1ZTGSJA(1) LAPACK routine (version 3.1) ZTGSJA(1)
2
6 ZTGSJA - the generalized singular value decomposition (GSVD) of two
7 complex upper triangular (or trapezoidal) matrices A and B
8
10 SUBROUTINE ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
11 TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
12 WORK, NCYCLE, INFO )
13 CHARACTER JOBQ, JOBU, JOBV
15 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, NCYCLE, P
17 DOUBLE PRECISION TOLA, TOLB
19 DOUBLE PRECISION ALPHA( * ), BETA( * )
21 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ),
23 V( LDV, * ), WORK( * )
24
26 ZTGSJA computes the generalized singular value decomposition (GSVD) of
27 two complex upper triangular (or trapezoidal) matrices A and B.
28 On entry, it is assumed that matrices A and B have the following forms,
30 which may be obtained by the preprocessing subroutine ZGGSVP from a
31 general M-by-N matrix A and P-by-N matrix B:
32 N-K-L K L
34 A = K ( 0 A12 A13 ) if M-K-L >= 0;
35 L ( 0 0 A23 )
36 M-K-L ( 0 0 0 )
37 N-K-L K L
39 A = K ( 0 A12 A13 ) if M-K-L < 0;
40 M-K ( 0 0 A23 )
41 N-K-L K L
43 B = L ( 0 0 B13 )
44 P-L ( 0 0 0 )
45 where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular upper
47 triangular; A23 is L-by-L upper triangular if M-K-L >= 0, otherwise A23
48 is (M-K)-by-L upper trapezoidal.
49 On exit,
51 U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),
53 where U, V and Q are unitary matrices, Z' denotes the conjugate trans‐
55 pose of Z, R is a nonsingular upper triangular matrix, and D1 and D2
56 are ``diagonal'' matrices, which are of the following structures:
57 If M-K-L >= 0,
59 K L
61 D1 = K ( I 0 )
62 L ( 0 C )
63 M-K-L ( 0 0 )
64 K L
66 D2 = L ( 0 S )
67 P-L ( 0 0 )
68 N-K-L K L
70 ( 0 R ) = K ( 0 R11 R12 ) K
71 L ( 0 0 R22 ) L
72 where
74 C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
76 S = diag( BETA(K+1), ... , BETA(K+L) ),
77 C**2 + S**2 = I.
78 R is stored in A(1:K+L,N-K-L+1:N) on exit.
80 If M-K-L < 0,
82 K M-K K+L-M
84 D1 = K ( I 0 0 )
85 M-K ( 0 C 0 )
86 K M-K K+L-M
88 D2 = M-K ( 0 S 0 )
89 K+L-M ( 0 0 I )
90 P-L ( 0 0 0 )
91 N-K-L K M-K K+L-M
93 M-K ( 0 0 R22 R23 )
95 K+L-M ( 0 0 0 R33 )
96 where
98 C = diag( ALPHA(K+1), ... , ALPHA(M) ),
99 S = diag( BETA(K+1), ... , BETA(M) ),
100 C**2 + S**2 = I.
101 R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
103 ( 0 R22 R23 )
104 in B(M-K+1:L,N+M-K-L+1:N) on exit.
105 The computation of the unitary transformation matrices U, V or Q is
107 optional. These matrices may either be formed explicitly, or they may
108 be postmultiplied into input matrices U1, V1, or Q1.
109
112 JOBU (input) CHARACTER*1
113 = 'U': U must contain a unitary matrix U1 on entry, and the
114 product U1*U is returned; = 'I': U is initialized to the unit
115 matrix, and the unitary matrix U is returned; = 'N': U is not
116 computed.
117 JOBV (input) CHARACTER*1
119 = 'V': V must contain a unitary matrix V1 on entry, and the
120 product V1*V is returned; = 'I': V is initialized to the unit
121 matrix, and the unitary matrix V is returned; = 'N': V is not
122 computed.
123 JOBQ (input) CHARACTER*1
125 = 'Q': Q must contain a unitary matrix Q1 on entry, and the
126 product Q1*Q is returned; = 'I': Q is initialized to the unit
127 matrix, and the unitary matrix Q is returned; = 'N': Q is not
128 computed.
129 M (input) INTEGER
131 The number of rows of the matrix A. M >= 0.
132 P (input) INTEGER
134 The number of rows of the matrix B. P >= 0.
135 N (input) INTEGER
137 The number of columns of the matrices A and B. N >= 0.
138 K (input) INTEGER
140 L (input) INTEGER K and L specify the subblocks in the
141 input matrices A and B:
142 A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N) of A
143 and B, whose GSVD is going to be computed by ZTGSJA. See Fur‐
144 ther details.
