1ZLALSA(1) LAPACK routine (version 3.2) ZLALSA(1)
2
6 ZLALSA - is an itermediate step in solving the least squares problem by
7 computing the SVD of the coefficient matrix in compact form (The singu‐
8 lar vectors are computed as products of simple orthorgonal matrices.)
9
11 SUBROUTINE ZLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, LDU,
12 VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
13 PERM, GIVNUM, C, S, RWORK, IWORK, INFO )
14 INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS, SML‐
16 SIZ
17 INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), K( *
19 ), PERM( LDGCOL, * )
20 DOUBLE PRECISION C( * ), DIFL( LDU, * ), DIFR( LDU, * ),
22 GIVNUM( LDU, * ), POLES( LDU, * ), RWORK( * ), S( *
23 ), U( LDU, * ), VT( LDU, * ), Z( LDU, * )
24 COMPLEX*16 B( LDB, * ), BX( LDBX, * )
26
28 ZLALSA is an itermediate step in solving the least squares problem by
29 computing the SVD of the coefficient matrix in compact form (The singu‐
30 lar vectors are computed as products of simple orthorgonal matrices.).
31 If ICOMPQ = 0, ZLALSA applies the inverse of the left singular vector
32 matrix of an upper bidiagonal matrix to the right hand side; and if
33 ICOMPQ = 1, ZLALSA applies the right singular vector matrix to the
34 right hand side. The singular vector matrices were generated in compact
35 form by ZLALSA.
36
38 ICOMPQ (input) INTEGER Specifies whether the left or the right singular
39 vector matrix is involved. = 0: Left singular vector matrix
40 = 1: Right singular vector matrix SMLSIZ (input) INTEGER The maximum
41 size of the subproblems at the bottom of the computation tree.
42 N (input) INTEGER
44 The row and column dimensions of the upper bidiagonal matrix.
45 NRHS (input) INTEGER
47 The number of columns of B and BX. NRHS must be at least 1.
48 B (input/output) COMPLEX*16 array, dimension ( LDB, NRHS )
50 On input, B contains the right hand sides of the least squares
51 problem in rows 1 through M. On output, B contains the solution
52 X in rows 1 through N.
53 LDB (input) INTEGER
55 The leading dimension of B in the calling subprogram. LDB must
56 be at least max(1,MAX( M, N ) ).
57 BX (output) COMPLEX*16 array, dimension ( LDBX, NRHS )
59 On exit, the result of applying the left or right singular vec‐
60 tor matrix to B.
61 LDBX (input) INTEGER
63 The leading dimension of BX.
64 U (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
66 On entry, U contains the left singular vector matrices of all
67 subproblems at the bottom level.
68 LDU (input) INTEGER, LDU = > N.
70 The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
71 GIVNUM, and Z.
72 VT (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
74 On entry, VT' contains the right singular vector matrices of all
75 subproblems at the bottom level.
76 K (input) INTEGER array, dimension ( N ).
78 DIFL (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ).
80 where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
81 DIFR (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
83 On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record distances
84 between singular values on the I-th level and singular values on
85 the (I -1)-th level, and DIFR(*, 2 * I) record the normalizing
86 factors of the right singular vectors matrices of subproblems on
87 I-th level.
88 Z (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ).
90 On entry, Z(1, I) contains the components of the deflation-
91 adjusted updating row vector for subproblems on the I-th level.
92 POLES (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
94 On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
95 singular values involved in the secular equations on the I-th
96 level. GIVPTR (input) INTEGER array, dimension ( N ). On
97 entry, GIVPTR( I ) records the number of Givens rotations per‐
98 formed on the I-th problem on the computation tree. GIVCOL
99 (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ). On
100 entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the loca‐
101 tions of Givens rotations performed on the I-th level on the
102 computation tree. LDGCOL (input) INTEGER, LDGCOL = > N. The
103 leading dimension of arrays GIVCOL and PERM.
104 PERM (input) INTEGER array, dimension ( LDGCOL, NLVL ).
106 On entry, PERM(*, I) records permutations done on the I-th level
107 of the computation tree. GIVNUM (input) DOUBLE PRECISION array,
108 dimension ( LDU, 2 * NLVL ). On entry, GIVNUM(*, 2 *I -1 : 2 *
109 I) records the C- and S- values of Givens rotations performed on
110 the I-th level on the computation tree.
111 C (input) DOUBLE PRECISION array, dimension ( N ).
113 On entry, if the I-th subproblem is not square, C( I ) contains
114 the C-value of a Givens rotation related to the right null space
115 of the I-th subproblem.
116 S (input) DOUBLE PRECISION array, dimension ( N ).
118 On entry, if the I-th subproblem is not square, S( I ) contains
119 the S-value of a Givens rotation related to the right null space
120 of the I-th subproblem.
121 RWORK (workspace) DOUBLE PRECISION array, dimension at least
123 max ( N, (SMLSZ+1)*NRHS*3 ).
124 IWORK (workspace) INTEGER array.
126 The dimension must be at least 3 * N
127 INFO (output) INTEGER
129 = 0: successful exit.
130 < 0: if INFO = -i, the i-th argument had an illegal value.
131
133 Based on contributions by
134 Ming Gu and Ren-Cang Li, Computer Science Division, University of
135 California at Berkeley, USA
136 Osni Marques, LBNL/NERSC, USA
137 LAPACK routine (version 3.2) November 2008 ZLALSA(1)