1 SUBROUTINE ZGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
2 $ INFO )
3 *
4 * -- LAPACK driver routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 INTEGER INFO, LDA, LDB, LWORK, M, N, P
11 * ..
12 * .. Array Arguments ..
13 COMPLEX*16 A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
14 $ X( * ), Y( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * ZGGGLM solves a general Gauss-Markov linear model (GLM) problem:
21 *
22 * minimize || y ||_2 subject to d = A*x + B*y
23 * x
24 *
25 * where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
26 * given N-vector. It is assumed that M <= N <= M+P, and
27 *
28 * rank(A) = M and rank( A B ) = N.
29 *
30 * Under these assumptions, the constrained equation is always
31 * consistent, and there is a unique solution x and a minimal 2-norm
32 * solution y, which is obtained using a generalized QR factorization
33 * of the matrices (A, B) given by
34 *
35 * A = Q*(R), B = Q*T*Z.
36 * (0)
37 *
38 * In particular, if matrix B is square nonsingular, then the problem
39 * GLM is equivalent to the following weighted linear least squares
40 * problem
41 *
42 * minimize || inv(B)*(d-A*x) ||_2
43 * x
44 *
45 * where inv(B) denotes the inverse of B.
46 *
47 * Arguments
48 * =========
49 *
50 * N (input) INTEGER
51 * The number of rows of the matrices A and B. N >= 0.
52 *
53 * M (input) INTEGER
54 * The number of columns of the matrix A. 0 <= M <= N.
55 *
56 * P (input) INTEGER
57 * The number of columns of the matrix B. P >= N-M.
58 *
59 * A (input/output) COMPLEX*16 array, dimension (LDA,M)
60 * On entry, the N-by-M matrix A.
61 * On exit, the upper triangular part of the array A contains
62 * the M-by-M upper triangular matrix R.
63 *
64 * LDA (input) INTEGER
65 * The leading dimension of the array A. LDA >= max(1,N).
66 *
67 * B (input/output) COMPLEX*16 array, dimension (LDB,P)
68 * On entry, the N-by-P matrix B.
69 * On exit, if N <= P, the upper triangle of the subarray
70 * B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
71 * if N > P, the elements on and above the (N-P)th subdiagonal
72 * contain the N-by-P upper trapezoidal matrix T.
73 *
74 * LDB (input) INTEGER
75 * The leading dimension of the array B. LDB >= max(1,N).
76 *
77 * D (input/output) COMPLEX*16 array, dimension (N)
78 * On entry, D is the left hand side of the GLM equation.
79 * On exit, D is destroyed.
80 *
81 * X (output) COMPLEX*16 array, dimension (M)
82 * Y (output) COMPLEX*16 array, dimension (P)
83 * On exit, X and Y are the solutions of the GLM problem.
84 *
85 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
86 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
87 *
88 * LWORK (input) INTEGER
89 * The dimension of the array WORK. LWORK >= max(1,N+M+P).
90 * For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
91 * where NB is an upper bound for the optimal blocksizes for
92 * ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.
93 *
94 * If LWORK = -1, then a workspace query is assumed; the routine
95 * only calculates the optimal size of the WORK array, returns
96 * this value as the first entry of the WORK array, and no error
97 * message related to LWORK is issued by XERBLA.
98 *
99 * INFO (output) INTEGER
100 * = 0: successful exit.
101 * < 0: if INFO = -i, the i-th argument had an illegal value.
102 * = 1: the upper triangular factor R associated with A in the
103 * generalized QR factorization of the pair (A, B) is
104 * singular, so that rank(A) < M; the least squares
105 * solution could not be computed.
106 * = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal
107 * factor T associated with B in the generalized QR
108 * factorization of the pair (A, B) is singular, so that
109 * rank( A B ) < N; the least squares solution could not
110 * be computed.
111 *
112 * ===================================================================
113 *
114 * .. Parameters ..
115 COMPLEX*16 CZERO, CONE
116 PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
117 $ CONE = ( 1.0D+0, 0.0D+0 ) )
118 * ..
