*> \brief \b DPBSTF
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBSTF computes a split Cholesky factorization of a real
*> symmetric positive definite band matrix A.
*>
*> This routine is designed to be used in conjunction with DSBGST.
*>
*> The factorization has the form A = S**T*S where S is a band matrix
*> of the same bandwidth as A and the following structure:
*>
*> S = ( U )
*> ( M L )
*>
*> where U is upper triangular of order m = (n+kd)/2, and L is lower
*> triangular of order n-m.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \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 superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION 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 factor S from the split Cholesky
*> factorization A = S**T*S. See Further Details.
*> \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 = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the factorization could not be completed,
*> because the updated element a(i,i) was negative; the
*> matrix A is not positive definite.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup doubleOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> N = 7, KD = 2:
*>
*> S = ( s11 s12 s13 )
*> ( s22 s23 s24 )
*> ( s33 s34 )
*> ( s44 )
*> ( s53 s54 s55 )
*> ( s64 s65 s66 )
*> ( s75 s76 s77 )
*>
*> If UPLO = 'U', the array AB holds:
*>
*> on entry: on exit:
*>
*> * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
*> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
*>
*> If UPLO = 'L', the array AB holds:
*>
*> on entry: on exit:
*>
*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
*> a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
*> a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *
*>
*> Array elements marked * are not used by the routine.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPBSTF( 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 ..
DOUBLE PRECISION AB( LDAB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, KLD, KM, M
DOUBLE PRECISION AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSYR, 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( 'DPBSTF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
KLD = MAX( 1, LDAB-1 )
*
* Set the splitting point m.
*
M = ( N+KD ) / 2
*
IF( UPPER ) THEN
*
* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
*
DO 10 J = N, M + 1, -1
*
* Compute s(j,j) and test for non-positive-definiteness.
*
AJJ = AB( KD+1, J )
IF( AJJ.LE.ZERO )
$ GO TO 50
AJJ = SQRT( AJJ )
AB( KD+1, J ) = AJJ
KM = MIN( J-1, KD )
*
* Compute elements j-km:j-1 of the j-th column and update the
* the leading submatrix within the band.
*
CALL DSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
CALL DSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
$ AB( KD+1, J-KM ), KLD )
10 CONTINUE
*
* Factorize the updated submatrix A(1:m,1:m) as U**T*U.
*
DO 20 J = 1, M
*
* Compute s(j,j) and test for non-positive-definiteness.
*
AJJ = AB( KD+1, J )
IF( AJJ.LE.ZERO )
$ GO TO 50
AJJ = SQRT( AJJ )
AB( KD+1, J ) = AJJ
KM = MIN( KD, M-J )
*
* Compute elements j+1:j+km of the j-th row and update the
* trailing submatrix within the band.
*
IF( KM.GT.0 ) THEN
CALL DSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
CALL DSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
$ AB( KD+1, J+1 ), KLD )
END IF
20 CONTINUE
ELSE
*
* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
*
DO 30 J = N, M + 1, -1
*
* Compute s(j,j) and test for non-positive-definiteness.
*
AJJ = AB( 1, J )
IF( AJJ.LE.ZERO )
$ GO TO 50
AJJ = SQRT( AJJ )
AB( 1, J ) = AJJ
KM = MIN( J-1, KD )
*
* Compute elements j-km:j-1 of the j-th row and update the
* trailing submatrix within the band.
*
CALL DSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
CALL DSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
$ AB( 1, J-KM ), KLD )
30 CONTINUE
*
* Factorize the updated submatrix A(1:m,1:m) as U**T*U.
*
DO 40 J = 1, M
*
* Compute s(j,j) and test for non-positive-definiteness.
*
AJJ = AB( 1, J )
IF( AJJ.LE.ZERO )
$ GO TO 50
AJJ = SQRT( AJJ )
AB( 1, J ) = AJJ
KM = MIN( KD, M-J )
*
* Compute elements j+1:j+km of the j-th column and update the
* trailing submatrix within the band.
*
IF( KM.GT.0 ) THEN
CALL DSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
CALL DSYR( 'Lower', KM, -ONE, AB( 2, J ), 1,
$ AB( 1, J+1 ), KLD )
END IF
40 CONTINUE
END IF
RETURN
*
50 CONTINUE
INFO = J
RETURN
*
* End of DPBSTF
*
END