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*> \brief \b SGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SGBTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgbtf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgbtf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgbtf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, KL, KU, LDAB, M, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* REAL AB( LDAB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGBTF2 computes an LU factorization of a real m-by-n band matrix A
*> using partial pivoting with row interchanges.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is REAL array, dimension (LDAB,N)
*> On entry, the matrix A in band storage, in rows KL+1 to
*> 2*KL+KU+1; rows 1 to KL of the array need not be set.
*> The j-th column of A is stored in the j-th column of the
*> array AB as follows:
*> AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
*>
*> On exit, details of the factorization: U is stored as an
*> upper triangular band matrix with KL+KU superdiagonals in
*> rows 1 to KL+KU+1, and the multipliers used during the
*> factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
*> See below for further details.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (min(M,N))
*> The pivot indices; for 1 <= i <= min(M,N), row i of the
*> matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
*> has been completed, but the factor U is exactly
*> singular, and division by zero will occur if it is used
*> to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup realGBcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> M = N = 6, KL = 2, KU = 1:
*>
*> On entry: On exit:
*>
*> * * * + + + * * * u14 u25 u36
*> * * + + + + * * u13 u24 u35 u46
*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*> a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
*> a31 a42 a53 a64 * * m31 m42 m53 m64 * *
*>
*> Array elements marked * are not used by the routine; elements marked
*> + need not be set on entry, but are required by the routine to store
*> elements of U, because of fill-in resulting from the row
*> interchanges.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER INFO, KL, KU, LDAB, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
REAL AB( LDAB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, JP, JU, KM, KV
* ..
* .. External Functions ..
INTEGER ISAMAX
EXTERNAL ISAMAX
* ..
* .. External Subroutines ..
EXTERNAL SGER, SSCAL, SSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* KV is the number of superdiagonals in the factor U, allowing for
* fill-in.
*
KV = KU + KL
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KL+KV+1 ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGBTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Gaussian elimination with partial pivoting
*
* Set fill-in elements in columns KU+2 to KV to zero.
*
DO 20 J = KU + 2, MIN( KV, N )
DO 10 I = KV - J + 2, KL
AB( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
*
* JU is the index of the last column affected by the current stage
* of the factorization.
*
JU = 1
*
DO 40 J = 1, MIN( M, N )
*
* Set fill-in elements in column J+KV to zero.
*
IF( J+KV.LE.N ) THEN
DO 30 I = 1, KL
AB( I, J+KV ) = ZERO
30 CONTINUE
END IF
*
* Find pivot and test for singularity. KM is the number of
* subdiagonal elements in the current column.
*
KM = MIN( KL, M-J )
JP = ISAMAX( KM+1, AB( KV+1, J ), 1 )
IPIV( J ) = JP + J - 1
IF( AB( KV+JP, J ).NE.ZERO ) THEN
JU = MAX( JU, MIN( J+KU+JP-1, N ) )
*
* Apply interchange to columns J to JU.
*
IF( JP.NE.1 )
$ CALL SSWAP( JU-J+1, AB( KV+JP, J ), LDAB-1,
$ AB( KV+1, J ), LDAB-1 )
*
IF( KM.GT.0 ) THEN
*
* Compute multipliers.
*
CALL SSCAL( KM, ONE / AB( KV+1, J ), AB( KV+2, J ), 1 )
*
* Update trailing submatrix within the band.
*
IF( JU.GT.J )
$ CALL SGER( KM, JU-J, -ONE, AB( KV+2, J ), 1,
$ AB( KV, J+1 ), LDAB-1, AB( KV+1, J+1 ),
$ LDAB-1 )
END IF
ELSE
*
* If pivot is zero, set INFO to the index of the pivot
* unless a zero pivot has already been found.
*
IF( INFO.EQ.0 )
$ INFO = J
END IF
40 CONTINUE
RETURN
*
* End of SGBTF2
*
END
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