1ZGGSVP ‐ 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 ZGGSVP( 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 DOUBLE PRECISION TOLA, TOLB
8 INTEGER IWORK( * )
9 DOUBLE PRECISION RWORK( * )
10 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), TAU( * ),
11U( LDU, * ), V( LDV, * ), WORK( * ) ZGGSVP computes unitary ma‐
12trices U, 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
32ZGGSVD.
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*16 array, dimension (LDA,N) On entry, the
44M‐by‐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*16 array, dimension (LDB,N) On entry, the
48P‐by‐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) DOUBLE PRECISION TOLB (input) DOUBLE PRECISION TOLA and
52TOLB are the thresholds to determine the effective numerical rank
53of matrix B and a subblock of A. Generally, they are set to TOLA
54= MAX(M,N)*norm(A)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS. 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*16 array, dimension (LDU,M)
60If JOBU = 'U', U contains the unitary matrix U. If JOBU = 'N', U
61is not referenced. LDU (input) INTEGER The leading dimension
62of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 other‐
63wise. V (output) COMPLEX*16 array, dimension (LDV,M) If
64JOBV = 'V', V contains the unitary matrix V. If JOBV = 'N', V is
65not referenced. LDV (input) INTEGER The leading dimension of
66the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
67Q (output) COMPLEX*16 array, dimension (LDQ,N) If JOBQ =
68'Q', Q contains the unitary matrix Q. If JOBQ = 'N', Q is not
69referenced. LDQ (input) INTEGER The leading dimension of the
70array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
71IWORK (workspace) INTEGER array, dimension (N) RWORK
72(workspace) DOUBLE PRECISION array, dimension (2*N) TAU
73(workspace) COMPLEX*16 array, dimension (N) WORK (workspace)
74COMPLEX*16 array, dimension (max(3*N,M,P)) INFO (output) INTE‐
75GER = 0: successful exit
76< 0: if INFO = ‐i, the i‐th argument had an illegal value. The
77subroutine uses LAPACK subroutine ZGEQPF for the QR factorization
78with column pivoting to detect the effective numerical rank of
79the a matrix. It may be replaced by a better rank determination
80strategy.
81