL = 2, KU = 1:
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*>
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*> On entry: On exit:
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*>
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*> * * * + + + * * * u14 u25 u36
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*> * * + + + + * * u13 u24 u35 u46
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*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
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*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
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*> a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
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*> a31 a42 a53 a64 * * m31 m42 m53 m64 * *
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*>
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*> Array elements marked * are not used by the routine; elements marked
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*> + need not be set on entry, but are required by the routine to store
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*> elements of U because of fill-in resulting from the row interchanges.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE SGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
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*
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* -- LAPACK computational routine (version 3.4.0) --
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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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* November 2011
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*
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* .. Scalar Arguments ..
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INTEGER INFO, KL, KU, LDAB, M, N
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* ..
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* .. Array Arguments ..
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INTEGER IPIV( * )
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REAL AB( LDAB, * )
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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, ZERO
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PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
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INTEGER NBMAX, LDWORK
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PARAMETER ( NBMAX = 64, LDWORK = NBMAX+1 )
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* ..
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* .. Local Scalars ..
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INTEGER I, I2, I3, II, IP, J, J2, J3, JB, JJ, JM, JP,
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$ JU, K2, KM, KV, NB, NW
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REAL TEMP
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* ..
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* .. Local Arrays ..
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REAL WORK13( LDWORK, NBMAX ),
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$ WORK31( LDWORK, NBMAX )
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* ..
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* .. External Functions ..
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INTEGER ILAENV, ISAMAX
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EXTERNAL ILAENV, ISAMAX
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* ..
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* .. External Subroutines ..
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EXTERNAL SCOPY, SGBTF2, SGEMM, SGER, SLASWP, SSCAL,
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$ SSWAP, STRSM, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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* KV is the number of superdiagonals in the factor U, allowing for
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* fill-in
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*
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KV = KU + KL
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*
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* Test the input parameters.
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*
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INFO = 0
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IF( M.LT.0 ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( KL.LT.0 ) THEN
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INFO = -3
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ELSE IF( KU.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDAB.LT.KL+KV+1 ) THEN
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INFO = -6
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SGBTRF', -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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IF( M.EQ.0 .OR. N.EQ.0 )
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$ RETURN
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*
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* Determine the block size for this environment
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*
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NB = ILAENV( 1, 'SGBTRF', ' ', M, N, KL, KU )
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*
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* The block size must not exceed the limit set by the size of the
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* local arrays WORK13 and WORK31.
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*
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NB = MIN( NB, NBMAX )
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*
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IF( NB.LE.1 .OR. NB.GT.KL ) THEN
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*
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* Use unblocked code
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*
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CALL SGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
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ELSE
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*
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* Use blocked code
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*
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* Zero the superdiagonal elements of the work array WORK13
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*
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DO 20 J = 1, NB
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DO 10 I = 1, J - 1
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WORK13( I, J ) = ZERO
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10 CONTINUE
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20 CONTINUE
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*
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* Zero the subdiagonal elements of the work array WORK31
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*
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DO 40 J = 1, NB
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DO 30 I = J + 1, NB
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WORK31( I, J ) = ZERO
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30 CONTINUE
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40 CONTINUE
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*
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* Gaussian elimination with partial pivoting
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*
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* Set fill-in elements in columns KU+2 to KV to zero
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*
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DO 60 J = KU + 2, MIN( KV, N )
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DO 50 I = KV - J + 2, KL
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AB( I, J ) = ZERO
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50 CONTINUE
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60 CONTINUE
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*
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* JU is the index of the last column affected by the current
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* stage of the factorization
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*
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JU = 1
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*
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DO 180 J = 1, MIN( M, N ), NB
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JB = MIN( NB, MIN( M, N )-J+1 )
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*
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* The active part of the matrix is partitioned
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*
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* A11 A12 A13
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* A21 A22 A23
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* A31 A32 A33
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*
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* Here A11, A21 and A31 denote the current block of JB columns
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* which is about to be factorized. The number of rows in the
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* partitioning are JB, I2, I3 respectively, and the numbers
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* of columns are JB, J2, J3. The superdiagonal elements of A13
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* and the subdiagonal elements of A31 lie outside the band.