119 * .. Local Scalars ..
120 LOGICAL LQUERY
121 INTEGER I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3,
122 $ NB4, NP
123 * ..
124 * .. External Subroutines ..
125 EXTERNAL XERBLA, ZCOPY, ZGEMV, ZGGQRF, ZTRTRS, ZUNMQR,
126 $ ZUNMRQ
127 * ..
128 * .. External Functions ..
129 INTEGER ILAENV
130 EXTERNAL ILAENV
131 * ..
132 * .. Intrinsic Functions ..
133 INTRINSIC INT, MAX, MIN
134 * ..
135 * .. Executable Statements ..
136 *
137 * Test the input parameters
138 *
139 INFO = 0
140 NP = MIN( N, P )
141 LQUERY = ( LWORK.EQ.-1 )
142 IF( N.LT.0 ) THEN
143 INFO = -1
144 ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
145 INFO = -2
146 ELSE IF( P.LT.0 .OR. P.LT.N-M ) THEN
147 INFO = -3
148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
149 INFO = -5
150 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
151 INFO = -7
152 END IF
153 *
154 * Calculate workspace
155 *
156 IF( INFO.EQ.0) THEN
157 IF( N.EQ.0 ) THEN
158 LWKMIN = 1
159 LWKOPT = 1
160 ELSE
161 NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, M, -1, -1 )
162 NB2 = ILAENV( 1, 'ZGERQF', ' ', N, M, -1, -1 )
163 NB3 = ILAENV( 1, 'ZUNMQR', ' ', N, M, P, -1 )
164 NB4 = ILAENV( 1, 'ZUNMRQ', ' ', N, M, P, -1 )
165 NB = MAX( NB1, NB2, NB3, NB4 )
166 LWKMIN = M + N + P
167 LWKOPT = M + NP + MAX( N, P )*NB
168 END IF
169 WORK( 1 ) = LWKOPT
170 *
171 IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
172 INFO = -12
173 END IF
174 END IF
175 *
176 IF( INFO.NE.0 ) THEN
177 CALL XERBLA( 'ZGGGLM', -INFO )
178 RETURN
179 ELSE IF( LQUERY ) THEN
180 RETURN
181 END IF
182 *
183 * Quick return if possible
184 *
185 IF( N.EQ.0 )
186 $ RETURN
187 *
188 * Compute the GQR factorization of matrices A and B:
189 *
190 * Q**H*A = ( R11 ) M, Q**H*B*Z**H = ( T11 T12 ) M
191 * ( 0 ) N-M ( 0 T22 ) N-M
192 * M M+P-N N-M
193 *
194 * where R11 and T22 are upper triangular, and Q and Z are
195 * unitary.
196 *
197 CALL ZGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ),
198 $ WORK( M+NP+1 ), LWORK-M-NP, INFO )
199 LOPT = WORK( M+NP+1 )
200 *
201 * Update left-hand-side vector d = Q**H*d = ( d1 ) M
202 * ( d2 ) N-M
203 *
204 CALL ZUNMQR( 'Left', 'Conjugate transpose', N, 1, M, A, LDA, WORK,
205 $ D, MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
206 LOPT = MAX( LOPT, INT( WORK( M+NP+1 ) ) )
207 *
208 * Solve T22*y2 = d2 for y2
209 *
210 IF( N.GT.M ) THEN
211 CALL ZTRTRS( 'Upper', 'No transpose', 'Non unit', N-M, 1,
212 $ B( M+1, M+P-N+1 ), LDB, D( M+1 ), N-M, INFO )
213 *
214 IF( INFO.GT.0 ) THEN
215 INFO = 1
216 RETURN
217 END IF
218 *
219 CALL ZCOPY( N-M, D( M+1 ), 1, Y( M+P-N+1 ), 1 )
220 END IF
221 *
222 * Set y1 = 0
223 *
224 DO 10 I = 1, M + P - N
225 Y( I ) = CZERO
226 10 CONTINUE
227 *
228 * Update d1 = d1 - T12*y2
229 *
230 CALL ZGEMV( 'No transpose', M, N-M, -CONE, B( 1, M+P-N+1 ), LDB,
231 $ Y( M+P-N+1 ), 1, CONE, D, 1 )
232 *
233 * Solve triangular system: R11*x = d1
234 *
235 IF( M.GT.0 ) THEN
236 CALL ZTRTRS( 'Upper', 'No Transpose', 'Non unit', M, 1, A, LDA,
