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SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, |
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$ DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, |
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$ PERM, GIVNUM, C, S, WORK, IWORK, INFO ) |
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* |
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* -- LAPACK auxiliary routine (version 3.2.2) -- |
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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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* June 2010 |
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* |
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* .. Scalar Arguments .. |
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INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE |
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* .. |
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* .. Array Arguments .. |
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INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), |
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$ K( * ), PERM( LDGCOL, * ) |
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DOUBLE PRECISION C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ), |
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$ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ), |
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$ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ), |
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$ Z( LDU, * ) |
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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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* Using a divide and conquer approach, DLASDA computes the singular |
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* value decomposition (SVD) of a real upper bidiagonal N-by-M matrix |
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* B with diagonal D and offdiagonal E, where M = N + SQRE. The |
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* algorithm computes the singular values in the SVD B = U * S * VT. |
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* The orthogonal matrices U and VT are optionally computed in |
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* compact form. |
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* |
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* A related subroutine, DLASD0, computes the singular values and |
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* the singular vectors in explicit form. |
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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 |
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* in compact form, as follows |
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* = 0: Compute singular values only. |
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* = 1: Compute singular vectors of upper bidiagonal |
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* matrix in compact form. |
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* |
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* SMLSIZ (input) INTEGER |
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* The maximum size of the subproblems at the bottom of the |
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* computation tree. |
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* |
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* N (input) INTEGER |
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* The row dimension of the upper bidiagonal matrix. This is |
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* also the dimension of the main diagonal array D. |
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* |
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* SQRE (input) INTEGER |
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* Specifies the column dimension of the bidiagonal matrix. |
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* = 0: The bidiagonal matrix has column dimension M = N; |
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* = 1: The bidiagonal matrix has column dimension M = N + 1. |
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* |
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* D (input/output) DOUBLE PRECISION array, dimension ( N ) |
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* On entry D contains the main diagonal of the bidiagonal |
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* matrix. On exit D, if INFO = 0, contains its singular values. |
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* |
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* E (input) DOUBLE PRECISION array, dimension ( M-1 ) |
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* Contains the subdiagonal entries of the bidiagonal matrix. |
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* On exit, E has been destroyed. |
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* |
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* U (output) DOUBLE PRECISION array, |
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* dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced |
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* if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left |
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* singular vector matrices of all subproblems at the bottom |
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* level. |
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* |
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* LDU (input) INTEGER, LDU = > N. |
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* The leading dimension of arrays U, VT, DIFL, DIFR, POLES, |
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* GIVNUM, and Z. |
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* |
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* VT (output) DOUBLE PRECISION array, |
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* dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced |
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* if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right |
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* singular vector matrices of all subproblems at the bottom |
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* level. |
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* |
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* K (output) INTEGER array, |
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* dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. |
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* If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th |
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* secular equation on the computation tree. |
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* |
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* DIFL (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ), |
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* where NLVL = floor(log_2 (N/SMLSIZ))). |
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* |
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* DIFR (output) DOUBLE PRECISION array, |
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* dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and |
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* dimension ( N ) if ICOMPQ = 0. |
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* If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) |
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* record distances between singular values on the I-th |
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* level and singular values on the (I -1)-th level, and |
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* DIFR(1:N, 2 * I ) contains the normalizing factors for |
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* the right singular vector matrix. See DLASD8 for details. |
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* |
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* Z (output) DOUBLE PRECISION array, |
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* dimension ( LDU, NLVL ) if ICOMPQ = 1 and |
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* dimension ( N ) if ICOMPQ = 0. |
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* The first K elements of Z(1, I) contain the components of |
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* the deflation-adjusted updating row vector for subproblems |
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* on the I-th level. |
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* |
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* POLES (output) DOUBLE PRECISION array, |
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* dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced |
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* if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and |
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* POLES(1, 2*I) contain the new and old singular values |
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* involved in the secular equations on the I-th level. |
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* |
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* GIVPTR (output) INTEGER array, |
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* dimension ( N ) if ICOMPQ = 1, and not referenced if |
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* ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records |
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* the number of Givens rotations performed on the I-th |
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* problem on the computation tree. |
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* |
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* GIVCOL (output) INTEGER array, |
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* dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not |