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*
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I2 = MIN( KL-JB, M-J-JB+1 )
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I3 = MIN( JB, M-J-KL+1 )
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*
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* J2 and J3 are computed after JU has been updated.
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*
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* Factorize the current block of JB columns
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*
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DO 80 JJ = J, J + JB - 1
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*
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* Set fill-in elements in column JJ+KV to zero
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*
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IF( JJ+KV.LE.N ) THEN
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DO 70 I = 1, KL
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AB( I, JJ+KV ) = ZERO
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70 CONTINUE
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END IF
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*
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* Find pivot and test for singularity. KM is the number of
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* subdiagonal elements in the current column.
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*
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KM = MIN( KL, M-JJ )
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JP = ISAMAX( KM+1, AB( KV+1, JJ ), 1 )
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IPIV( JJ ) = JP + JJ - J
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IF( AB( KV+JP, JJ ).NE.ZERO ) THEN
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JU = MAX( JU, MIN( JJ+KU+JP-1, N ) )
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IF( JP.NE.1 ) THEN
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*
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* Apply interchange to columns J to J+JB-1
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*
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IF( JP+JJ-1.LT.J+KL ) THEN
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*
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CALL SSWAP( JB, AB( KV+1+JJ-J, J ), LDAB-1,
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$ AB( KV+JP+JJ-J, J ), LDAB-1 )
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ELSE
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*
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* The interchange affects columns J to JJ-1 of A31
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* which are stored in the work array WORK31
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*
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CALL SSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
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$ WORK31( JP+JJ-J-KL, 1 ), LDWORK )
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CALL SSWAP( J+JB-JJ, AB( KV+1, JJ ), LDAB-1,
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$ AB( KV+JP, JJ ), LDAB-1 )
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END IF
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END IF
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*
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* Compute multipliers
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*
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CALL SSCAL( KM, ONE / AB( KV+1, JJ ), AB( KV+2, JJ ),
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$ 1 )
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*
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* Update trailing submatrix within the band and within
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* the current block. JM is the index of the last column
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* which needs to be updated.
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*
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JM = MIN( JU, J+JB-1 )
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IF( JM.GT.JJ )
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$ CALL SGER( KM, JM-JJ, -ONE, AB( KV+2, JJ ), 1,
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$ AB( KV, JJ+1 ), LDAB-1,
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$ AB( KV+1, JJ+1 ), LDAB-1 )
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ELSE
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*
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* If pivot is zero, set INFO to the index of the pivot
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* unless a zero pivot has already been found.
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*
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IF( INFO.EQ.0 )
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$ INFO = JJ
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END IF
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*
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* Copy current column of A31 into the work array WORK31
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*
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NW = MIN( JJ-J+1, I3 )
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IF( NW.GT.0 )
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$ CALL SCOPY( NW, AB( KV+KL+1-JJ+J, JJ ), 1,
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$ WORK31( 1, JJ-J+1 ), 1 )
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80 CONTINUE
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IF( J+JB.LE.N ) THEN
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*
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* Apply the row interchanges to the other blocks.
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*
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J2 = MIN( JU-J+1, KV ) - JB
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J3 = MAX( 0, JU-J-KV+1 )
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*
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* Use SLASWP to apply the row interchanges to A12, A22, and
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* A32.
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*
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CALL SLASWP( J2, AB( KV+1-JB, J+JB ), LDAB-1, 1, JB,
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$ IPIV( J ), 1 )
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*
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* Adjust the pivot indices.
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*
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DO 90 I = J, J + JB - 1
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IPIV( I ) = IPIV( I ) + J - 1
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90 CONTINUE
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*
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* Apply the row interchanges to A13, A23, and A33
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* columnwise.