237 $ D, M, INFO )
238 *
239 IF( INFO.GT.0 ) THEN
240 INFO = 2
241 RETURN
242 END IF
243 *
244 * Copy D to X
245 *
246 CALL ZCOPY( M, D, 1, X, 1 )
247 END IF
248 *
249 * Backward transformation y = Z**H *y
250 *
251 CALL ZUNMRQ( 'Left', 'Conjugate transpose', P, 1, NP,
252 $ B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y,
253 $ MAX( 1, P ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
254 WORK( 1 ) = M + NP + MAX( LOPT, INT( WORK( M+NP+1 ) ) )
255 *
256 RETURN
257 *
258 * End of ZGGGLM
259 *
260 END
2 $ INFO )
3 *
4 * -- LAPACK driver routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 INTEGER INFO, LDA, LDB, LWORK, M, N, P
11 * ..
12 * .. Array Arguments ..
13 COMPLEX*16 A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
14 $ X( * ), Y( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * ZGGGLM solves a general Gauss-Markov linear model (GLM) problem:
21 *
22 * minimize || y ||_2 subject to d = A*x + B*y
23 * x
24 *
25 * where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
26 * given N-vector. It is assumed that M <= N <= M+P, and
27 *
28 * rank(A) = M and rank( A B ) = N.
29 *
30 * Under these assumptions, the constrained equation is always
31 * consistent, and there is a unique solution x and a minimal 2-norm
32 * solution y, which is obtained using a generalized QR factorization
33 * of the matrices (A, B) given by
34 *
35 * A = Q*(R), B = Q*T*Z.
36 * (0)
37 *
38 * In particular, if matrix B is square nonsingular, then the problem
39 * GLM is equivalent to the following weighted linear least squares
40 * problem
41 *
42 * minimize || inv(B)*(d-A*x) ||_2
43 * x
44 *
45 * where inv(B) denotes the inverse of B.
46 *
47 * Arguments
48 * =========
49 *
50 * N (input) INTEGER
51 * The number of rows of the matrices A and B. N >= 0.
52 *
53 * M (input) INTEGER
54 * The number of columns of the matrix A. 0 <= M <= N.
55 *
56 * P (input) INTEGER
57 * The number of columns of the matrix B. P >= N-M.
58 *
59 * A (input/output) COMPLEX*16 array, dimension (LDA,M)
60 * On entry, the N-by-M matrix A.
61 * On exit, the upper triangular part of the array A contains
62 * the M-by-M upper triangular matrix R.
63 *
64 * LDA (input) INTEGER
65 * The leading dimension of the array A. LDA >= max(1,N).
66 *
67 * B (input/output) COMPLEX*16 array, dimension (LDB,P)
68 * On entry, the N-by-P matrix B.
69 * On exit, if N <= P, the upper triangle of the subarray
70 * B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
71 * if N > P, the elements on and above the (N-P)th subdiagonal
72 * contain the N-by-P upper trapezoidal matrix T.
73 *
74 * LDB (input) INTEGER
75 * The leading dimension of the array B. LDB >= max(1,N).
76 *
77 * D (input/output) COMPLEX*16 array, dimension (N)
78 * On entry, D is the left hand side of the GLM equation.
79 * On exit, D is destroyed.
80 *
81 * X (output) COMPLEX*16 array, dimension (M)
82 * Y (output) COMPLEX*16 array, dimension (P)
83 * On exit, X and Y are the solutions of the GLM problem.
