*> \brief \b SPBTRF * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SPBTRF + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SPBTRF( UPLO, N, KD, AB, LDAB, INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, KD, LDAB, N * .. * .. Array Arguments .. * REAL AB( LDAB, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SPBTRF 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 and L is lower triangular. *> \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 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 = -i, the i-th argument had an illegal value *> > 0: if INFO = i, the leading minor of order i 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. * *> \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 * *> \par Contributors: * ================== *> *> Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989 * * ===================================================================== SUBROUTINE SPBTRF( UPLO, N, KD, AB, LDAB, INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. 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 ) INTEGER NBMAX, LDWORK PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 ) * .. * .. Local Scalars .. INTEGER I, I2, I3, IB, II, J, JJ, NB * .. * .. Local Arrays .. REAL WORK( LDWORK, NBMAX ) * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV * .. * .. External Subroutines .. EXTERNAL SGEMM, SPBTF2, SPOTF2, SSYRK, STRSM, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MIN * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF( ( .NOT.LSAME( UPLO, 'U' ) ) .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( 'SPBTRF', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * * Determine the block size for this environment * NB = ILAENV( 1, 'SPBTRF', UPLO, N, KD, -1, -1 ) * * The block size must not exceed the semi-bandwidth KD, and must not * exceed the limit set by the size of the local array WORK. * NB = MIN( NB, NBMAX ) * IF( NB.LE.1 .OR. NB.GT.KD ) THEN * * Use unblocked code * CALL SPBTF2( UPLO, N, KD, AB, LDAB, INFO ) ELSE * * Use blocked code * IF( LSAME( UPLO, 'U' ) ) THEN * * Compute the Cholesky factorization of a symmetric band * matrix, given the upper triangle of the matrix in band * storage. * * Zero the upper triangle of the work array. * DO 20 J = 1, NB DO 10 I = 1, J - 1 WORK( I, J ) = ZERO 10 CONTINUE 20 CONTINUE * * Process the band matrix one diagonal block at a time. * DO 70 I = 1, N, NB IB = MIN( NB, N-I+1 ) * * Factorize the diagonal block * CALL SPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II ) IF( II.NE.0 ) THEN INFO = I + II - 1 GO TO 150 END IF IF( I+IB.LE.N ) THEN * * Update the relevant part of the trailing submatrix. * If A11 denotes the diagonal block which has just been * factorized, then we need to update the remaining * blocks in the diagram: * * A11 A12 A13 * A22 A23 * A33 * * The numbers of rows and columns in the partitioning * are IB, I2, I3 respectively. The blocks A12, A22 and * A23 are empty if IB = KD. The upper triangle of A13 * lies outside the band. * I2 = MIN( KD-IB, N-I-IB+1 ) I3 = MIN( IB, N-I-KD+1 ) * IF( I2.GT.0 ) THEN * * Update A12 * CALL STRSM( 'Left', 'Upper', 'Transpose', \$ 'Non-unit', IB, I2, ONE, AB( KD+1, I ), \$ LDAB-1, AB( KD+1-IB, I+IB ), LDAB-1 ) * * Update A22 * CALL SSYRK( 'Upper', 'Transpose', I2, IB, -ONE, \$ AB( KD+1-IB, I+IB ), LDAB-1, ONE, \$ AB( KD+1, I+IB ), LDAB-1 ) END IF * IF( I3.GT.0 ) THEN * * Copy the lower triangle of A13 into the work array. * DO 40 JJ = 1, I3 DO 30 II = JJ, IB WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 ) 30 CONTINUE 40 CONTINUE * * Update A13 (in the work array). * CALL STRSM( 'Left', 'Upper', 'Transpose', \$ 'Non-unit', IB, I3, ONE, AB( KD+1, I ), \$ LDAB-1, WORK, LDWORK ) * * Update A23 * IF( I2.GT.0 ) \$ CALL SGEMM( 'Transpose', 'No Transpose', I2, I3, \$ IB, -ONE, AB( KD+1-IB, I+IB ), \$ LDAB-1, WORK, LDWORK, ONE, \$ AB( 1+IB, I+KD ), LDAB-1 ) * * Update A33 * CALL SSYRK( 'Upper', 'Transpose', I3, IB, -ONE, \$ WORK, LDWORK, ONE, AB( KD+1, I+KD ), \$ LDAB-1 ) * * Copy the lower triangle of A13 back into place. * DO 60 JJ = 1, I3 DO 50 II = JJ, IB AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ ) 50 CONTINUE 60 CONTINUE END IF END IF 70 CONTINUE ELSE * * Compute the Cholesky factorization of a symmetric band * matrix, given the lower triangle of the matrix in band * storage. * * Zero the lower triangle of the work array. * DO 90 J = 1, NB DO 80 I = J + 1, NB WORK( I, J ) = ZERO 80 CONTINUE 90 CONTINUE * * Process the band matrix one diagonal block at a time. * DO 140 I = 1, N, NB IB = MIN( NB, N-I+1 ) * * Factorize the diagonal block * CALL SPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II ) IF( II.NE.0 ) THEN INFO = I + II - 1 GO TO 150 END IF IF( I+IB.LE.N ) THEN * * Update the relevant part of the trailing submatrix. * If A11 denotes the diagonal block which has just been * factorized, then we need to update the remaining * blocks in the diagram: * * A11 * A21 A22 * A31 A32 A33 * * The numbers of rows and columns in the partitioning * are IB, I2, I3 respectively. The blocks A21, A22 and * A32 are empty if IB = KD. The lower triangle of A31 * lies outside the band. * I2 = MIN( KD-IB, N-I-IB+1 ) I3 = MIN( IB, N-I-KD+1 ) * IF( I2.GT.0 ) THEN * * Update A21 * CALL STRSM( 'Right', 'Lower', 'Transpose', \$ 'Non-unit', I2, IB, ONE, AB( 1, I ), \$ LDAB-1, AB( 1+IB, I ), LDAB-1 ) * * Update A22 * CALL SSYRK( 'Lower', 'No Transpose', I2, IB, -ONE, \$ AB( 1+IB, I ), LDAB-1, ONE, \$ AB( 1, I+IB ), LDAB-1 ) END IF * IF( I3.GT.0 ) THEN * * Copy the upper triangle of A31 into the work array. * DO 110 JJ = 1, IB DO 100 II = 1, MIN( JJ, I3 ) WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 ) 100 CONTINUE 110 CONTINUE * * Update A31 (in the work array). * CALL STRSM( 'Right', 'Lower', 'Transpose', \$ 'Non-unit', I3, IB, ONE, AB( 1, I ), \$ LDAB-1, WORK, LDWORK ) * * Update A32 * IF( I2.GT.0 ) \$ CALL SGEMM( 'No transpose', 'Transpose', I3, I2, \$ IB, -ONE, WORK, LDWORK, \$ AB( 1+IB, I ), LDAB-1, ONE, \$ AB( 1+KD-IB, I+IB ), LDAB-1 ) * * Update A33 * CALL SSYRK( 'Lower', 'No Transpose', I3, IB, -ONE, \$ WORK, LDWORK, ONE, AB( 1, I+KD ), \$ LDAB-1 ) * * Copy the upper triangle of A31 back into place. * DO 130 JJ = 1, IB DO 120 II = 1, MIN( JJ, I3 ) AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ ) 120 CONTINUE 130 CONTINUE END IF END IF 140 CONTINUE END IF END IF RETURN * 150 CONTINUE RETURN * * End of SPBTRF * END