1ZGESVD(1) LAPACK driver routine (version 3.1) ZGESVD(1)
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6 ZGESVD - the singular value decomposition (SVD) of a complex M-by-N
7 matrix A, optionally computing the left and/or right singular vectors
8
10 SUBROUTINE ZGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT,
11 WORK, LWORK, RWORK, INFO )
12 CHARACTER JOBU, JOBVT
14 INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
16 DOUBLE PRECISION RWORK( * ), S( * )
18 COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
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22 ZGESVD computes the singular value decomposition (SVD) of a complex M-
23 by-N matrix A, optionally computing the left and/or right singular vec‐
24 tors. The SVD is written
25 A = U * SIGMA * conjugate-transpose(V)
27 where SIGMA is an M-by-N matrix which is zero except for its min(m,n)
29 diagonal elements, U is an M-by-M unitary matrix, and V is an N-by-N
30 unitary matrix. The diagonal elements of SIGMA are the singular values
31 of A; they are real and non-negative, and are returned in descending
32 order. The first min(m,n) columns of U and V are the left and right
33 singular vectors of A.
34 Note that the routine returns V**H, not V.
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39 JOBU (input) CHARACTER*1
40 Specifies options for computing all or part of the matrix U:
41 = 'A': all M columns of U are returned in array U:
42 = 'S': the first min(m,n) columns of U (the left singular vec‐
43 tors) are returned in the array U; = 'O': the first min(m,n)
44 columns of U (the left singular vectors) are overwritten on the
45 array A; = 'N': no columns of U (no left singular vectors) are
46 computed.
47 JOBVT (input) CHARACTER*1
49 Specifies options for computing all or part of the matrix V**H:
50 = 'A': all N rows of V**H are returned in the array VT;
51 = 'S': the first min(m,n) rows of V**H (the right singular
52 vectors) are returned in the array VT; = 'O': the first
53 min(m,n) rows of V**H (the right singular vectors) are over‐
54 written on the array A; = 'N': no rows of V**H (no right sin‐
55 gular vectors) are computed.
56 JOBVT and JOBU cannot both be 'O'.
58 M (input) INTEGER
60 The number of rows of the input matrix A. M >= 0.
61 N (input) INTEGER
63 The number of columns of the input matrix A. N >= 0.
64 A (input/output) COMPLEX*16 array, dimension (LDA,N)
66 On entry, the M-by-N matrix A. On exit, if JOBU = 'O', A is
67 overwritten with the first min(m,n) columns of U (the left sin‐
68 gular vectors, stored columnwise); if JOBVT = 'O', A is over‐
69 written with the first min(m,n) rows of V**H (the right singu‐
70 lar vectors, stored rowwise); if JOBU .ne. 'O' and JOBVT .ne.
71 'O', the contents of A are destroyed.
72 LDA (input) INTEGER
74 The leading dimension of the array A. LDA >= max(1,M).
75 S (output) DOUBLE PRECISION array, dimension (min(M,N))
77 The singular values of A, sorted so that S(i) >= S(i+1).
78 U (output) COMPLEX*16 array, dimension (LDU,UCOL)
80 (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'. If JOBU
81 = 'A', U contains the M-by-M unitary matrix U; if JOBU = 'S', U
82 contains the first min(m,n) columns of U (the left singular
83 vectors, stored columnwise); if JOBU = 'N' or 'O', U is not
84 referenced.
85 LDU (input) INTEGER
87 The leading dimension of the array U. LDU >= 1; if JOBU = 'S'
88 or 'A', LDU >= M.
89 VT (output) COMPLEX*16 array, dimension (LDVT,N)
91 If JOBVT = 'A', VT contains the N-by-N unitary matrix V**H; if
92 JOBVT = 'S', VT contains the first min(m,n) rows of V**H (the
93 right singular vectors, stored rowwise); if JOBVT = 'N' or 'O',
94 VT is not referenced.
95 LDVT (input) INTEGER
97 The leading dimension of the array VT. LDVT >= 1; if JOBVT =
98 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
99 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
101 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
102 LWORK (input) INTEGER
104 The dimension of the array WORK. LWORK >=
105 MAX(1,2*MIN(M,N)+MAX(M,N)). For good performance, LWORK should
106 generally be larger.
107 If LWORK = -1, then a workspace query is assumed; the routine
109 only calculates the optimal size of the WORK array, returns
110 this value as the first entry of the WORK array, and no error
111 message related to LWORK is issued by XERBLA.
112 RWORK (workspace) DOUBLE PRECISION array, dimension (5*min(M,N))
114 On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) contains the uncon‐
115 verged superdiagonal elements of an upper bidiagonal matrix B
116 whose diagonal is in S (not necessarily sorted). B satisfies A
117 = U * B * VT, so it has the same singular values as A, and sin‐
118 gular vectors related by U and VT.
119 INFO (output) INTEGER
121 = 0: successful exit.
122 < 0: if INFO = -i, the i-th argument had an illegal value.
123 > 0: if ZBDSQR did not converge, INFO specifies how many
124 superdiagonals of an intermediate bidiagonal form B did not
125 converge to zero. See the description of RWORK above for
126 details.
127 LAPACK driver routine (version 3.N1o)vember 2006 ZGESVD(1)