1ZGESVD(1) LAPACK driver routine (version 3.2) ZGESVD(1)
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6 ZGESVD - computes the singular value decomposition (SVD) of a complex
7 M-by-N matrix A, optionally computing the left and/or right singular
8 vectors
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11 SUBROUTINE ZGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT,
12 WORK, LWORK, RWORK, INFO )
13 CHARACTER JOBU, JOBVT
15 INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
17 DOUBLE PRECISION RWORK( * ), S( * )
19 COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
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23 ZGESVD computes the singular value decomposition (SVD) of a complex M-
24 by-N matrix A, optionally computing the left and/or right singular vec‐
25 tors. The SVD is written
26 A = U * SIGMA * conjugate-transpose(V)
27 where SIGMA is an M-by-N matrix which is zero except for its min(m,n)
28 diagonal elements, U is an M-by-M unitary matrix, and V is an N-by-N
29 unitary matrix. The diagonal elements of SIGMA are the singular values
30 of A; they are real and non-negative, and are returned in descending
31 order. The first min(m,n) columns of U and V are the left and right
32 singular vectors of A.
33 Note that the routine returns V**H, not V.
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36 JOBU (input) CHARACTER*1
37 Specifies options for computing all or part of the matrix U:
38 = 'A': all M columns of U are returned in array U:
39 = 'S': the first min(m,n) columns of U (the left singular vec‐
40 tors) are returned in the array U; = 'O': the first min(m,n)
41 columns of U (the left singular vectors) are overwritten on the
42 array A; = 'N': no columns of U (no left singular vectors) are
43 computed.
44 JOBVT (input) CHARACTER*1
46 Specifies options for computing all or part of the matrix V**H:
47 = 'A': all N rows of V**H are returned in the array VT;
48 = 'S': the first min(m,n) rows of V**H (the right singular
49 vectors) are returned in the array VT; = 'O': the first
50 min(m,n) rows of V**H (the right singular vectors) are over‐
51 written on the array A; = 'N': no rows of V**H (no right sin‐
52 gular vectors) are computed. JOBVT and JOBU cannot both be
53 'O'.
54 M (input) INTEGER
56 The number of rows of the input matrix A. M >= 0.
57 N (input) INTEGER
59 The number of columns of the input matrix A. N >= 0.
60 A (input/output) COMPLEX*16 array, dimension (LDA,N)
62 On entry, the M-by-N matrix A. On exit, if JOBU = 'O', A is
63 overwritten with the first min(m,n) columns of U (the left sin‐
64 gular vectors, stored columnwise); if JOBVT = 'O', A is over‐
65 written with the first min(m,n) rows of V**H (the right singu‐
66 lar vectors, stored rowwise); if JOBU .ne. 'O' and JOBVT .ne.
67 'O', the contents of A are destroyed.
68 LDA (input) INTEGER
70 The leading dimension of the array A. LDA >= max(1,M).
71 S (output) DOUBLE PRECISION array, dimension (min(M,N))
73 The singular values of A, sorted so that S(i) >= S(i+1).
74 U (output) COMPLEX*16 array, dimension (LDU,UCOL)
76 (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'. If JOBU
77 = 'A', U contains the M-by-M unitary matrix U; if JOBU = 'S', U
78 contains the first min(m,n) columns of U (the left singular
79 vectors, stored columnwise); if JOBU = 'N' or 'O', U is not
80 referenced.
81 LDU (input) INTEGER
83 The leading dimension of the array U. LDU >= 1; if JOBU = 'S'
84 or 'A', LDU >= M.
85 VT (output) COMPLEX*16 array, dimension (LDVT,N)
87 If JOBVT = 'A', VT contains the N-by-N unitary matrix V**H; if
88 JOBVT = 'S', VT contains the first min(m,n) rows of V**H (the
89 right singular vectors, stored rowwise); if JOBVT = 'N' or 'O',
90 VT is not referenced.
91 LDVT (input) INTEGER
93 The leading dimension of the array VT. LDVT >= 1; if JOBVT =
94 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
95 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
97 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
98 LWORK (input) INTEGER
100 The dimension of the array WORK. LWORK >=
101 MAX(1,2*MIN(M,N)+MAX(M,N)). For good performance, LWORK should
102 generally be larger. If LWORK = -1, then a workspace query is
103 assumed; the routine only calculates the optimal size of the
104 WORK array, returns this value as the first entry of the WORK
105 array, and no error message related to LWORK is issued by
106 XERBLA.
107 RWORK (workspace) DOUBLE PRECISION array, dimension (5*min(M,N))
109 On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) contains the uncon‐
110 verged superdiagonal elements of an upper bidiagonal matrix B
111 whose diagonal is in S (not necessarily sorted). B satisfies A
112 = U * B * VT, so it has the same singular values as A, and sin‐
113 gular vectors related by U and VT.
114 INFO (output) INTEGER
116 = 0: successful exit.
117 < 0: if INFO = -i, the i-th argument had an illegal value.
118 > 0: if ZBDSQR did not converge, INFO specifies how many
119 superdiagonals of an intermediate bidiagonal form B did not
120 converge to zero. See the description of RWORK above for
121 details.
122 LAPACK driver routine (version 3.N2o)vember 2008 ZGESVD(1)