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275 lines
7.8 KiB
275 lines
7.8 KiB
2 years ago
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*> \brief \b SGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the 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 SGBTF2 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgbtf2.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/sgbtf2.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/sgbtf2.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 SGBTF2( 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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*> SGBTF2 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 unblocked version of the algorithm, calling Level 2 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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*> \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
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*> interchanges.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE SGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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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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* ..
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* .. Local Scalars ..
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INTEGER I, J, JP, JU, KM, KV
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* ..
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* .. External Functions ..
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INTEGER ISAMAX
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EXTERNAL ISAMAX
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* ..
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* .. External Subroutines ..
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EXTERNAL SGER, SSCAL, SSWAP, 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( 'SGBTF2', -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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* 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 20 J = KU + 2, MIN( KV, N )
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DO 10 I = KV - J + 2, KL
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AB( I, J ) = ZERO
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10 CONTINUE
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20 CONTINUE
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*
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* JU is the index of the last column affected by the current stage
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* of the factorization.
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*
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JU = 1
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*
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DO 40 J = 1, MIN( M, N )
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*
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* Set fill-in elements in column J+KV to zero.
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*
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IF( J+KV.LE.N ) THEN
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DO 30 I = 1, KL
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AB( I, J+KV ) = ZERO
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30 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-J )
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JP = ISAMAX( KM+1, AB( KV+1, J ), 1 )
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IPIV( J ) = JP + J - 1
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IF( AB( KV+JP, J ).NE.ZERO ) THEN
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JU = MAX( JU, MIN( J+KU+JP-1, N ) )
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*
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* Apply interchange to columns J to JU.
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*
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IF( JP.NE.1 )
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$ CALL SSWAP( JU-J+1, AB( KV+JP, J ), LDAB-1,
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$ AB( KV+1, J ), LDAB-1 )
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*
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IF( KM.GT.0 ) THEN
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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, J ), AB( KV+2, J ), 1 )
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*
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* Update trailing submatrix within the band.
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*
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IF( JU.GT.J )
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$ CALL SGER( KM, JU-J, -ONE, AB( KV+2, J ), 1,
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$ AB( KV, J+1 ), LDAB-1, AB( KV+1, J+1 ),
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$ LDAB-1 )
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END IF
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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 = J
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END IF
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40 CONTINUE
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RETURN
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*
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* End of SGBTF2
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*
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END
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