SUBROUTINE DDBTRF( M, N, KL, KU, AB, LDAB, INFO ) * * -- ScaLAPACK auxiliary routine (version 2.0) -- * Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver * * Written by Andrew J. Cleary, University of Tennessee. * August, 1996. * Modified from DGBTRF: * -- LAPACK routine (preliminary version) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * August 6, 1991 * * .. Scalar Arguments .. INTEGER INFO, KL, KU, LDAB, M, N * .. * .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ) * .. * * Purpose * ======= * * Ddbtrf computes an LU factorization of a real m-by-n band matrix A * without using partial pivoting or row interchanges. * * This is the blocked version of the algorithm, calling Level 3 BLAS. * * Arguments * ========= * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * N (input) INTEGER * The number of columns of the matrix A. N >= 0. * * KL (input) INTEGER * The number of subdiagonals within the band of A. KL >= 0. * * KU (input) INTEGER * The number of superdiagonals within the band of A. KU >= 0. * * AB (input/output) REAL array, dimension (LDAB,N) * On entry, the matrix A in band storage, in rows KL+1 to * 2*KL+KU+1; rows 1 to KL of the array need not be set. * The j-th column of A is stored in the j-th column of the * array AB as follows: * AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) * * On exit, details of the factorization: U is stored as an * upper triangular band matrix with KL+KU superdiagonals in * rows 1 to KL+KU+1, and the multipliers used during the * factorization are stored in rows KL+KU+2 to 2*KL+KU+1. * See below for further details. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= 2*KL+KU+1. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = +i, U(i,i) is exactly zero. The factorization * has been completed, but the factor U is exactly * singular, and division by zero will occur if it is used * to solve a system of equations. * * Further Details * =============== * * The band storage scheme is illustrated by the following example, when * M = N = 6, KL = 2, KU = 1: * * On entry: On exit: * * * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 * a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 * a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * * a31 a42 a53 a64 * * m31 m42 m53 m64 * * * * Array elements marked * are not used by the routine. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0 ) PARAMETER ( ZERO = 0.0D+0 ) INTEGER NBMAX, LDWORK PARAMETER ( NBMAX = 64, LDWORK = NBMAX+1 ) * .. * .. Local Scalars .. INTEGER I, I2, I3, II, J, J2, J3, JB, JJ, JM, JP, $ JU, KM, KV, NB, NW * .. * .. Local Arrays .. DOUBLE PRECISION WORK13( LDWORK, NBMAX ), $ WORK31( LDWORK, NBMAX ) * .. * .. External Functions .. INTEGER ILAENV, ISAMAX EXTERNAL ILAENV, ISAMAX * .. * .. External Subroutines .. EXTERNAL DCOPY, DDBTF2, DGEMM, DGER, DSCAL, $ DSWAP, DTRSM, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * * KV is the number of superdiagonals in the factor U * KV = KU * * Test the input parameters. * INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KL.LT.0 ) THEN INFO = -3 ELSE IF( KU.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.MIN( MIN( KL+KV+1,M ),N ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DDBTRF', -INFO ) RETURN END IF * * Quick return if possible * IF( M.EQ.0 .OR. N.EQ.0 ) $ RETURN * * Determine the block size for this environment * NB = ILAENV( 1, 'DDBTRF', ' ', M, N, KL, KU ) * * The block size must not exceed the limit set by the size of the * local arrays WORK13 and WORK31. * NB = MIN( NB, NBMAX ) * IF( NB.LE.1 .OR. NB.GT.KL ) THEN * * Use unblocked code * CALL DDBTF2( M, N, KL, KU, AB, LDAB, INFO ) ELSE * * Use blocked code * * Zero the superdiagonal elements