1SLASD1(1) LAPACK auxiliary routine (version 3.2) SLASD1(1)
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6 SLASD1 - computes the SVD of an upper bidiagonal N-by-M matrix B,
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9 SUBROUTINE SLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
10 IDXQ, IWORK, WORK, INFO )
11 INTEGER INFO, LDU, LDVT, NL, NR, SQRE
13 REAL ALPHA, BETA
15 INTEGER IDXQ( * ), IWORK( * )
17 REAL D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
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21 SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N
22 = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0. A
23 related subroutine SLASD7 handles the case in which the singular values
24 (and the singular vectors in factored form) are desired. SLASD1 com‐
25 putes the SVD as follows:
26 ( D1(in) 0 0 0 )
27 B = U(in) * ( Z1' a Z2' b ) * VT(in)
28 ( 0 0 D2(in) 0 )
29 = U(out) * ( D(out) 0) * VT(out)
30 where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
31 with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros else‐
32 where; and the entry b is empty if SQRE = 0.
33 The left singular vectors of the original matrix are stored in U, and
34 the transpose of the right singular vectors are stored in VT, and the
35 singular values are in D. The algorithm consists of three stages:
36 The first stage consists of deflating the size of the problem
37 when there are multiple singular values or when there are zeros in
38 the Z vector. For each such occurence the dimension of the
39 secular equation problem is reduced by one. This stage is
40 performed by the routine SLASD2.
41 The second stage consists of calculating the updated
42 singular values. This is done by finding the square roots of the
43 roots of the secular equation via the routine SLASD4 (as called
44 by SLASD3). This routine also calculates the singular vectors of
45 the current problem.
46 The final stage consists of computing the updated singular vectors
47 directly using the updated singular values. The singular vectors
48 for the current problem are multiplied with the singular vectors
49 from the overall problem.
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52 NL (input) INTEGER
53 The row dimension of the upper block. NL >= 1.
54 NR (input) INTEGER
56 The row dimension of the lower block. NR >= 1.
57 SQRE (input) INTEGER
59 = 0: the lower block is an NR-by-NR square matrix.
60 = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The
61 bidiagonal matrix has row dimension N = NL + NR + 1, and column
62 dimension M = N + SQRE.
63 D (input/output) REAL array, dimension (NL+NR+1).
65 N = NL+NR+1 On entry D(1:NL,1:NL) contains the singular values
66 of the
67 upper block; and D(NL+2:N) contains the singular values of
68 the lower block. On exit D(1:N) contains the singular values of
69 the modified matrix.
70 ALPHA (input/output) REAL
72 Contains the diagonal element associated with the added row.
73 BETA (input/output) REAL
75 Contains the off-diagonal element associated with the added row.
76 U (input/output) REAL array, dimension (LDU,N)
78 On entry U(1:NL, 1:NL) contains the left singular vectors of
79 the upper block; U(NL+2:N, NL+2:N) contains the left singular
80 vectors of the lower block. On exit U contains the left singular
81 vectors of the bidiagonal matrix.
82 LDU (input) INTEGER
84 The leading dimension of the array U. LDU >= max( 1, N ).
85 VT (input/output) REAL array, dimension (LDVT,M)
87 where M = N + SQRE. On entry VT(1:NL+1, 1:NL+1)' contains the
88 right singular
89 vectors of the upper block; VT(NL+2:M, NL+2:M)' contains the
90 right singular vectors of the lower block. On exit VT' contains
91 the right singular vectors of the bidiagonal matrix.
92 LDVT (input) INTEGER
94 The leading dimension of the array VT. LDVT >= max( 1, M ).
95 IDXQ (output) INTEGER array, dimension (N)
97 This contains the permutation which will reintegrate the subprob‐
98 lem just solved back into sorted order, i.e. D( IDXQ( I = 1, N )
99 ) will be in ascending order.
100 IWORK (workspace) INTEGER array, dimension (4*N)
102 WORK (workspace) REAL array, dimension (3*M**2+2*M)
104 INFO (output) INTEGER
106 = 0: successful exit.
107 < 0: if INFO = -i, the i-th argument had an illegal value.
108 > 0: if INFO = 1, an singular value did not converge
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111 Based on contributions by
112 Ming Gu and Huan Ren, Computer Science Division, University of
113 California at Berkeley, USA
114 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 SLASD1(1)