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SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, |
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$ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, |
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$ POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO ) |
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* |
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* -- LAPACK routine (version 3.2) -- |
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* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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* November 2006 |
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* |
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* .. Scalar Arguments .. |
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INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL, |
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$ LDGNUM, NL, NR, NRHS, SQRE |
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DOUBLE PRECISION C, S |
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* .. |
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* .. Array Arguments .. |
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INTEGER GIVCOL( LDGCOL, * ), PERM( * ) |
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DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ), |
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$ DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ), |
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$ POLES( LDGNUM, * ), WORK( * ), Z( * ) |
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* .. |
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* |
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* Purpose |
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* ======= |
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* |
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* DLALS0 applies back the multiplying factors of either the left or the |
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* right singular vector matrix of a diagonal matrix appended by a row |
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* to the right hand side matrix B in solving the least squares problem |
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* using the divide-and-conquer SVD approach. |
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* |
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* For the left singular vector matrix, three types of orthogonal |
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* matrices are involved: |
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* |
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* (1L) Givens rotations: the number of such rotations is GIVPTR; the |
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* pairs of columns/rows they were applied to are stored in GIVCOL; |
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* and the C- and S-values of these rotations are stored in GIVNUM. |
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* |
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* (2L) Permutation. The (NL+1)-st row of B is to be moved to the first |
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* row, and for J=2:N, PERM(J)-th row of B is to be moved to the |
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* J-th row. |
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* |
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* (3L) The left singular vector matrix of the remaining matrix. |
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* |
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* For the right singular vector matrix, four types of orthogonal |
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* matrices are involved: |
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* |
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* (1R) The right singular vector matrix of the remaining matrix. |
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* |
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* (2R) If SQRE = 1, one extra Givens rotation to generate the right |
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* null space. |
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* |
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* (3R) The inverse transformation of (2L). |
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* |
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* (4R) The inverse transformation of (1L). |
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* |
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* Arguments |
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* ========= |
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* |
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* ICOMPQ (input) INTEGER |
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* Specifies whether singular vectors are to be computed in |
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* factored form: |
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* = 0: Left singular vector matrix. |
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* = 1: Right singular vector matrix. |
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* |
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* NL (input) INTEGER |
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* The row dimension of the upper block. NL >= 1. |
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* |
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* NR (input) INTEGER |
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* The row dimension of the lower block. NR >= 1. |
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* |
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* SQRE (input) INTEGER |
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* = 0: the lower block is an NR-by-NR square matrix. |
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* = 1: the lower block is an NR-by-(NR+1) rectangular matrix. |
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* |
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* The bidiagonal matrix has row dimension N = NL + NR + 1, |
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* and column dimension M = N + SQRE. |
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* |
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* NRHS (input) INTEGER |
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* The number of columns of B and BX. NRHS must be at least 1. |
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* |
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* B (input/output) DOUBLE PRECISION array, dimension ( LDB, NRHS ) |
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* On input, B contains the right hand sides of the least |
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* squares problem in rows 1 through M. On output, B contains |
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* the solution X in rows 1 through N. |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of B. LDB must be at least |
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* max(1,MAX( M, N ) ). |
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* |
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* BX (workspace) DOUBLE PRECISION array, dimension ( LDBX, NRHS ) |
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* |
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* LDBX (input) INTEGER |
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* The leading dimension of BX. |
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* |
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* PERM (input) INTEGER array, dimension ( N ) |
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* The permutations (from deflation and sorting) applied |
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* to the two blocks. |
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* |
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* GIVPTR (input) INTEGER |
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* The number of Givens rotations which took place in this |
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* subproblem. |
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* |
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* GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 ) |
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* Each pair of numbers indicates a pair of rows/columns |
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* involved in a Givens rotation. |
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* |
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* LDGCOL (input) INTEGER |
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* The leading dimension of GIVCOL, must be at least N. |
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* |
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* GIVNUM (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) |
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* Each number indicates the C or S value used in the |
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* corresponding Givens rotation. |
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* |
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* LDGNUM (input) INTEGER |
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* The leading dimension of arrays DIFR, POLES and |
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* GIVNUM, must be at least K. |
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* |
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* POLES (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) |
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* On entry, POLES(1:K, 1) contains the new singular |
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* values obtained from solving the secular equation, and |