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* referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, |
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* GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations |
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* of Givens rotations performed on the I-th level on the |
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* computation tree. |
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* |
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* LDGCOL (input) INTEGER, LDGCOL = > N. |
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* The leading dimension of arrays GIVCOL and PERM. |
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* |
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* PERM (output) INTEGER array, |
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* dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced |
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* if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records |
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* permutations done on the I-th level of the computation tree. |
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* |
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* GIVNUM (output) DOUBLE PRECISION array, |
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* dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not |
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* referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, |
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* GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S- |
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* values of Givens rotations performed on the I-th level on |
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* the computation tree. |
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* |
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* C (output) DOUBLE PRECISION array, |
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* dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. |
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* If ICOMPQ = 1 and the I-th subproblem is not square, on exit, |
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* C( I ) contains the C-value of a Givens rotation related to |
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* the right null space of the I-th subproblem. |
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* |
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* S (output) DOUBLE PRECISION array, dimension ( N ) if |
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* ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 |
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* and the I-th subproblem is not square, on exit, S( I ) |
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* contains the S-value of a Givens rotation related to |
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* the right null space of the I-th subproblem. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension |
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* (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)). |
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* |
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* IWORK (workspace) INTEGER array. |
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* Dimension must be at least (7 * N). |
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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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* > 0: if INFO = 1, a singular value did not converge |
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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 Huan Ren, Computer Science Division, University of |
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* California at Berkeley, USA |
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* |
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* ===================================================================== |
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* |
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* .. Parameters .. |
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DOUBLE PRECISION ZERO, ONE |
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) |
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* .. |
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* .. Local Scalars .. |
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INTEGER I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK, |
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$ J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML, |
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$ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU, |
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$ NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI |
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DOUBLE PRECISION ALPHA, BETA |
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* .. |
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* .. External Subroutines .. |
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EXTERNAL DCOPY, DLASD6, DLASDQ, DLASDT, DLASET, XERBLA |
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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( SMLSIZ.LT.3 ) THEN |
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INFO = -2 |
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ELSE IF( N.LT.0 ) 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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ELSE IF( LDU.LT.( N+SQRE ) ) THEN |
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INFO = -8 |
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ELSE IF( LDGCOL.LT.N ) THEN |
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INFO = -17 |
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END IF |
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IF( INFO.NE.0 ) THEN |
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CALL XERBLA( 'DLASDA', -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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* |
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* If the input matrix is too small, call DLASDQ to find the SVD. |
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* |
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IF( N.LE.SMLSIZ ) THEN |
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IF( ICOMPQ.EQ.0 ) THEN |
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CALL DLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU, |
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$ U, LDU, WORK, INFO ) |
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ELSE |
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CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U, LDU, |
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$ U, LDU, WORK, INFO ) |
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END IF |
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RETURN |
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END IF |
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* |
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* Book-keeping and set up the computation tree. |
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* |
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INODE = 1 |
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NDIML = INODE + N |
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NDIMR = NDIML + N |
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IDXQ = NDIMR + N |
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IWK = IDXQ + N |
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* |
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NCC = 0 |
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NRU = 0 |
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* |
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SMLSZP = SMLSIZ + 1 |
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VF = 1 |
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VL = VF + M |
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NWORK1 = VL + M |
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NWORK2 = NWORK1 + SMLSZP*SMLSZP |
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* |
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CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ), |
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$ IWORK( NDIMR ), SMLSIZ ) |
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* |
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* for the nodes on bottom level of the tree, solve |
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* their subproblems by DLASDQ. |
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* |
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NDB1 = ( ND+1 ) / 2 |
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DO 30 I = NDB1, ND |
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* |
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* IC : center row of each node |
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* NL : number of rows of left subproblem |
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* NR : number of rows of right subproblem |
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* NLF: starting row of the left subproblem |
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* NRF: starting row of the right subproblem |
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* |
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I1 = I - 1 |
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IC = IWORK( INODE+I1 ) |
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NL = IWORK( NDIML+I1 ) |
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NLP1 = NL + 1 |
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NR = IWORK( NDIMR+I1 ) |
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NLF = IC - NL |
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NRF = IC + 1 |
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IDXQI = IDXQ + NLF - 2 |
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VFI = VF + NLF - 1 |