145 A (input/output) COMPLEX*16 array, dimension (LDA,N)
147 On entry, the M-by-N matrix A. On exit, A(N-K+1:N,1:MIN(K+L,M)
148 ) contains the triangular matrix R or part of R. See Purpose
149 for details.
150 LDA (input) INTEGER
152 The leading dimension of the array A. LDA >= max(1,M).
153 B (input/output) COMPLEX*16 array, dimension (LDB,N)
155 On entry, the P-by-N matrix B. On exit, if necessary, B(M-
156 K+1:L,N+M-K-L+1:N) contains a part of R. See Purpose for
157 details.
158 LDB (input) INTEGER
160 The leading dimension of the array B. LDB >= max(1,P).
161 TOLA (input) DOUBLE PRECISION
163 TOLB (input) DOUBLE PRECISION TOLA and TOLB are the conver‐
164 gence criteria for the Jacobi- Kogbetliantz iteration proce‐
165 dure. Generally, they are the same as used in the preprocessing
166 step, say TOLA = MAX(M,N)*norm(A)*MAZHEPS, TOLB =
167 MAX(P,N)*norm(B)*MAZHEPS.
168 ALPHA (output) DOUBLE PRECISION array, dimension (N)
170 BETA (output) DOUBLE PRECISION array, dimension (N) On exit,
171 ALPHA and BETA contain the generalized singular value pairs of
172 A and B; ALPHA(1:K) = 1,
173 BETA(1:K) = 0, and if M-K-L >= 0, ALPHA(K+1:K+L) = diag(C),
174 BETA(K+1:K+L) = diag(S), or if M-K-L < 0, ALPHA(K+1:M)= C,
175 ALPHA(M+1:K+L)= 0
176 BETA(K+1:M) = S, BETA(M+1:K+L) = 1. Furthermore, if K+L < N,
177 ALPHA(K+L+1:N) = 0
178 BETA(K+L+1:N) = 0.
179 U (input/output) COMPLEX*16 array, dimension (LDU,M)
181 On entry, if JOBU = 'U', U must contain a matrix U1 (usually
182 the unitary matrix returned by ZGGSVP). On exit, if JOBU =
183 'I', U contains the unitary matrix U; if JOBU = 'U', U contains
184 the product U1*U. If JOBU = 'N', U is not referenced.
185 LDU (input) INTEGER
187 The leading dimension of the array U. LDU >= max(1,M) if JOBU =
188 'U'; LDU >= 1 otherwise.
189 V (input/output) COMPLEX*16 array, dimension (LDV,P)
191 On entry, if JOBV = 'V', V must contain a matrix V1 (usually
192 the unitary matrix returned by ZGGSVP). On exit, if JOBV =
193 'I', V contains the unitary matrix V; if JOBV = 'V', V contains
194 the product V1*V. If JOBV = 'N', V is not referenced.
195 LDV (input) INTEGER
197 The leading dimension of the array V. LDV >= max(1,P) if JOBV =
198 'V'; LDV >= 1 otherwise.
199 Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
201 On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
202 the unitary matrix returned by ZGGSVP). On exit, if JOBQ =
203 'I', Q contains the unitary matrix Q; if JOBQ = 'Q', Q contains
204 the product Q1*Q. If JOBQ = 'N', Q is not referenced.
205 LDQ (input) INTEGER
207 The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ =
208 'Q'; LDQ >= 1 otherwise.
209 WORK (workspace) COMPLEX*16 array, dimension (2*N)
211 NCYCLE (output) INTEGER
213 The number of cycles required for convergence.
214 INFO (output) INTEGER
216 = 0: successful exit
217 < 0: if INFO = -i, the i-th argument had an illegal value.
218 = 1: the procedure does not converge after MAXIT cycles.
219
221 MAXIT INTEGER
222 MAXIT specifies the total loops that the iterative procedure
223 may take. If after MAXIT cycles, the routine fails to converge,
224 we return INFO = 1.
225 Further Details ===============
227 ZTGSJA essentially uses a variant of Kogbetliantz algorithm to
229 reduce min(L,M-K)-by-L triangular (or trapezoidal) matrix A23
230 and L-by-L matrix B13 to the form:
231 U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
233 where U1, V1 and Q1 are unitary matrix, and Z' is the conjugate
235 transpose of Z. C1 and S1 are diagonal matrices satisfying
236 C1**2 + S1**2 = I,
238 and R1 is an L-by-L nonsingular upper triangular matrix.
240 LAPACK routine (version 3.1) November 2006 ZTGSJA(1)