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*> \brief \b SPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SPBTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/spbtf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/spbtf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/spbtf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SPBTF2( UPLO, N, KD, AB, LDAB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
* REAL AB( LDAB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SPBTF2 computes the Cholesky factorization of a real symmetric
*> positive definite band matrix A.
*>
*> The factorization has the form
*> A = U**T * U , if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix, U**T is the transpose of U, and
*> L is lower triangular.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of super-diagonals of the matrix A if UPLO = 'U',
*> or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is REAL array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*>
*> On exit, if INFO = 0, the triangular factor U or L from the
*> Cholesky factorization A = U**T*U or A = L*L**T of the band
*> matrix A, in the same storage format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> > 0: if INFO = k, the leading minor of order k is not
*> positive definite, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup realOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> N = 6, KD = 2, and UPLO = 'U':
*>
*> On entry: On exit:
*>
*> * * a13 a24 a35 a46 * * 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
*>
*> Similarly, if UPLO = 'L' the format of A is as follows:
*>
*> On entry: On exit:
*>
*> a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
*> a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
*> a31 a42 a53 a64 * * l31 l42 l53 l64 * *
*>
*> Array elements marked * are not used by the routine.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SPBTF2( UPLO, N, KD, AB, LDAB, 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 ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
REAL AB( LDAB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, KLD, KN
REAL AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL SSCAL, SSYR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SPBTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
KLD = MAX( 1, LDAB-1 )
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U**T*U.
*
DO 10 J = 1, N
*
* Compute U(J,J) and test for non-positive-definiteness.
*
AJJ = AB( KD+1, J )
IF( AJJ.LE.ZERO )
$ GO TO 30
AJJ = SQRT( AJJ )
AB( KD+1, J ) = AJJ
*
* Compute elements J+1:J+KN of row J and update the
* trailing submatrix within the band.
*
KN = MIN( KD, N-J )
IF( KN.GT.0 ) THEN
CALL SSCAL( KN, ONE / AJJ, AB( KD, J+1 ), KLD )
CALL SSYR( 'Upper', KN, -ONE, AB( KD, J+1 ), KLD,
$ AB( KD+1, J+1 ), KLD )
END IF
10 CONTINUE
ELSE
*
* Compute the Cholesky factorization A = L*L**T.
*
DO 20 J = 1, N
*
* Compute L(J,J) and test for non-positive-definiteness.
*
AJJ = AB( 1, J )
IF( AJJ.LE.ZERO )
$ GO TO 30
AJJ = SQRT( AJJ )
AB( 1, J ) = AJJ
*
* Compute elements J+1:J+KN of column J and update the
* trailing submatrix within the band.
*
KN = MIN( KD, N-J )
IF( KN.GT.0 ) THEN
CALL SSCAL( KN, ONE / AJJ, AB( 2, J ), 1 )
CALL SSYR( 'Lower', KN, -ONE, AB( 2, J ), 1,
$ AB( 1, J+1 ), KLD )
END IF
20 CONTINUE
END IF
RETURN
*
30 CONTINUE
INFO = J
RETURN
*
* End of SPBTF2
*
END
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