433 lines
13 KiB
Fortran
433 lines
13 KiB
Fortran
*> \brief \b SPBTRF
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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 SPBTRF + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/spbtrf.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/spbtrf.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/spbtrf.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 SPBTRF( UPLO, N, KD, AB, LDAB, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER UPLO
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* INTEGER INFO, KD, LDAB, N
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* ..
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* .. Array Arguments ..
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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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*> SPBTRF computes the Cholesky factorization of a real symmetric
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*> positive definite band matrix A.
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*>
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*> The factorization has the form
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*> A = U**T * U, if UPLO = 'U', or
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*> A = L * L**T, if UPLO = 'L',
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*> where U is an upper triangular matrix and L is lower triangular.
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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] UPLO
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*> \verbatim
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*> UPLO is CHARACTER*1
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*> = 'U': Upper triangle of A is stored;
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*> = 'L': Lower triangle of A is stored.
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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 A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] KD
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*> \verbatim
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*> KD is INTEGER
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*> The number of superdiagonals of the matrix A if UPLO = 'U',
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*> or the number of subdiagonals if UPLO = 'L'. KD >= 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 upper or lower triangle of the symmetric band
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*> matrix A, stored in the first KD+1 rows of the array. The
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*> j-th column of A is stored in the j-th column of the array AB
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*> as follows:
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*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
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*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
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*>
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*> On exit, if INFO = 0, the triangular factor U or L from the
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*> Cholesky factorization A = U**T*U or A = L*L**T of the band
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*> matrix A, in the same storage format as A.
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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 >= KD+1.
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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, the leading principal minor of order i
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*> is not positive, and the factorization could not be
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*> completed.
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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 realOTHERcomputational
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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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*> N = 6, KD = 2, and UPLO = 'U':
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*>
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*> On entry: On exit:
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*>
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*> * * a13 a24 a35 a46 * * 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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*>
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*> Similarly, if UPLO = 'L' the format of A is as follows:
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*>
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*> On entry: On exit:
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*>
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*> a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
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*> a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
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*> a31 a42 a53 a64 * * l31 l42 l53 l64 * *
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*>
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*> Array elements marked * are not used by the routine.
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*> \endverbatim
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*
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*> \par Contributors:
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* ==================
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*>
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*> Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
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*
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* =====================================================================
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SUBROUTINE SPBTRF( UPLO, N, KD, AB, LDAB, 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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CHARACTER UPLO
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INTEGER INFO, KD, LDAB, N
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* ..
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* .. Array Arguments ..
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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 = 32, LDWORK = NBMAX+1 )
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* ..
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* .. Local Scalars ..
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INTEGER I, I2, I3, IB, II, J, JJ, NB
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* ..
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* .. Local Arrays ..
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REAL WORK( LDWORK, NBMAX )
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILAENV
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EXTERNAL LSAME, ILAENV
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* ..
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* .. External Subroutines ..
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EXTERNAL SGEMM, SPBTF2, SPOTF2, SSYRK, STRSM, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MIN
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* ..
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* .. Executable Statements ..
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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( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
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$ ( .NOT.LSAME( UPLO, 'L' ) ) ) 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( KD.LT.0 ) THEN
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INFO = -3
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ELSE IF( LDAB.LT.KD+1 ) THEN
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INFO = -5
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SPBTRF', -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( 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, 'SPBTRF', UPLO, N, KD, -1, -1 )
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*
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* The block size must not exceed the semi-bandwidth KD, and must not
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* exceed the limit set by the size of the local array WORK.
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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.KD ) THEN
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*
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* Use unblocked code
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*
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CALL SPBTF2( UPLO, N, KD, AB, LDAB, 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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IF( LSAME( UPLO, 'U' ) ) THEN
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*
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* Compute the Cholesky factorization of a symmetric band
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* matrix, given the upper triangle of the matrix in band
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* storage.
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*
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* Zero the upper triangle of the work array.
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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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WORK( I, J ) = ZERO
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10 CONTINUE
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20 CONTINUE
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*
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* Process the band matrix one diagonal block at a time.
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*
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DO 70 I = 1, N, NB
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IB = MIN( NB, N-I+1 )
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*
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* Factorize the diagonal block
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*
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CALL SPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
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IF( II.NE.0 ) THEN
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INFO = I + II - 1
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GO TO 150
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END IF
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IF( I+IB.LE.N ) THEN
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*
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* Update the relevant part of the trailing submatrix.
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* If A11 denotes the diagonal block which has just been
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* factorized, then we need to update the remaining
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* blocks in the diagram:
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*
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* A11 A12 A13
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* A22 A23
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* A33
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*
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* The numbers of rows and columns in the partitioning
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* are IB, I2, I3 respectively. The blocks A12, A22 and
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* A23 are empty if IB = KD. The upper triangle of A13
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* lies outside the band.
