1DLASD1(1) LAPACK auxiliary routine (version 3.2) DLASD1(1)
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6 DLASD1 - computes the SVD of an upper bidiagonal N-by-M matrix B,
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9 SUBROUTINE DLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
10 IDXQ, IWORK, WORK, INFO )
11 INTEGER INFO, LDU, LDVT, NL, NR, SQRE
13 DOUBLE PRECISION ALPHA, BETA
15 INTEGER IDXQ( * ), IWORK( * )
17 DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK(
19 * )
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22 DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N
23 = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0. A
24 related subroutine DLASD7 handles the case in which the singular values
25 (and the singular vectors in factored form) are desired. DLASD1 com‐
26 putes the SVD as follows:
27 ( D1(in) 0 0 0 )
28 B = U(in) * ( Z1' a Z2' b ) * VT(in)
29 ( 0 0 D2(in) 0 )
30 = U(out) * ( D(out) 0) * VT(out)
31 where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
32 with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros else‐
33 where; and the entry b is empty if SQRE = 0.
34 The left singular vectors of the original matrix are stored in U, and
35 the transpose of the right singular vectors are stored in VT, and the
36 singular values are in D. The algorithm consists of three stages:
37 The first stage consists of deflating the size of the problem
38 when there are multiple singular values or when there are zeros in
39 the Z vector. For each such occurence the dimension of the
40 secular equation problem is reduced by one. This stage is
41 performed by the routine DLASD2.
42 The second stage consists of calculating the updated
43 singular values. This is done by finding the square roots of the
44 roots of the secular equation via the routine DLASD4 (as called
45 by DLASD3). This routine also calculates the singular vectors of
46 the current problem.
47 The final stage consists of computing the updated singular vectors
48 directly using the updated singular values. The singular vectors
49 for the current problem are multiplied with the singular vectors
50 from the overall problem.
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53 NL (input) INTEGER
54 The row dimension of the upper block. NL >= 1.
55 NR (input) INTEGER
57 The row dimension of the lower block. NR >= 1.
58 SQRE (input) INTEGER
60 = 0: the lower block is an NR-by-NR square matrix.
61 = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The
62 bidiagonal matrix has row dimension N = NL + NR + 1, and column
63 dimension M = N + SQRE.
64 D (input/output) DOUBLE PRECISION array,
66 dimension (N = NL+NR+1). On entry D(1:NL,1:NL) contains the
67 singular values of the
68 upper block; and D(NL+2:N) contains the singular values of
69 the lower block. On exit D(1:N) contains the singular values of
70 the modified matrix.
71 ALPHA (input/output) DOUBLE PRECISION
73 Contains the diagonal element associated with the added row.
74 BETA (input/output) DOUBLE PRECISION
76 Contains the off-diagonal element associated with the added row.
77 U (input/output) DOUBLE PRECISION array, dimension(LDU,N)
79 On entry U(1:NL, 1:NL) contains the left singular vectors of
80 the upper block; U(NL+2:N, NL+2:N) contains the left singular
81 vectors of the lower block. On exit U contains the left singular
82 vectors of the bidiagonal matrix.
83 LDU (input) INTEGER
85 The leading dimension of the array U. LDU >= max( 1, N ).
86 VT (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
88 where M = N + SQRE. On entry VT(1:NL+1, 1:NL+1)' contains the
89 right singular
90 vectors of the upper block; VT(NL+2:M, NL+2:M)' contains the
91 right singular vectors of the lower block. On exit VT' contains
92 the right singular vectors of the bidiagonal matrix.
93 LDVT (input) INTEGER
95 The leading dimension of the array VT. LDVT >= max( 1, M ).
96 IDXQ (output) INTEGER array, dimension(N)
98 This contains the permutation which will reintegrate the subprob‐
99 lem just solved back into sorted order, i.e. D( IDXQ( I = 1, N )
100 ) will be in ascending order.
101 IWORK (workspace) INTEGER array, dimension( 4 * N )
103 WORK (workspace) DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )
105 INFO (output) INTEGER
107 = 0: successful exit.
108 < 0: if INFO = -i, the i-th argument had an illegal value.
109 > 0: if INFO = 1, an singular value did not converge
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112 Based on contributions by
113 Ming Gu and Huan Ren, Computer Science Division, University of
114 California at Berkeley, USA
115 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 DLASD1(1)