1DLASD1(1) LAPACK auxiliary routine (version 3.1) DLASD1(1)
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6 DLASD1 - 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.
24 A related subroutine DLASD7 handles the case in which the singular val‐
26 ues (and the singular vectors in factored form) are desired.
27 DLASD1 computes the SVD as follows:
29 ( D1(in) 0 0 0 )
31 B = U(in) * ( Z1' a Z2' b ) * VT(in)
32 ( 0 0 D2(in) 0 )
33 = U(out) * ( D(out) 0) * VT(out)
35 where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
37 with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros else‐
38 where; and the entry b is empty if SQRE = 0.
39 The left singular vectors of the original matrix are stored in U, and
41 the transpose of the right singular vectors are stored in VT, and the
42 singular values are in D. The algorithm consists of three stages:
43 The first stage consists of deflating the size of the problem
45 when there are multiple singular values or when there are zeros in
46 the Z vector. For each such occurence the dimension of the
47 secular equation problem is reduced by one. This stage is
48 performed by the routine DLASD2.
49 The second stage consists of calculating the updated
51 singular values. This is done by finding the square roots of the
52 roots of the secular equation via the routine DLASD4 (as called
53 by DLASD3). This routine also calculates the singular vectors of
54 the current problem.
55 The final stage consists of computing the updated singular vectors
57 directly using the updated singular values. The singular vectors
58 for the current problem are multiplied with the singular vectors
59 from the overall problem.
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63 NL (input) INTEGER
64 The row dimension of the upper block. NL >= 1.
65 NR (input) INTEGER
67 The row dimension of the lower block. NR >= 1.
68 SQRE (input) INTEGER
70 = 0: the lower block is an NR-by-NR square matrix.
71 = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
72 The bidiagonal matrix has row dimension N = NL + NR + 1, and
74 column dimension M = N + SQRE.
75 D (input/output) DOUBLE PRECISION array,
77 dimension (N = NL+NR+1). On entry D(1:NL,1:NL) contains the
78 singular values of the
79 upper block; and D(NL+2:N) contains the singular values of
80 the lower block. On exit D(1:N) contains the singular values of
81 the modified matrix.
82 ALPHA (input/output) DOUBLE PRECISION
84 Contains the diagonal element associated with the added row.
85 BETA (input/output) DOUBLE PRECISION
87 Contains the off-diagonal element associated with the added row.
88 U (input/output) DOUBLE PRECISION array, dimension(LDU,N)
90 On entry U(1:NL, 1:NL) contains the left singular vectors of
91 the upper block; U(NL+2:N, NL+2:N) contains the left singular
92 vectors of the lower block. On exit U contains the left singular
93 vectors of the bidiagonal matrix.
94 LDU (input) INTEGER
96 The leading dimension of the array U. LDU >= max( 1, N ).
97 VT (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
99 where M = N + SQRE. On entry VT(1:NL+1, 1:NL+1)' contains the
100 right singular
101 vectors of the upper block; VT(NL+2:M, NL+2:M)' contains the
102 right singular vectors of the lower block. On exit VT' contains
103 the right singular vectors of the bidiagonal matrix.
104 LDVT (input) INTEGER
106 The leading dimension of the array VT. LDVT >= max( 1, M ).
107 IDXQ (output) INTEGER array, dimension(N)
109 This contains the permutation which will reintegrate the subprob‐
110 lem just solved back into sorted order, i.e. D( IDXQ( I = 1, N )
111 ) will be in ascending order.
112 IWORK (workspace) INTEGER array, dimension( 4 * N )
114 WORK (workspace) DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )
116 INFO (output) INTEGER
118 = 0: successful exit.
119 < 0: if INFO = -i, the i-th argument had an illegal value.
120 > 0: if INFO = 1, an singular value did not converge
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123 Based on contributions by
124 Ming Gu and Huan Ren, Computer Science Division, University of
125 California at Berkeley, USA
126 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 DLASD1(1)