1SBDSQR(1) LAPACK routine (version 3.2) SBDSQR(1)
2
6 SBDSQR - computes the singular values and, optionally, the right and/or
7 left singular vectors from the singular value decomposition (SVD) of a
8 real N-by-N (upper or lower) bidiagonal matrix B using the implicit
9 zero-shift QR algorithm
10
12 SUBROUTINE SBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C,
13 LDC, WORK, INFO )
14 CHARACTER UPLO
16 INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
18 REAL C( LDC, * ), D( * ), E( * ), U( LDU, * ), VT( LDVT,
20 * ), WORK( * )
21
23 SBDSQR computes the singular values and, optionally, the right and/or
24 left singular vectors from the singular value decomposition (SVD) of a
25 real N-by-N (upper or lower) bidiagonal matrix B using the implicit
26 zero-shift QR algorithm. The SVD of B has the form
27 B = Q * S * P**T
28 where S is the diagonal matrix of singular values, Q is an orthogonal
29 matrix of left singular vectors, and P is an orthogonal matrix of right
30 singular vectors. If left singular vectors are requested, this subrou‐
31 tine actually returns U*Q instead of Q, and, if right singular vectors
32 are requested, this subroutine returns P**T*VT instead of P**T, for
33 given real input matrices U and VT. When U and VT are the orthogonal
34 matrices that reduce a general matrix A to bidiagonal form: A =
35 U*B*VT, as computed by SGEBRD, then
36 A = (U*Q) * S * (P**T*VT)
37 is the SVD of A. Optionally, the subroutine may also compute Q**T*C
38 for a given real input matrix C.
39 See "Computing Small Singular Values of Bidiagonal Matrices With Guar‐
40 anteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Work‐
41 ing Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp.
42 873-912, Sept 1990) and
43 "Accurate singular values and differential qd algorithms," by B. Par‐
44 lett and V. Fernando, Technical Report CPAM-554, Mathematics Depart‐
45 ment, University of California at Berkeley, July 1992 for a detailed
46 description of the algorithm.
47
49 UPLO (input) CHARACTER*1
50 = 'U': B is upper bidiagonal;
51 = 'L': B is lower bidiagonal.
52 N (input) INTEGER
54 The order of the matrix B. N >= 0.
55 NCVT (input) INTEGER
57 The number of columns of the matrix VT. NCVT >= 0.
58 NRU (input) INTEGER
60 The number of rows of the matrix U. NRU >= 0.
61 NCC (input) INTEGER
63 The number of columns of the matrix C. NCC >= 0.
64 D (input/output) REAL array, dimension (N)
66 On entry, the n diagonal elements of the bidiagonal matrix B.
67 On exit, if INFO=0, the singular values of B in decreasing
68 order.
69 E (input/output) REAL array, dimension (N-1)
71 On entry, the N-1 offdiagonal elements of the bidiagonal matrix
72 B. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
73 will contain the diagonal and superdiagonal elements of a bidi‐
74 agonal matrix orthogonally equivalent to the one given as
75 input.
76 VT (input/output) REAL array, dimension (LDVT, NCVT)
78 On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten
79 by P**T * VT. Not referenced if NCVT = 0.
80 LDVT (input) INTEGER
82 The leading dimension of the array VT. LDVT >= max(1,N) if
83 NCVT > 0; LDVT >= 1 if NCVT = 0.
84 U (input/output) REAL array, dimension (LDU, N)
86 On entry, an NRU-by-N matrix U. On exit, U is overwritten by U
87 * Q. Not referenced if NRU = 0.
88 LDU (input) INTEGER
90 The leading dimension of the array U. LDU >= max(1,NRU).
91 C (input/output) REAL array, dimension (LDC, NCC)
93 On entry, an N-by-NCC matrix C. On exit, C is overwritten by
94 Q**T * C. Not referenced if NCC = 0.
95 LDC (input) INTEGER
97 The leading dimension of the array C. LDC >= max(1,N) if NCC >
98 0; LDC >=1 if NCC = 0.
99 WORK (workspace) REAL array, dimension (4*N)
101 INFO (output) INTEGER
103 = 0: successful exit
104 < 0: If INFO = -i, the i-th argument had an illegal value
105 > 0: if NCVT = NRU = NCC = 0, = 1, a split was marked by a pos‐
106 itive value in E = 2, current block of Z not diagonalized after
107 30*N iterations (in inner while loop) = 3, termination crite‐
108 rion of outer while loop not met (program created more than N
109 unreduced blocks) else NCVT = NRU = NCC = 0, the algorithm did
110 not converge; D and E contain the elements of a bidiagonal
111 matrix which is orthogonally similar to the input matrix B; if
112 INFO = i, i elements of E have not converged to zero.
113
115 TOLMUL REAL, default = max(10,min(100,EPS**(-1/8)))
116 TOLMUL controls the convergence criterion of the QR loop. If
117 it is positive, TOLMUL*EPS is the desired relative precision in
118 the computed singular values. If it is negative, abs(TOL‐
119 MUL*EPS*sigma_max) is the desired absolute accuracy in the com‐
120 puted singular values (corresponds to relative accuracy
121 abs(TOLMUL*EPS) in the largest singular value. abs(TOLMUL)
122 should be between 1 and 1/EPS, and preferably between 10 (for
123 fast convergence) and .1/EPS (for there to be some accuracy in
124 the results). Default is to lose at either one eighth or 2 of
125 the available decimal digits in each computed singular value
126 (whichever is smaller).
127 MAXITR INTEGER, default = 6
129 MAXITR controls the maximum number of passes of the algorithm
130 through its inner loop. The algorithms stops (and so fails to
131 converge) if the number of passes through the inner loop
132 exceeds MAXITR*N**2.
133 LAPACK routine (version 3.2) November 2008 SBDSQR(1)