of the work array WORK13 * DO 20 J = 1, NB DO 10 I = 1, J - 1 WORK13( I, J ) = ZERO 10 CONTINUE 20 CONTINUE * * Zero the subdiagonal elements of the work array WORK31 * DO 40 J = 1, NB DO 30 I = J + 1, NB WORK31( I, J ) = ZERO 30 CONTINUE 40 CONTINUE * * JU is the index of the last column affected by the current * stage of the factorization * JU = 1 * DO 180 J = 1, MIN( M, N ), NB JB = MIN( NB, MIN( M, N )-J+1 ) * * The active part of the matrix is partitioned * * A11 A12 A13 * A21 A22 A23 * A31 A32 A33 * * Here A11, A21 and A31 denote the current block of JB columns * which is about to be factorized. The number of rows in the * partitioning are JB, I2, I3 respectively, and the numbers * of columns are JB, J2, J3. The superdiagonal elements of A13 * and the subdiagonal elements of A31 lie outside the band. * I2 = MIN( KL-JB, M-J-JB+1 ) I3 = MIN( JB, M-J-KL+1 ) * * J2 and J3 are computed after JU has been updated. * * Factorize the current block of JB columns * DO 80 JJ = J, J + JB - 1 * * Find pivot and test for singularity. KM is the number of * subdiagonal elements in the current column. * KM = MIN( KL, M-JJ ) JP = 1 IF( AB( KV+JP, JJ ).NE.ZERO ) THEN JU = MAX( JU, MIN( JJ+KU+JP-1, N ) ) * * Compute multipliers * CALL DSCAL( KM, ONE / AB( KV+1, JJ ), AB( KV+2, JJ ), $ 1 ) * * Update trailing submatrix within the band and within * the current block. JM is the index of the last column * which needs to be updated. * JM = MIN( JU, J+JB-1 ) IF( JM.GT.JJ ) THEN CALL DGER( KM, JM-JJ, -ONE, AB( KV+2, JJ ), 1, $ AB( KV, JJ+1 ), LDAB-1, $ AB( KV+1, JJ+1 ), LDAB-1 ) END IF END IF * * Copy current column of A31 into the work array WORK31 * NW = MIN( JJ-J+1, I3 ) IF( NW.GT.0 ) $ CALL DCOPY( NW, AB( KV+KL+1-JJ+J, JJ ), 1, $ WORK31( 1, JJ-J+1 ), 1 ) 80 CONTINUE IF( J+JB.LE.N ) THEN * * Apply the row interchanges to the other blocks. * J2 = MIN( JU-J+1, KV ) - JB J3 = MAX( 0, JU-J-KV+1 ) * * Update the relevant part of the trailing submatrix * IF( J2.GT.0 ) THEN * * Update A12 * CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', $ JB, J2, ONE, AB( KV+1, J ), LDAB-1, $ AB( KV+1-JB, J+JB ), LDAB-1 ) * IF( I2.GT.0 ) THEN * * Update A22 * CALL DGEMM( 'No transpose', 'No transpose', I2, J2, $ JB, -ONE, AB( KV+1+JB, J ), LDAB-1, $ AB( KV+1-JB, J+JB ), LDAB-1, ONE, $ AB( KV+1, J+JB ), LDAB-1 ) END IF * IF( I3.GT.0 ) THEN * * Update A32 * CALL DGEMM( 'No transpose', 'No transpose', I3, J2, $ JB, -ONE, WORK31, LDWORK, $ AB( KV+1-JB, J+JB ), LDAB-1, ONE, $ AB( KV+KL+1-JB, J+JB ), LDAB-1 ) END IF END IF * IF( J3.GT.0 ) THEN * * Copy the lower triangle of A13 into the work array * WORK13 * DO 130 JJ = 1, J3 DO 120 II = JJ, JB WORK13( II, JJ ) = AB( II-JJ+1, JJ+J+KV-1 ) 120 CONTINUE 130 CONTINUE * * Update A13 in the work array * CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', $ JB, J3, ONE, AB( KV+1, J ), LDAB-1, $ WORK13, LDWORK ) * IF( I2.GT.0 ) THEN * * Update A23 * CALL DGEMM( 'No transpose', 'No transpose', I2, J3, $ JB, -ONE, AB( KV+1+JB, J ), LDAB-1, $ WORK13, LDWORK, ONE, AB( 1+JB, J+KV ), $ LDAB-1 ) END IF * IF( I3.GT.0 ) THEN * * Update A33 * CALL DGEMM( 'No transpose', 'No transpose', I3, J3, $ JB, -ONE, WORK31, LDWORK, WORK13, $ LDWORK, ONE, AB( 1+KL, J+KV ), LDAB-1 ) END IF * * Copy the lower triangle of A13 back into place * DO 150 JJ = 1, J3 DO 140 II = JJ, JB AB( II-JJ+1, JJ+J+KV-1 ) = WORK13( II, JJ ) 140 CONTINUE 150 CONTINUE END IF ELSE END IF * * copy the upper triangle of A31 back into place * DO 170 JJ = J + JB - 1, J, -1 * * Copy the current column of A31 back into place * NW = MIN( I3, JJ-J+1 ) IF( NW.GT.0 ) $ CALL DCOPY( NW, WORK31( 1, JJ-J+1 ), 1, $ AB( KV+KL+1-JJ+J, JJ ), 1 ) 170 CONTINUE 180 CONTINUE END IF * RETURN * * End of DDBTRF * END