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316 lines
8.9 KiB
316 lines
8.9 KiB
*> \brief \b SPBSTF
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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 SPBSTF + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/spbstf.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/spbstf.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/spbstf.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 SPBSTF( 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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*> SPBSTF computes a split Cholesky factorization of a real
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*> symmetric positive definite band matrix A.
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*>
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*> This routine is designed to be used in conjunction with SSBGST.
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*>
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*> The factorization has the form A = S**T*S where S is a band matrix
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*> of the same bandwidth as A and the following structure:
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*>
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*> S = ( U )
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*> ( M L )
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*>
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*> where U is upper triangular of order m = (n+kd)/2, and L is lower
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*> triangular of order n-m.
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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 factor S from the split Cholesky
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*> factorization A = S**T*S. See 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 >= 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 factorization could not be completed,
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*> because the updated element a(i,i) was negative; the
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*> matrix A is not positive definite.
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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 = 7, KD = 2:
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*>
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*> S = ( s11 s12 s13 )
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*> ( s22 s23 s24 )
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*> ( s33 s34 )
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*> ( s44 )
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*> ( s53 s54 s55 )
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*> ( s64 s65 s66 )
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*> ( s75 s76 s77 )
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*>
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*> If UPLO = 'U', the array AB holds:
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*>
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*> on entry: on exit:
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*>
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*> * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
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*> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
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*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
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*>
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*> If UPLO = 'L', the array AB holds:
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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 a77 s11 s22 s33 s44 s55 s66 s77
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*> a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
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*> a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *
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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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* =====================================================================
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SUBROUTINE SPBSTF( 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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* ..
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* .. Local Scalars ..
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LOGICAL UPPER
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INTEGER J, KLD, KM, M
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REAL AJJ
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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* ..
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* .. External Subroutines ..
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EXTERNAL SSCAL, SSYR, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN, SQRT
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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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UPPER = LSAME( UPLO, 'U' )
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IF( .NOT.UPPER .AND. .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( 'SPBSTF', -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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KLD = MAX( 1, LDAB-1 )
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*
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* Set the splitting point m.
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*
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M = ( N+KD ) / 2
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*
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IF( UPPER ) THEN
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*
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* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
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*
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DO 10 J = N, M + 1, -1
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*
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* Compute s(j,j) and test for non-positive-definiteness.
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*
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AJJ = AB( KD+1, J )
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IF( AJJ.LE.ZERO )
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$ GO TO 50
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AJJ = SQRT( AJJ )
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AB( KD+1, J ) = AJJ
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KM = MIN( J-1, KD )
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*
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* Compute elements j-km:j-1 of the j-th column and update the
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* the leading submatrix within the band.
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*
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CALL SSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
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CALL SSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
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$ AB( KD+1, J-KM ), KLD )
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10 CONTINUE
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*
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* Factorize the updated submatrix A(1:m,1:m) as U**T*U.
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*
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DO 20 J = 1, M
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*
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* Compute s(j,j) and test for non-positive-definiteness.
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*
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AJJ = AB( KD+1, J )
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IF( AJJ.LE.ZERO )
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$ GO TO 50
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AJJ = SQRT( AJJ )
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AB( KD+1, J ) = AJJ
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KM = MIN( KD, M-J )
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*
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* Compute elements j+1:j+km of the j-th row and update the
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* trailing submatrix within the band.
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*
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IF( KM.GT.0 ) THEN
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CALL SSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
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CALL SSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
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$ AB( KD+1, J+1 ), KLD )
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END IF
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20 CONTINUE
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ELSE
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*
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* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
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*
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DO 30 J = N, M + 1, -1
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*
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* Compute s(j,j) and test for non-positive-definiteness.
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*
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AJJ = AB( 1, J )
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IF( AJJ.LE.ZERO )
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$ GO TO 50
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AJJ = SQRT( AJJ )
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AB( 1, J ) = AJJ
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KM = MIN( J-1, KD )
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*
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* Compute elements j-km:j-1 of the j-th row and update the
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* trailing submatrix within the band.
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*
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CALL SSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
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CALL SSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
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$ AB( 1, J-KM ), KLD )
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30 CONTINUE
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*
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* Factorize the updated submatrix A(1:m,1:m) as U**T*U.
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*
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DO 40 J = 1, M
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*
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* Compute s(j,j) and test for non-positive-definiteness.
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*
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AJJ = AB( 1, J )
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IF( AJJ.LE.ZERO )
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$ GO TO 50
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AJJ = SQRT( AJJ )
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AB( 1, J ) = AJJ
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KM = MIN( KD, M-J )
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*
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* Compute elements j+1:j+km of the j-th column and update the
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* trailing submatrix within the band.
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*
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IF( KM.GT.0 ) THEN
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CALL SSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
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CALL SSYR( 'Lower', KM, -ONE, AB( 2, J ), 1,
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$ AB( 1, J+1 ), KLD )
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END IF
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40 CONTINUE
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END IF
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RETURN
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*
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50 CONTINUE
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INFO = J
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RETURN
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*
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* End of SPBSTF
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*
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END
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