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*
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K2 = J - 1 + JB + J2
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DO 110 I = 1, J3
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JJ = K2 + I
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DO 100 II = J + I - 1, J + JB - 1
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IP = IPIV( II )
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IF( IP.NE.II ) THEN
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TEMP = AB( KV+1+II-JJ, JJ )
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AB( KV+1+II-JJ, JJ ) = AB( KV+1+IP-JJ, JJ )
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AB( KV+1+IP-JJ, JJ ) = TEMP
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END IF
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100 CONTINUE
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110 CONTINUE
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*
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* Update the relevant part of the trailing submatrix
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*
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IF( J2.GT.0 ) THEN
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*
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* Update A12
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*
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CALL STRSM( 'Left', 'Lower', 'No transpose', 'Unit',
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$ JB, J2, ONE, AB( KV+1, J ), LDAB-1,
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$ AB( KV+1-JB, J+JB ), LDAB-1 )
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*
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IF( I2.GT.0 ) THEN
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*
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* Update A22
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*
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CALL SGEMM( 'No transpose', 'No transpose', I2, J2,
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$ JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
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$ AB( KV+1-JB, J+JB ), LDAB-1, ONE,
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$ AB( KV+1, J+JB ), LDAB-1 )
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END IF
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*
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IF( I3.GT.0 ) THEN
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*
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* Update A32
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*
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CALL SGEMM( 'No transpose', 'No transpose', I3, J2,
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$ JB, -ONE, WORK31, LDWORK,
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$ AB( KV+1-JB, J+JB ), LDAB-1, ONE,
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$ AB( KV+KL+1-JB, J+JB ), LDAB-1 )
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END IF
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END IF
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*
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IF( J3.GT.0 ) THEN
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*
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* Copy the lower triangle of A13 into the work array
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* WORK13
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*
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DO 130 JJ = 1, J3
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DO 120 II = JJ, JB
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WORK13( II, JJ ) = AB( II-JJ+1, JJ+J+KV-1 )
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120 CONTINUE
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130 CONTINUE
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*
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* Update A13 in the work array
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*
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CALL STRSM( 'Left', 'Lower', 'No transpose', 'Unit',
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$ JB, J3, ONE, AB( KV+1, J ), LDAB-1,
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$ WORK13, LDWORK )
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*
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IF( I2.GT.0 ) THEN
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*
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* Update A23
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*
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CALL SGEMM( 'No transpose', 'No transpose', I2, J3,
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$ JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
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$ WORK13, LDWORK, ONE, AB( 1+JB, J+KV ),
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$ LDAB-1 )
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END IF
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*
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IF( I3.GT.0 ) THEN
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*
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* Update A33
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*
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CALL SGEMM( 'No transpose', 'No transpose', I3, J3,
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$ JB, -ONE, WORK31, LDWORK, WORK13,
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$ LDWORK, ONE, AB( 1+KL, J+KV ), LDAB-1 )
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END IF
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*
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* Copy the lower triangle of A13 back into place
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*
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DO 150 JJ = 1, J3
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DO 140 II = JJ, JB
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AB( II-JJ+1, JJ+J+KV-1 ) = WORK13( II, JJ )
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140 CONTINUE
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150 CONTINUE
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END IF
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ELSE
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*
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* Adjust the pivot indices.
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*
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DO 160 I = J, J + JB - 1
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IPIV( I ) = IPIV( I ) + J - 1
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160 CONTINUE
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END IF
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*
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* Partially undo the interchanges in the current block to
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* restore the upper triangular form of A31 and copy the upper
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* triangle of A31 back into place
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*
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DO 170 JJ = J + JB - 1, J, -1
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JP = IPIV( JJ ) - JJ + 1
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IF( JP.NE.1 ) THEN
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*
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* Apply interchange to columns J to JJ-1
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*
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IF( JP+JJ-1.LT.J+KL ) THEN
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*
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* The interchange does not affect A31
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*
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CALL SSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
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$ AB( KV+JP+JJ-J, J ), LDAB-1 )
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ELSE
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*
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* The interchange does affect A31
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*
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CALL SSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
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$ WORK31( JP+JJ-J-KL, 1 ), LDWORK )
|
|
END IF
|
|
END IF
|
|
*
|
|
* Copy the current column of A31 back into place
|
|
*
|
|
NW = MIN( I3, JJ-J+1 )
|
|
IF( NW.GT.0 )
|
|
$ CALL SCOPY( NW, WORK31( 1, JJ-J+1 ), 1,
|
|
$ AB( KV+KL+1-JJ+J, JJ ), 1 )
|
|
170 CONTINUE
|
|
180 CONTINUE
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of SGBTRF
|
|
*
|
|
END
|