1SLASD3(1) LAPACK auxiliary routine (version 3.2) SLASD3(1)
2
6 SLASD3 - finds all the square roots of the roots of the secular equa‐
7 tion, as defined by the values in D and Z
8
10 SUBROUTINE SLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
11 LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO )
12 INTEGER INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
14 INTEGER CTOT( * ), IDXC( * )
16 REAL D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ), U2(
18 LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), Z( * )
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21 SLASD3 finds all the square roots of the roots of the secular equation,
22 as defined by the values in D and Z. It makes the appropriate calls to
23 SLASD4 and then updates the singular vectors by matrix multiplication.
24 This code makes very mild assumptions about floating point arithmetic.
25 It will work on machines with a guard digit in add/subtract, or on
26 those binary machines without guard digits which subtract like the Cray
27 XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on
28 hexadecimal or decimal machines without guard digits, but we know of
29 none.
30 SLASD3 is called from SLASD1.
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33 NL (input) INTEGER
34 The row dimension of the upper block. NL >= 1.
35 NR (input) INTEGER
37 The row dimension of the lower block. NR >= 1.
38 SQRE (input) INTEGER
40 = 0: the lower block is an NR-by-NR square matrix.
41 = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The
42 bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N
43 columns.
44 K (input) INTEGER
46 The size of the secular equation, 1 =< K = < N.
47 D (output) REAL array, dimension(K)
49 On exit the square roots of the roots of the secular equation,
50 in ascending order.
51 Q (workspace) REAL array,
53 dimension at least (LDQ,K).
54 LDQ (input) INTEGER
56 The leading dimension of the array Q. LDQ >= K. DSIGMA
57 (input/output) REAL array, dimension(K) The first K elements of
58 this array contain the old roots of the deflated updating prob‐
59 lem. These are the poles of the secular equation.
60 U (output) REAL array, dimension (LDU, N)
62 The last N - K columns of this matrix contain the deflated left
63 singular vectors.
64 LDU (input) INTEGER
66 The leading dimension of the array U. LDU >= N.
67 U2 (input) REAL array, dimension (LDU2, N)
69 The first K columns of this matrix contain the non-deflated left
70 singular vectors for the split problem.
71 LDU2 (input) INTEGER
73 The leading dimension of the array U2. LDU2 >= N.
74 VT (output) REAL array, dimension (LDVT, M)
76 The last M - K columns of VT' contain the deflated right singu‐
77 lar vectors.
78 LDVT (input) INTEGER
80 The leading dimension of the array VT. LDVT >= N.
81 VT2 (input/output) REAL array, dimension (LDVT2, N)
83 The first K columns of VT2' contain the non-deflated right sin‐
84 gular vectors for the split problem.
85 LDVT2 (input) INTEGER
87 The leading dimension of the array VT2. LDVT2 >= N.
88 IDXC (input) INTEGER array, dimension (N)
90 The permutation used to arrange the columns of U (and rows of
91 VT) into three groups: the first group contains non-zero
92 entries only at and above (or before) NL +1; the second contains
93 non-zero entries only at and below (or after) NL+2; and the
94 third is dense. The first column of U and the row of VT are
95 treated separately, however. The rows of the singular vectors
96 found by SLASD4 must be likewise permuted before the matrix mul‐
97 tiplies can take place.
98 CTOT (input) INTEGER array, dimension (4)
100 A count of the total number of the various types of columns in U
101 (or rows in VT), as described in IDXC. The fourth column type is
102 any column which has been deflated.
103 Z (input/output) REAL array, dimension (K)
105 The first K elements of this array contain the components of the
106 deflation-adjusted updating row vector.
107 INFO (output) INTEGER
109 = 0: successful exit.
110 < 0: if INFO = -i, the i-th argument had an illegal value.
111 > 0: if INFO = 1, an singular value did not converge
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114 Based on contributions by
115 Ming Gu and Huan Ren, Computer Science Division, University of
116 California at Berkeley, USA
117 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 SLASD3(1)