1CGGSVP(1) LAPACK routine (version 3.2) CGGSVP(1)
2
6 CGGSVP - 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 CGGSVP( 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 REAL TOLA, TOLB
19 INTEGER IWORK( * )
21 REAL RWORK( * )
23 COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), TAU( * ), U(
25 LDU, * ), V( LDV, * ), WORK( * )
26
28 CGGSVP 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 CGGSVD.
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 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 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) REAL
82 TOLB (input) REAL TOLA and TOLB are the thresholds to deter‐
83 mine the effective numerical rank of matrix B and a subblock of
84 A. Generally, they are set to TOLA = MAX(M,N)*norm(A)*MACHEPS,
85 TOLB = MAX(P,N)*norm(B)*MACHEPS. The size of TOLA and TOLB may
86 affect the size of backward errors of the decomposition.
87 K (output) INTEGER
89 L (output) INTEGER On exit, K and L specify the dimension
90 of the subblocks described in Purpose section. K + L = effec‐
91 tive numerical rank of (A',B')'.
92 U (output) COMPLEX array, dimension (LDU,M)
94 If JOBU = 'U', U contains the unitary matrix U. If JOBU = 'N',
95 U is not referenced.
96 LDU (input) INTEGER
98 The leading dimension of the array U. LDU >= max(1,M) if JOBU =
99 'U'; LDU >= 1 otherwise.
100 V (output) COMPLEX array, dimension (LDV,P)
102 If JOBV = 'V', V contains the unitary matrix V. If JOBV = 'N',
103 V is not referenced.
104 LDV (input) INTEGER
106 The leading dimension of the array V. LDV >= max(1,P) if JOBV =
107 'V'; LDV >= 1 otherwise.
108 Q (output) COMPLEX array, dimension (LDQ,N)
110 If JOBQ = 'Q', Q contains the unitary matrix Q. If JOBQ = 'N',
111 Q is not referenced.
112 LDQ (input) INTEGER
114 The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ =
115 'Q'; LDQ >= 1 otherwise.
116 IWORK (workspace) INTEGER array, dimension (N)
118 RWORK (workspace) REAL array, dimension (2*N)
120 TAU (workspace) COMPLEX array, dimension (N)
122 WORK (workspace) COMPLEX array, dimension (max(3*N,M,P))
124 INFO (output) INTEGER
126 = 0: successful exit
127 < 0: if INFO = -i, the i-th argument had an illegal value.
128
130 The subroutine uses LAPACK subroutine CGEQPF for the QR factorization
131 with column pivoting to detect the effective numerical rank of the a
132 matrix. It may be replaced by a better rank determination strategy.
133 LAPACK routine (version 3.2) November 2008 CGGSVP(1)