1ZGESDD(1) LAPACK driver routine (version 3.1) ZGESDD(1)
2
6 ZGESDD - the singular value decomposition (SVD) of a complex M-by-N
7 matrix A, optionally computing the left and/or right singular vectors,
8 by using divide-and-conquer method
9
11 SUBROUTINE ZGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
12 LWORK, RWORK, IWORK, INFO )
13 CHARACTER JOBZ
15 INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
17 INTEGER IWORK( * )
19 DOUBLE PRECISION RWORK( * ), S( * )
21 COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
23
25 ZGESDD computes the singular value decomposition (SVD) of a complex M-
26 by-N matrix A, optionally computing the left and/or right singular vec‐
27 tors, by using divide-and-conquer method. The SVD is written
28 A = U * SIGMA * conjugate-transpose(V)
30 where SIGMA is an M-by-N matrix which is zero except for its min(m,n)
32 diagonal elements, U is an M-by-M unitary matrix, and V is an N-by-N
33 unitary matrix. The diagonal elements of SIGMA are the singular values
34 of A; they are real and non-negative, and are returned in descending
35 order. The first min(m,n) columns of U and V are the left and right
36 singular vectors of A.
37 Note that the routine returns VT = V**H, not V.
39 The divide and conquer algorithm makes very mild assumptions about
41 floating point arithmetic. It will work on machines with a guard digit
42 in add/subtract, or on those binary machines without guard digits which
43 subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
44 conceivably fail on hexadecimal or decimal machines without guard dig‐
45 its, but we know of none.
46
49 JOBZ (input) CHARACTER*1
50 Specifies options for computing all or part of the matrix U:
51 = 'A': all M columns of U and all N rows of V**H are returned
52 in the arrays U and VT; = 'S': the first min(M,N) columns of U
53 and the first min(M,N) rows of V**H are returned in the arrays
54 U and VT; = 'O': If M >= N, the first N columns of U are over‐
55 written in the array A and all rows of V**H are returned in the
56 array VT; otherwise, all columns of U are returned in the array
57 U and the first M rows of V**H are overwritten in the array A;
58 = 'N': no columns of U or rows of V**H are computed.
59 M (input) INTEGER
61 The number of rows of the input matrix A. M >= 0.
62 N (input) INTEGER
64 The number of columns of the input matrix A. N >= 0.
65 A (input/output) COMPLEX*16 array, dimension (LDA,N)
67 On entry, the M-by-N matrix A. On exit, if JOBZ = 'O', A is
68 overwritten with the first N columns of U (the left singular
69 vectors, stored columnwise) if M >= N; A is overwritten with
70 the first M rows of V**H (the right singular vectors, stored
71 rowwise) otherwise. if JOBZ .ne. 'O', the contents of A are
72 destroyed.
73 LDA (input) INTEGER
75 The leading dimension of the array A. LDA >= max(1,M).
76 S (output) DOUBLE PRECISION array, dimension (min(M,N))
78 The singular values of A, sorted so that S(i) >= S(i+1).
79 U (output) COMPLEX*16 array, dimension (LDU,UCOL)
81 UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N; UCOL = min(M,N)
82 if JOBZ = 'S'. If JOBZ = 'A' or JOBZ = 'O' and M < N, U con‐
83 tains the M-by-M unitary matrix U; if JOBZ = 'S', U contains
84 the first min(M,N) columns of U (the left singular vectors,
85 stored columnwise); if JOBZ = 'O' and M >= N, or JOBZ = 'N', U
86 is not referenced.
87 LDU (input) INTEGER
89 The leading dimension of the array U. LDU >= 1; if JOBZ = 'S'
90 or 'A' or JOBZ = 'O' and M < N, LDU >= M.
91 VT (output) COMPLEX*16 array, dimension (LDVT,N)
93 If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the N-by-N
94 unitary matrix V**H; if JOBZ = 'S', VT contains the first
95 min(M,N) rows of V**H (the right singular vectors, stored row‐
96 wise); if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not refer‐
97 enced.
98 LDVT (input) INTEGER
100 The leading dimension of the array VT. LDVT >= 1; if JOBZ =
101 'A' or JOBZ = 'O' and M >= N, LDVT >= N; if JOBZ = 'S', LDVT >=
102 min(M,N).
103 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
105 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
106 LWORK (input) INTEGER
108 The dimension of the array WORK. LWORK >= 1. if JOBZ = 'N',
109 LWORK >= 2*min(M,N)+max(M,N). if JOBZ = 'O', LWORK >=
110 2*min(M,N)*min(M,N)+2*min(M,N)+max(M,N). if JOBZ = 'S' or 'A',
111 LWORK >= min(M,N)*min(M,N)+2*min(M,N)+max(M,N). For good per‐
112 formance, LWORK should generally be larger.
113 If LWORK = -1, a workspace query is assumed. The optimal size
115 for the WORK array is calculated and stored in WORK(1), and no
116 other work except argument checking is performed.
117 RWORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
119 If JOBZ = 'N', LRWORK >= 5*min(M,N). Otherwise, LRWORK >=
120 5*min(M,N)*min(M,N) + 7*min(M,N)
121 IWORK (workspace) INTEGER array, dimension (8*min(M,N))
123 INFO (output) INTEGER
125 = 0: successful exit.
126 < 0: if INFO = -i, the i-th argument had an illegal value.
127 > 0: The updating process of DBDSDC did not converge.
128
130 Based on contributions by
131 Ming Gu and Huan Ren, Computer Science Division, University of
132 California at Berkeley, USA
133 LAPACK driver routine (version 3.N1o)vember 2006 ZGESDD(1)