1CGGSVP ‐ unitary matrices U, V and Q such that N‐K‐L K L
2U'*A*Q = K ( 0 A12 A13 ) if M‐K‐L >= 0 SUBROUTINE CGGSVP( JOBU,
3JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V,
4LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, INFO )
5 CHARACTER JOBQ, JOBU, JOBV
6 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
7 REAL TOLA, TOLB
8 INTEGER IWORK( * )
9 REAL RWORK( * )
10 COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), TAU( * ), U(
11LDU, * ), V( LDV, * ), WORK( * ) CGGSVP computes unitary matrices
12U, V and Q such that
13 L ( 0 0 A23 )
14 M‐K‐L ( 0 0 0 )
15 N‐K‐L K L
17 = K ( 0 A12 A13 ) if M‐K‐L < 0;
18 M‐K ( 0 0 A23 )
19 N‐K‐L K L
21 V'*B*Q = L ( 0 0 B13 )
22 P‐L ( 0 0 0 )
23where the K‐by‐K matrix A12 and L‐by‐L matrix B13 are nonsingular
25upper triangular; A23 is L‐by‐L upper triangular if M‐K‐L >= 0,
26otherwise A23 is (M‐K)‐by‐L upper trapezoidal. K+L = the effec‐
27tive numerical rank of the (M+P)‐by‐N matrix (A',B')'. Z' de‐
28notes the conjugate transpose of Z.
29This decomposition is the preprocessing step for computing the
31Generalized Singular Value Decomposition (GSVD), see subroutine
32CGGSVD.
33JOBU (input) CHARACTER*1 = 'U': Unitary matrix U is computed;
35= 'N': U is not computed. JOBV (input) CHARACTER*1
36= 'V': Unitary matrix V is computed;
37= 'N': V is not computed. JOBQ (input) CHARACTER*1
38= 'Q': Unitary matrix Q is computed;
39= 'N': Q is not computed. M (input) INTEGER The number of
40rows of the matrix A. M >= 0. P (input) INTEGER The num‐
41ber of rows of the matrix B. P >= 0. N (input) INTEGER
42The number of columns of the matrices A and B. N >= 0. A
43(input/output) COMPLEX array, dimension (LDA,N) On entry, the M‐
44by‐N matrix A. On exit, A contains the triangular (or trape‐
45zoidal) matrix described in the Purpose section. LDA (input)
46INTEGER The leading dimension of the array A. LDA >= max(1,M). B
47(input/output) COMPLEX array, dimension (LDB,N) On entry, the P‐
48by‐N matrix B. On exit, B contains the triangular matrix de‐
49scribed in the Purpose section. LDB (input) INTEGER The
50leading dimension of the array B. LDB >= max(1,P). TOLA (in‐
51put) REAL TOLB (input) REAL TOLA and TOLB are the thresholds
52to determine the effective numerical rank of matrix B and a sub‐
53block of A. Generally, they are set to TOLA =
54MAX(M,N)*norm(A)*MACHEPS, TOLB = MAX(P,N)*norm(B)*MACHEPS. The
55size of TOLA and TOLB may affect the size of backward errors of
56the decomposition. K (output) INTEGER L (output) IN‐
57TEGER On exit, K and L specify the dimension of the subblocks de‐
58scribed in Purpose section. K + L = effective numerical rank of
59(A',B')'. U (output) COMPLEX array, dimension (LDU,M) If
60JOBU = 'U', U contains the unitary matrix U. If JOBU = 'N', U is
61not referenced. LDU (input) INTEGER The leading dimension of
62the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
63V (output) COMPLEX array, dimension (LDV,M) If JOBV = 'V',
64V contains the unitary matrix V. If JOBV = 'N', V is not refer‐
65enced. LDV (input) INTEGER The leading dimension of the ar‐
66ray V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise. Q
67(output) COMPLEX array, dimension (LDQ,N) If JOBQ = 'Q', Q con‐
68tains the unitary matrix Q. If JOBQ = 'N', Q is not referenced.
69LDQ (input) INTEGER The leading dimension of the array Q. LDQ
70>= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise. IWORK
71(workspace) INTEGER array, dimension (N) RWORK (workspace) REAL
72array, dimension (2*N) TAU (workspace) COMPLEX array, dimen‐
73sion (N) WORK (workspace) COMPLEX array, dimension
74(max(3*N,M,P)) INFO (output) INTEGER = 0: successful exit
75< 0: if INFO = ‐i, the i‐th argument had an illegal value. The
76subroutine uses LAPACK subroutine CGEQPF for the QR factorization
77with column pivoting to detect the effective numerical rank of
78the a matrix. It may be replaced by a better rank determination
79strategy.
80