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VLI = VL + NLF - 1 |
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SQREI = 1 |
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IF( ICOMPQ.EQ.0 ) THEN |
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CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ), |
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$ SMLSZP ) |
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CALL DLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ), |
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$ E( NLF ), WORK( NWORK1 ), SMLSZP, |
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$ WORK( NWORK2 ), NL, WORK( NWORK2 ), NL, |
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$ WORK( NWORK2 ), INFO ) |
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ITEMP = NWORK1 + NL*SMLSZP |
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CALL DCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 ) |
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CALL DCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 ) |
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ELSE |
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CALL DLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU ) |
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CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU ) |
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CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ), |
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$ E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU, |
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$ U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO ) |
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CALL DCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 ) |
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CALL DCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 ) |
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END IF |
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IF( INFO.NE.0 ) THEN |
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RETURN |
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END IF |
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DO 10 J = 1, NL |
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IWORK( IDXQI+J ) = J |
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10 CONTINUE |
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IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN |
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SQREI = 0 |
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ELSE |
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SQREI = 1 |
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END IF |
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IDXQI = IDXQI + NLP1 |
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VFI = VFI + NLP1 |
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VLI = VLI + NLP1 |
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NRP1 = NR + SQREI |
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IF( ICOMPQ.EQ.0 ) THEN |
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CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ), |
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$ SMLSZP ) |
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CALL DLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ), |
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$ E( NRF ), WORK( NWORK1 ), SMLSZP, |
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$ WORK( NWORK2 ), NR, WORK( NWORK2 ), NR, |
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$ WORK( NWORK2 ), INFO ) |
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ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP |
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CALL DCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 ) |
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CALL DCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 ) |
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ELSE |
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CALL DLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU ) |
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CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU ) |
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CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ), |
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$ E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU, |
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$ U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO ) |
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CALL DCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 ) |
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CALL DCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 ) |
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END IF |
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IF( INFO.NE.0 ) THEN |
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RETURN |
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END IF |
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DO 20 J = 1, NR |
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IWORK( IDXQI+J ) = J |
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20 CONTINUE |
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30 CONTINUE |
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* |
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* Now conquer each subproblem bottom-up. |
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* |
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J = 2**NLVL |
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DO 50 LVL = NLVL, 1, -1 |
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LVL2 = LVL*2 - 1 |
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* |
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* Find the first node LF and last node LL on |
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* the current level LVL. |
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* |
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IF( LVL.EQ.1 ) THEN |
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LF = 1 |
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LL = 1 |
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ELSE |
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LF = 2**( LVL-1 ) |
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LL = 2*LF - 1 |
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END IF |
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DO 40 I = LF, LL |
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IM1 = I - 1 |
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IC = IWORK( INODE+IM1 ) |
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NL = IWORK( NDIML+IM1 ) |
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NR = IWORK( NDIMR+IM1 ) |
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NLF = IC - NL |
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NRF = IC + 1 |
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IF( I.EQ.LL ) THEN |
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SQREI = SQRE |
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ELSE |
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SQREI = 1 |
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END IF |
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VFI = VF + NLF - 1 |
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VLI = VL + NLF - 1 |
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IDXQI = IDXQ + NLF - 1 |
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ALPHA = D( IC ) |
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BETA = E( IC ) |
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IF( ICOMPQ.EQ.0 ) THEN |
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CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ), |
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$ WORK( VFI ), WORK( VLI ), ALPHA, BETA, |
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$ IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL, |
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$ LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z, |
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$ K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ), |
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$ IWORK( IWK ), INFO ) |
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ELSE |
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J = J - 1 |
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CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ), |
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$ WORK( VFI ), WORK( VLI ), ALPHA, BETA, |
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$ IWORK( IDXQI ), PERM( NLF, LVL ), |
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$ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL, |
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$ GIVNUM( NLF, LVL2 ), LDU, |
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$ POLES( NLF, LVL2 ), DIFL( NLF, LVL ), |
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$ DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ), |
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$ C( J ), S( J ), WORK( NWORK1 ), |
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$ IWORK( IWK ), INFO ) |
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END IF |
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IF( INFO.NE.0 ) THEN |
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RETURN |
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END IF |
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40 CONTINUE |
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50 CONTINUE |
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* |
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RETURN |
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* |
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* End of DLASDA |
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* |
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END |