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* POLES(1:K, 2) is an array containing the poles in the secular |
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* equation. |
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* |
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* DIFL (input) DOUBLE PRECISION array, dimension ( K ). |
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* On entry, DIFL(I) is the distance between I-th updated |
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* (undeflated) singular value and the I-th (undeflated) old |
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* singular value. |
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* |
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* DIFR (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ). |
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* On entry, DIFR(I, 1) contains the distances between I-th |
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* updated (undeflated) singular value and the I+1-th |
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* (undeflated) old singular value. And DIFR(I, 2) is the |
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* normalizing factor for the I-th right singular vector. |
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* |
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* Z (input) DOUBLE PRECISION array, dimension ( K ) |
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* Contain the components of the deflation-adjusted updating row |
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* vector. |
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* |
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* K (input) INTEGER |
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* Contains the dimension of the non-deflated matrix, |
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* This is the order of the related secular equation. 1 <= K <=N. |
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* |
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* C (input) DOUBLE PRECISION |
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* C contains garbage if SQRE =0 and the C-value of a Givens |
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* rotation related to the right null space if SQRE = 1. |
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* |
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* S (input) DOUBLE PRECISION |
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* S contains garbage if SQRE =0 and the S-value of a Givens |
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* rotation related to the right null space if SQRE = 1. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension ( K ) |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit. |
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* < 0: if INFO = -i, the i-th argument had an illegal value. |
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* |
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* Further Details |
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* =============== |
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* |
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* Based on contributions by |
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* Ming Gu and Ren-Cang Li, Computer Science Division, University of |
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* California at Berkeley, USA |
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* Osni Marques, LBNL/NERSC, USA |
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* |
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* ===================================================================== |
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* |
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* .. Parameters .. |
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DOUBLE PRECISION ONE, ZERO, NEGONE |
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PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, NEGONE = -1.0D0 ) |
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* .. |
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* .. Local Scalars .. |
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INTEGER I, J, M, N, NLP1 |
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DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP |
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* .. |
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* .. External Subroutines .. |
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EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DROT, DSCAL, |
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$ XERBLA |
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* .. |
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* .. External Functions .. |
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DOUBLE PRECISION DLAMC3, DNRM2 |
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EXTERNAL DLAMC3, DNRM2 |
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* .. |
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* .. Intrinsic Functions .. |
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INTRINSIC MAX |
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* .. |
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* .. Executable Statements .. |
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* |
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* Test the input parameters. |
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* |
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INFO = 0 |
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* |
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IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN |
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INFO = -1 |
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ELSE IF( NL.LT.1 ) THEN |
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INFO = -2 |
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ELSE IF( NR.LT.1 ) THEN |
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INFO = -3 |
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ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN |
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INFO = -4 |
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END IF |
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* |
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N = NL + NR + 1 |
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* |
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IF( NRHS.LT.1 ) THEN |
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INFO = -5 |
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ELSE IF( LDB.LT.N ) THEN |
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INFO = -7 |
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ELSE IF( LDBX.LT.N ) THEN |
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INFO = -9 |
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ELSE IF( GIVPTR.LT.0 ) THEN |
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INFO = -11 |
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ELSE IF( LDGCOL.LT.N ) THEN |
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INFO = -13 |
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ELSE IF( LDGNUM.LT.N ) THEN |
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INFO = -15 |
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ELSE IF( K.LT.1 ) THEN |
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INFO = -20 |
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END IF |
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IF( INFO.NE.0 ) THEN |
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CALL XERBLA( 'DLALS0', -INFO ) |
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RETURN |
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END IF |
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* |
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M = N + SQRE |
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NLP1 = NL + 1 |
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* |
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IF( ICOMPQ.EQ.0 ) THEN |
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* |
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* Apply back orthogonal transformations from the left. |
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* |
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* Step (1L): apply back the Givens rotations performed. |
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* |
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DO 10 I = 1, GIVPTR |
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CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB, |
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$ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ), |
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$ GIVNUM( I, 1 ) ) |
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10 CONTINUE |
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* |
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* Step (2L): permute rows of B. |
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* |
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CALL DCOPY( NRHS, B( NLP1, 1 ), LDB, BX( 1, 1 ), LDBX ) |
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DO 20 I = 2, N |
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CALL DCOPY( NRHS, B( PERM( I ), 1 ), LDB, BX( I, 1 ), LDBX ) |
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20 CONTINUE |
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* |
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* Step (3L): apply the inverse of the left singular vector |
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* matrix to BX. |
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* |
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IF( K.EQ.1 ) THEN |
