1SGGSVP(1) LAPACK routine (version 3.2) SGGSVP(1)
2
6 SGGSVP - computes orthogonal 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 SGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
11 TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU,
12 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 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), TAU( * ), U(
23 LDU, * ), V( LDV, * ), WORK( * )
24
26 SGGSVP computes orthogonal matrices U, V and Q such that
27 L ( 0 0 A23 )
28 M-K-L ( 0 0 0 )
29 N-K-L K L
30 = K ( 0 A12 A13 ) if M-K-L < 0;
31 M-K ( 0 0 A23 )
32 N-K-L K L
33 V'*B*Q = L ( 0 0 B13 )
34 P-L ( 0 0 0 )
35 where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular upper
36 triangular; A23 is L-by-L upper triangular if M-K-L >= 0, otherwise A23
37 is (M-K)-by-L upper trapezoidal. K+L = the effective numerical rank of
38 the (M+P)-by-N matrix (A',B')'. Z' denotes the transpose of Z.
39 This decomposition is the preprocessing step for computing the General‐
40 ized Singular Value Decomposition (GSVD), see subroutine SGGSVD.
41
43 JOBU (input) CHARACTER*1
44 = 'U': Orthogonal matrix U is computed;
45 = 'N': U is not computed.
46 JOBV (input) CHARACTER*1
48 = 'V': Orthogonal matrix V is computed;
49 = 'N': V is not computed.
50 JOBQ (input) CHARACTER*1
52 = 'Q': Orthogonal matrix Q is computed;
53 = 'N': Q is not computed.
54 M (input) INTEGER
56 The number of rows of the matrix A. M >= 0.
57 P (input) INTEGER
59 The number of rows of the matrix B. P >= 0.
60 N (input) INTEGER
62 The number of columns of the matrices A and B. N >= 0.
63 A (input/output) REAL array, dimension (LDA,N)
65 On entry, the M-by-N matrix A. On exit, A contains the trian‐
66 gular (or trapezoidal) matrix described in the Purpose section.
67 LDA (input) INTEGER
69 The leading dimension of the array A. LDA >= max(1,M).
70 B (input/output) REAL array, dimension (LDB,N)
72 On entry, the P-by-N matrix B. On exit, B contains the trian‐
73 gular matrix described in the Purpose section.
74 LDB (input) INTEGER
76 The leading dimension of the array B. LDB >= max(1,P).
77 TOLA (input) REAL
79 TOLB (input) REAL TOLA and TOLB are the thresholds to deter‐
80 mine the effective numerical rank of matrix B and a subblock of
81 A. Generally, they are set to TOLA = MAX(M,N)*norm(A)*MACHEPS,
82 TOLB = MAX(P,N)*norm(B)*MACHEPS. The size of TOLA and TOLB may
83 affect the size of backward errors of the decomposition.
84 K (output) INTEGER
86 L (output) INTEGER On exit, K and L specify the dimension
87 of the subblocks described in Purpose. K + L = effective
88 numerical rank of (A',B')'.
89 U (output) REAL array, dimension (LDU,M)
91 If JOBU = 'U', U contains the orthogonal matrix U. If JOBU =
92 'N', U is not referenced.
93 LDU (input) INTEGER
95 The leading dimension of the array U. LDU >= max(1,M) if JOBU =
96 'U'; LDU >= 1 otherwise.
97 V (output) REAL array, dimension (LDV,P)
99 If JOBV = 'V', V contains the orthogonal matrix V. If JOBV =
100 'N', V is not referenced.
101 LDV (input) INTEGER
103 The leading dimension of the array V. LDV >= max(1,P) if JOBV =
104 'V'; LDV >= 1 otherwise.
105 Q (output) REAL array, dimension (LDQ,N)
107 If JOBQ = 'Q', Q contains the orthogonal matrix Q. If JOBQ =
108 'N', Q is not referenced.
109 LDQ (input) INTEGER
111 The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ =
112 'Q'; LDQ >= 1 otherwise.
113 IWORK (workspace) INTEGER array, dimension (N)
115 TAU (workspace) REAL array, dimension (N)
117 WORK (workspace) REAL array, dimension (max(3*N,M,P))
119 INFO (output) INTEGER
121 = 0: successful exit
122 < 0: if INFO = -i, the i-th argument had an illegal value.
123
125 The subroutine uses LAPACK subroutine SGEQPF for the QR factorization
126 with column pivoting to detect the effective numerical rank of the a
127 matrix. It may be replaced by a better rank determination strategy.
128 LAPACK routine (version 3.2) November 2008 SGGSVP(1)