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SUBROUTINE SLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,
$ DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, $ PERM, GIVNUM, C, S, WORK, IWORK, INFO ) * * -- LAPACK auxiliary routine (version 3.2.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2010 * * .. Scalar Arguments .. INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE * .. * .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), $ K( * ), PERM( LDGCOL, * ) REAL C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ), $ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ), $ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ), $ Z( LDU, * ) * .. * * Purpose * ======= * * Using a divide and conquer approach, SLASDA computes the singular * value decomposition (SVD) of a real upper bidiagonal N-by-M matrix * B with diagonal D and offdiagonal E, where M = N + SQRE. The * algorithm computes the singular values in the SVD B = U * S * VT. * The orthogonal matrices U and VT are optionally computed in * compact form. * * A related subroutine, SLASD0, computes the singular values and * the singular vectors in explicit form. * * Arguments * ========= * * ICOMPQ (input) INTEGER * Specifies whether singular vectors are to be computed * in compact form, as follows * = 0: Compute singular values only. * = 1: Compute singular vectors of upper bidiagonal * matrix in compact form. * * SMLSIZ (input) INTEGER * The maximum size of the subproblems at the bottom of the * computation tree. * * N (input) INTEGER * The row dimension of the upper bidiagonal matrix. This is * also the dimension of the main diagonal array D. * * SQRE (input) INTEGER * Specifies the column dimension of the bidiagonal matrix. * = 0: The bidiagonal matrix has column dimension M = N; * = 1: The bidiagonal matrix has column dimension M = N + 1. * * D (input/output) REAL array, dimension ( N ) * On entry D contains the main diagonal of the bidiagonal * matrix. On exit D, if INFO = 0, contains its singular values. * * E (input) REAL array, dimension ( M-1 ) * Contains the subdiagonal entries of the bidiagonal matrix. * On exit, E has been destroyed. * * U (output) REAL array, * dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced * if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left * singular vector matrices of all subproblems at the bottom * level. * * LDU (input) INTEGER, LDU = > N. * The leading dimension of arrays U, VT, DIFL, DIFR, POLES, * GIVNUM, and Z. * * VT (output) REAL array, * dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced * if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right * singular vector matrices of all subproblems at the bottom * level. * * K (output) INTEGER array, dimension ( N ) * if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. * If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th * secular equation on the computation tree. * * DIFL (output) REAL array, dimension ( LDU, NLVL ), * where NLVL = floor(log_2 (N/SMLSIZ))). * * DIFR (output) REAL array, * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and * dimension ( N ) if ICOMPQ = 0. * If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) * record distances between singular values on the I-th * level and singular values on the (I -1)-th level, and * DIFR(1:N, 2 * I ) contains the normalizing factors for * the right singular vector matrix. See SLASD8 for details. * * Z (output) REAL array, * dimension ( LDU, NLVL ) if ICOMPQ = 1 and * dimension ( N ) if ICOMPQ = 0. * The first K elements of Z(1, I) contain the components of * the deflation-adjusted updating row vector for subproblems * on the I-th level. * * POLES (output) REAL array, * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced * if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and * POLES(1, 2*I) contain the new and old singular values * involved in the secular equations on the I-th level. * * GIVPTR (output) INTEGER array, * dimension ( N ) if ICOMPQ = 1, and not referenced if * ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records * the number of Givens rotations performed on the I-th * problem on the computation tree. * * GIVCOL (output) INTEGER array, * dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not * referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, * GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations * of Givens rotations performed on the I-th level on the * computation tree. * * LDGCOL (input) INTEGER, LDGCOL = > N. * The leading dimension of arrays GIVCOL and PERM. * * PERM (output) INTEGER array, dimension ( LDGCOL, NLVL ) * if ICOMPQ = 1, and not referenced * if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records * permutations done on the I-th level of the computation tree. * * GIVNUM (output) REAL array, * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not * referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, * GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S- * values of Givens rotations performed on the I-th level on * the computation tree. * * C (output) REAL array, * dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. * If ICOMPQ = 1 and the I-th subproblem is not square, on exit, * C( I ) contains the C-value of a Givens rotation related to * the right null space of the I-th subproblem. * * S (output) REAL array, dimension ( N ) if * ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 * and the I-th subproblem is not square, on exit, S( I ) * contains the S-value of a Givens rotation related to * the right null space of the I-th subproblem. * * WORK (workspace) REAL array, dimension * (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)). * * IWORK (workspace) INTEGER array, dimension (7*N). * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * > 0: if INFO = 1, a singular value did not converge * * Further Details * =============== * * Based on contributions by * Ming Gu and Huan Ren, Computer Science Division, University of * California at Berkeley, USA * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. INTEGER I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK, $ J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML, $ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU, $ NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI REAL ALPHA, BETA * .. * .. External Subroutines .. EXTERNAL SCOPY, SLASD6, SLASDQ, SLASDT, SLASET, XERBLA * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN INFO = -1 ELSE IF( SMLSIZ.LT.3 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN INFO = -4 ELSE IF( LDU.LT.( N+SQRE ) ) THEN INFO = -8 ELSE IF( LDGCOL.LT.N ) THEN INFO = -17 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLASDA', -INFO ) RETURN END IF * M = N + SQRE * * If the input matrix is too small, call SLASDQ to find the SVD. * IF( N.LE.SMLSIZ ) THEN IF( ICOMPQ.EQ.0 ) THEN CALL SLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU, $ U, LDU, WORK, INFO ) ELSE CALL SLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U, LDU, $ U, LDU, WORK, INFO ) END IF RETURN END IF * * Book-keeping and set up the computation tree. * INODE = 1 NDIML = INODE + N NDIMR = NDIML + N IDXQ = NDIMR + N IWK = IDXQ + N * NCC = 0 NRU = 0 * SMLSZP = SMLSIZ + 1 VF = 1 VL = VF + M NWORK1 = VL + M NWORK2 = NWORK1 + SMLSZP*SMLSZP * CALL SLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ), $ IWORK( NDIMR ), SMLSIZ ) * * for the nodes on bottom level of the tree, solve * their subproblems by SLASDQ. * NDB1 = ( ND+1 ) / 2 DO 30 I = NDB1, ND * * IC : center row of each node * NL : number of rows of left subproblem * NR : number of rows of right subproblem * NLF: starting row of the left subproblem * NRF: starting row of the right subproblem * I1 = I - 1 IC = IWORK( INODE+I1 ) NL = IWORK( NDIML+I1 ) NLP1 = NL + 1 NR = IWORK( NDIMR+I1 ) NLF = IC - NL NRF = IC + 1 IDXQI = IDXQ + NLF - 2 VFI = VF + NLF - 1 VLI = VL + NLF - 1 SQREI = 1 IF( ICOMPQ.EQ.0 ) THEN CALL SLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ), $ SMLSZP ) CALL SLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ), $ E( NLF ), WORK( NWORK1 ), SMLSZP, $ WORK( NWORK2 ), NL, WORK( NWORK2 ), NL, $ WORK( NWORK2 ), INFO ) ITEMP = NWORK1 + NL*SMLSZP CALL SCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 ) CALL SCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 ) ELSE CALL SLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU ) CALL SLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU ) CALL SLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ), $ E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU, $ U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO ) CALL SCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 ) CALL SCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 ) END IF IF( INFO.NE.0 ) THEN RETURN END IF DO 10 J = 1, NL IWORK( IDXQI+J ) = J 10 CONTINUE IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN SQREI = 0 ELSE SQREI = 1 END IF IDXQI = IDXQI + NLP1 VFI = VFI + NLP1 VLI = VLI + NLP1 NRP1 = NR + SQREI IF( ICOMPQ.EQ.0 ) THEN CALL SLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ), $ SMLSZP ) CALL SLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ), $ E( NRF ), WORK( NWORK1 ), SMLSZP, $ WORK( NWORK2 ), NR, WORK( NWORK2 ), NR, $ WORK( NWORK2 ), INFO ) ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP CALL SCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 ) CALL SCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 ) ELSE CALL SLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU ) CALL SLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU ) CALL SLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ), $ E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU, $ U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO ) CALL SCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 ) CALL SCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 ) END IF IF( INFO.NE.0 ) THEN RETURN END IF DO 20 J = 1, NR IWORK( IDXQI+J ) = J 20 CONTINUE 30 CONTINUE * * Now conquer each subproblem bottom-up. * J = 2**NLVL DO 50 LVL = NLVL, 1, -1 LVL2 = LVL*2 - 1 * * Find the first node LF and last node LL on * the current level LVL. * IF( LVL.EQ.1 ) THEN LF = 1 LL = 1 ELSE LF = 2**( LVL-1 ) LL = 2*LF - 1 END IF DO 40 I = LF, LL IM1 = I - 1 IC = IWORK( INODE+IM1 ) NL = IWORK( NDIML+IM1 ) NR = IWORK( NDIMR+IM1 ) NLF = IC - NL NRF = IC + 1 IF( I.EQ.LL ) THEN SQREI = SQRE ELSE SQREI = 1 END IF VFI = VF + NLF - 1 VLI = VL + NLF - 1 IDXQI = IDXQ + NLF - 1 ALPHA = D( IC ) BETA = E( IC ) IF( ICOMPQ.EQ.0 ) THEN CALL SLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ), $ WORK( VFI ), WORK( VLI ), ALPHA, BETA, $ IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL, $ LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z, $ K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ), $ IWORK( IWK ), INFO ) ELSE J = J - 1 CALL SLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ), $ WORK( VFI ), WORK( VLI ), ALPHA, BETA, $ IWORK( IDXQI ), PERM( NLF, LVL ), $ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL, $ GIVNUM( NLF, LVL2 ), LDU, $ POLES( NLF, LVL2 ), DIFL( NLF, LVL ), $ DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ), $ C( J ), S( J ), WORK( NWORK1 ), $ IWORK( IWK ), INFO ) END IF IF( INFO.NE.0 ) THEN RETURN END IF 40 CONTINUE 50 CONTINUE * RETURN * * End of SLASDA * END |