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CALL DCOPY( NRHS, BX, LDBX, B, LDB ) |
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IF( Z( 1 ).LT.ZERO ) THEN |
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CALL DSCAL( NRHS, NEGONE, B, LDB ) |
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END IF |
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ELSE |
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DO 50 J = 1, K |
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DIFLJ = DIFL( J ) |
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DJ = POLES( J, 1 ) |
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DSIGJ = -POLES( J, 2 ) |
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IF( J.LT.K ) THEN |
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DIFRJ = -DIFR( J, 1 ) |
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DSIGJP = -POLES( J+1, 2 ) |
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END IF |
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IF( ( Z( J ).EQ.ZERO ) .OR. ( POLES( J, 2 ).EQ.ZERO ) ) |
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$ THEN |
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WORK( J ) = ZERO |
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ELSE |
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WORK( J ) = -POLES( J, 2 )*Z( J ) / DIFLJ / |
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$ ( POLES( J, 2 )+DJ ) |
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END IF |
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DO 30 I = 1, J - 1 |
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IF( ( Z( I ).EQ.ZERO ) .OR. |
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$ ( POLES( I, 2 ).EQ.ZERO ) ) THEN |
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WORK( I ) = ZERO |
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ELSE |
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WORK( I ) = POLES( I, 2 )*Z( I ) / |
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$ ( DLAMC3( POLES( I, 2 ), DSIGJ )- |
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$ DIFLJ ) / ( POLES( I, 2 )+DJ ) |
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END IF |
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30 CONTINUE |
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DO 40 I = J + 1, K |
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IF( ( Z( I ).EQ.ZERO ) .OR. |
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$ ( POLES( I, 2 ).EQ.ZERO ) ) THEN |
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WORK( I ) = ZERO |
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ELSE |
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WORK( I ) = POLES( I, 2 )*Z( I ) / |
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$ ( DLAMC3( POLES( I, 2 ), DSIGJP )+ |
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$ DIFRJ ) / ( POLES( I, 2 )+DJ ) |
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END IF |
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40 CONTINUE |
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WORK( 1 ) = NEGONE |
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TEMP = DNRM2( K, WORK, 1 ) |
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CALL DGEMV( 'T', K, NRHS, ONE, BX, LDBX, WORK, 1, ZERO, |
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$ B( J, 1 ), LDB ) |
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CALL DLASCL( 'G', 0, 0, TEMP, ONE, 1, NRHS, B( J, 1 ), |
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$ LDB, INFO ) |
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50 CONTINUE |
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END IF |
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* |
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* Move the deflated rows of BX to B also. |
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* |
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IF( K.LT.MAX( M, N ) ) |
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$ CALL DLACPY( 'A', N-K, NRHS, BX( K+1, 1 ), LDBX, |
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$ B( K+1, 1 ), LDB ) |
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ELSE |
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* |
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* Apply back the right orthogonal transformations. |
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* |
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* Step (1R): apply back the new right singular vector matrix |
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* to B. |
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* |
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IF( K.EQ.1 ) THEN |
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CALL DCOPY( NRHS, B, LDB, BX, LDBX ) |
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ELSE |
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DO 80 J = 1, K |
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DSIGJ = POLES( J, 2 ) |
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IF( Z( J ).EQ.ZERO ) THEN |
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WORK( J ) = ZERO |
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ELSE |
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WORK( J ) = -Z( J ) / DIFL( J ) / |
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$ ( DSIGJ+POLES( J, 1 ) ) / DIFR( J, 2 ) |
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END IF |
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DO 60 I = 1, J - 1 |
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IF( Z( J ).EQ.ZERO ) THEN |
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WORK( I ) = ZERO |
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ELSE |
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WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I+1, |
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$ 2 ) )-DIFR( I, 1 ) ) / |
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$ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 ) |
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END IF |
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60 CONTINUE |
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DO 70 I = J + 1, K |
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IF( Z( J ).EQ.ZERO ) THEN |
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WORK( I ) = ZERO |
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ELSE |
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WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I, |
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$ 2 ) )-DIFL( I ) ) / |
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$ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 ) |
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END IF |
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70 CONTINUE |
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CALL DGEMV( 'T', K, NRHS, ONE, B, LDB, WORK, 1, ZERO, |
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$ BX( J, 1 ), LDBX ) |
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80 CONTINUE |
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END IF |
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* |
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* Step (2R): if SQRE = 1, apply back the rotation that is |
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* related to the right null space of the subproblem. |
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* |
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IF( SQRE.EQ.1 ) THEN |
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CALL DCOPY( NRHS, B( M, 1 ), LDB, BX( M, 1 ), LDBX ) |
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CALL DROT( NRHS, BX( 1, 1 ), LDBX, BX( M, 1 ), LDBX, C, S ) |
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END IF |
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IF( K.LT.MAX( M, N ) ) |
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$ CALL DLACPY( 'A', N-K, NRHS, B( K+1, 1 ), LDB, BX( K+1, 1 ), |
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$ LDBX ) |
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* |
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* Step (3R): permute rows of B. |
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* |
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CALL DCOPY( NRHS, BX( 1, 1 ), LDBX, B( NLP1, 1 ), LDB ) |
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IF( SQRE.EQ.1 ) THEN |
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CALL DCOPY( NRHS, BX( M, 1 ), LDBX, B( M, 1 ), LDB ) |
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END IF |
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DO 90 I = 2, N |
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CALL DCOPY( NRHS, BX( I, 1 ), LDBX, B( PERM( I ), 1 ), LDB ) |
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90 CONTINUE |
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* |
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* Step (4R): apply back the Givens rotations performed. |
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* |
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DO 100 I = GIVPTR, 1, -1 |
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CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB, |
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$ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ), |
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$ -GIVNUM( I, 1 ) ) |
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100 CONTINUE |
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END IF |
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* |
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RETURN |
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* |
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* End of DLALS0 |
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* |
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END |