1ZGGSVP(1) LAPACK routine (version 3.2) ZGGSVP(1)
2
6 ZGGSVP - computes unitary matrices U, V and Q such that N-K-L K L
7 U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0
8
10 SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
11 TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK,
12 TAU, WORK, INFO )
13 CHARACTER JOBQ, JOBU, JOBV
15 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
17 DOUBLE PRECISION TOLA, TOLB
19 INTEGER IWORK( * )
21 DOUBLE PRECISION RWORK( * )
23 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), TAU( * ), U(
25 LDU, * ), V( LDV, * ), WORK( * )
26
28 ZGGSVP computes unitary matrices U, V and Q such that
29 L ( 0 0 A23 )
30 M-K-L ( 0 0 0 )
31 N-K-L K L
32 = K ( 0 A12 A13 ) if M-K-L < 0;
33 M-K ( 0 0 A23 )
34 N-K-L K L
35 V'*B*Q = L ( 0 0 B13 )
36 P-L ( 0 0 0 )
37 where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular upper
38 triangular; A23 is L-by-L upper triangular if M-K-L >= 0, otherwise A23
39 is (M-K)-by-L upper trapezoidal. K+L = the effective numerical rank of
40 the (M+P)-by-N matrix (A',B')'. Z' denotes the conjugate transpose of
41 Z.
42 This decomposition is the preprocessing step for computing the General‐
43 ized Singular Value Decomposition (GSVD), see subroutine ZGGSVD.
44
46 JOBU (input) CHARACTER*1
47 = 'U': Unitary matrix U is computed;
48 = 'N': U is not computed.
49 JOBV (input) CHARACTER*1
51 = 'V': Unitary matrix V is computed;
52 = 'N': V is not computed.
53 JOBQ (input) CHARACTER*1
55 = 'Q': Unitary matrix Q is computed;
56 = 'N': Q is not computed.
57 M (input) INTEGER
59 The number of rows of the matrix A. M >= 0.
60 P (input) INTEGER
62 The number of rows of the matrix B. P >= 0.
63 N (input) INTEGER
65 The number of columns of the matrices A and B. N >= 0.
66 A (input/output) COMPLEX*16 array, dimension (LDA,N)
68 On entry, the M-by-N matrix A. On exit, A contains the trian‐
69 gular (or trapezoidal) matrix described in the Purpose section.
70 LDA (input) INTEGER
72 The leading dimension of the array A. LDA >= max(1,M).
73 B (input/output) COMPLEX*16 array, dimension (LDB,N)
75 On entry, the P-by-N matrix B. On exit, B contains the trian‐
76 gular matrix described in the Purpose section.
77 LDB (input) INTEGER
79 The leading dimension of the array B. LDB >= max(1,P).
80 TOLA (input) DOUBLE PRECISION
82 TOLB (input) DOUBLE PRECISION TOLA and TOLB are the thresh‐
83 olds to determine the effective numerical rank of matrix B and
84 a subblock of A. Generally, they are set to TOLA =
85 MAX(M,N)*norm(A)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS. The
86 size of TOLA and TOLB may affect the size of backward errors of
87 the decomposition.
88 K (output) INTEGER
90 L (output) INTEGER On exit, K and L specify the dimension
91 of the subblocks described in Purpose section. K + L = effec‐
92 tive numerical rank of (A',B')'.
93 U (output) COMPLEX*16 array, dimension (LDU,M)
95 If JOBU = 'U', U contains the unitary matrix U. If JOBU = 'N',
96 U is not referenced.
97 LDU (input) INTEGER
99 The leading dimension of the array U. LDU >= max(1,M) if JOBU =
100 'U'; LDU >= 1 otherwise.
101 V (output) COMPLEX*16 array, dimension (LDV,P)
103 If JOBV = 'V', V contains the unitary matrix V. If JOBV = 'N',
104 V is not referenced.
105 LDV (input) INTEGER
107 The leading dimension of the array V. LDV >= max(1,P) if JOBV =
108 'V'; LDV >= 1 otherwise.
109 Q (output) COMPLEX*16 array, dimension (LDQ,N)
111 If JOBQ = 'Q', Q contains the unitary matrix Q. If JOBQ = 'N',
112 Q is not referenced.
113 LDQ (input) INTEGER
115 The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ =
116 'Q'; LDQ >= 1 otherwise.
117 IWORK (workspace) INTEGER array, dimension (N)
119 RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
121 TAU (workspace) COMPLEX*16 array, dimension (N)
123 WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P))
125 INFO (output) INTEGER
127 = 0: successful exit
128 < 0: if INFO = -i, the i-th argument had an illegal value.
129
131 The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization
132 with column pivoting to detect the effective numerical rank of the a
133 matrix. It may be replaced by a better rank determination strategy.
134 LAPACK routine (version 3.2) November 2008 ZGGSVP(1)