*> \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 *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \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