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*
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I2 = MIN( KD-IB, N-I-IB+1 )
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I3 = MIN( IB, N-I-KD+1 )
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*
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IF( I2.GT.0 ) THEN
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*
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* Update A12
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*
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CALL STRSM( 'Left', 'Upper', 'Transpose',
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$ 'Non-unit', IB, I2, ONE, AB( KD+1, I ),
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$ LDAB-1, AB( KD+1-IB, I+IB ), LDAB-1 )
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*
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* Update A22
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*
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CALL SSYRK( 'Upper', 'Transpose', I2, IB, -ONE,
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$ AB( KD+1-IB, I+IB ), LDAB-1, ONE,
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$ AB( KD+1, I+IB ), 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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* Copy the lower triangle of A13 into the work array.
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*
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DO 40 JJ = 1, I3
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DO 30 II = JJ, IB
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WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 )
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30 CONTINUE
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40 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', 'Upper', 'Transpose',
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$ 'Non-unit', IB, I3, ONE, AB( KD+1, I ),
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$ LDAB-1, WORK, LDWORK )
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*
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* Update A23
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*
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IF( I2.GT.0 )
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$ CALL SGEMM( 'Transpose', 'No Transpose', I2, I3,
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$ IB, -ONE, AB( KD+1-IB, I+IB ),
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$ LDAB-1, WORK, LDWORK, ONE,
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$ AB( 1+IB, I+KD ), LDAB-1 )
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*
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* Update A33
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*
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CALL SSYRK( 'Upper', 'Transpose', I3, IB, -ONE,
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$ WORK, LDWORK, ONE, AB( KD+1, I+KD ),
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$ LDAB-1 )
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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 60 JJ = 1, I3
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DO 50 II = JJ, IB
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AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
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50 CONTINUE
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60 CONTINUE
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END IF
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END IF
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70 CONTINUE
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ELSE
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*
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* Compute the Cholesky factorization of a symmetric band
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* matrix, given the lower triangle of the matrix in band
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* storage.
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*
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* Zero the lower triangle of the work array.
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*
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DO 90 J = 1, NB
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DO 80 I = J + 1, NB
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WORK( I, J ) = ZERO
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80 CONTINUE
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90 CONTINUE
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*
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* Process the band matrix one diagonal block at a time.
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*
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DO 140 I = 1, N, NB
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IB = MIN( NB, N-I+1 )
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*
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* Factorize the diagonal block
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*
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CALL SPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
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IF( II.NE.0 ) THEN
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INFO = I + II - 1
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GO TO 150
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END IF
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IF( I+IB.LE.N ) THEN
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*
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* Update the relevant part of the trailing submatrix.
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* If A11 denotes the diagonal block which has just been
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* factorized, then we need to update the remaining
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* blocks in the diagram:
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*
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* A11
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* A21 A22
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* A31 A32 A33
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*
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* The numbers of rows and columns in the partitioning
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* are IB, I2, I3 respectively. The blocks A21, A22 and
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* A32 are empty if IB = KD. The lower triangle of A31
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* lies outside the band.
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*
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I2 = MIN( KD-IB, N-I-IB+1 )
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I3 = MIN( IB, N-I-KD+1 )
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*
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IF( I2.GT.0 ) THEN
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*
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* Update A21
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*
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CALL STRSM( 'Right', 'Lower', 'Transpose',
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$ 'Non-unit', I2, IB, ONE, AB( 1, I ),
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$ LDAB-1, AB( 1+IB, I ), LDAB-1 )
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*
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* Update A22
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*
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CALL SSYRK( 'Lower', 'No Transpose', I2, IB, -ONE,
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$ AB( 1+IB, I ), LDAB-1, ONE,
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$ AB( 1, I+IB ), 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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* Copy the upper triangle of A31 into the work array.
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*
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DO 110 JJ = 1, IB
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DO 100 II = 1, MIN( JJ, I3 )
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WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
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100 CONTINUE
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110 CONTINUE
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*
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* Update A31 (in the work array).
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*
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CALL STRSM( 'Right', 'Lower', 'Transpose',
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$ 'Non-unit', I3, IB, ONE, AB( 1, I ),
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$ LDAB-1, WORK, LDWORK )
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*
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* Update A32
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*
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IF( I2.GT.0 )
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$ CALL SGEMM( 'No transpose', 'Transpose', I3, I2,
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$ IB, -ONE, WORK, LDWORK,
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$ AB( 1+IB, I ), LDAB-1, ONE,
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$ AB( 1+KD-IB, I+IB ), LDAB-1 )
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*
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* Update A33
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*
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CALL SSYRK( 'Lower', 'No Transpose', I3, IB, -ONE,
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$ WORK, LDWORK, ONE, AB( 1, I+KD ),
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$ LDAB-1 )
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*
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* Copy the upper triangle of A31 back into place.
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*
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DO 130 JJ = 1, IB
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DO 120 II = 1, MIN( JJ, I3 )
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AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
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120 CONTINUE
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130 CONTINUE
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END IF
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END IF
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140 CONTINUE
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END IF
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END IF
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RETURN
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*
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150 CONTINUE
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RETURN
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*
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* End of SPBTRF
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*
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END
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