84 *
85 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
86 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
87 *
88 * LWORK (input) INTEGER
89 * The dimension of the array WORK. LWORK >= max(1,N+M+P).
90 * For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
91 * where NB is an upper bound for the optimal blocksizes for
92 * ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.
93 *
94 * If LWORK = -1, then a workspace query is assumed; the routine
95 * only calculates the optimal size of the WORK array, returns
96 * this value as the first entry of the WORK array, and no error
97 * message related to LWORK is issued by XERBLA.
98 *
99 * INFO (output) INTEGER
100 * = 0: successful exit.
101 * < 0: if INFO = -i, the i-th argument had an illegal value.
102 * = 1: the upper triangular factor R associated with A in the
103 * generalized QR factorization of the pair (A, B) is
104 * singular, so that rank(A) < M; the least squares
105 * solution could not be computed.
106 * = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal
107 * factor T associated with B in the generalized QR
108 * factorization of the pair (A, B) is singular, so that
109 * rank( A B ) < N; the least squares solution could not
110 * be computed.
111 *
112 * ===================================================================
113 *
114 * .. Parameters ..
115 COMPLEX*16 CZERO, CONE
116 PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
117 $ CONE = ( 1.0D+0, 0.0D+0 ) )
118 * ..
119 * .. Local Scalars ..
120 LOGICAL LQUERY
121 INTEGER I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3,
122 $ NB4, NP
123 * ..
124 * .. External Subroutines ..
125 EXTERNAL XERBLA, ZCOPY, ZGEMV, ZGGQRF, ZTRTRS, ZUNMQR,
126 $ ZUNMRQ
127 * ..
128 * .. External Functions ..
129 INTEGER ILAENV
130 EXTERNAL ILAENV
131 * ..
132 * .. Intrinsic Functions ..
133 INTRINSIC INT, MAX, MIN
134 * ..
135 * .. Executable Statements ..
136 *
137 * Test the input parameters
138 *
139 INFO = 0
140 NP = MIN( N, P )
141 LQUERY = ( LWORK.EQ.-1 )
142 IF( N.LT.0 ) THEN
143 INFO = -1
144 ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
145 INFO = -2
146 ELSE IF( P.LT.0 .OR. P.LT.N-M ) THEN
147 INFO = -3
148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
149 INFO = -5
150 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
151 INFO = -7
152 END IF
153 *
154 * Calculate workspace
155 *
156 IF( INFO.EQ.0) THEN
157 IF( N.EQ.0 ) THEN
158 LWKMIN = 1
159 LWKOPT = 1
160 ELSE
161 NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, M, -1, -1 )
162 NB2 = ILAENV( 1, 'ZGERQF', ' ', N, M, -1, -1 )
163 NB3 = ILAENV( 1, 'ZUNMQR', ' ', N, M, P, -1 )
164 NB4 = ILAENV( 1, 'ZUNMRQ', ' ', N, M, P, -1 )
165 NB = MAX( NB1, NB2, NB3, NB4 )
166 LWKMIN = M + N + P
167 LWKOPT = M + NP + MAX( N, P )*NB
168 END IF
169 WORK( 1 ) = LWKOPT
170 *
171 IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
172 INFO = -12
173 END IF
174 END IF
175 *
176 IF( INFO.NE.0 ) THEN
177 CALL XERBLA( 'ZGGGLM', -INFO )
178 RETURN
179 ELSE IF( LQUERY ) THEN
180 RETURN
181 END IF
182 *
183 * Quick return if possible
184 *
185 IF( N.EQ.0 )
186 $ RETURN
187 *
188 * Compute the GQR factorization of matrices A and B:
189 *
190 * Q**H*A = ( R11 ) M, Q**H*B*Z**H = ( T11 T12 ) M
191 * ( 0 ) N-M ( 0 T22 ) N-M
192 * M M+P-N N-M
193 *
194 * where R11 and T22 are upper triangular, and Q and Z are
195 * unitary.
196 *
197 CALL ZGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ),
198 $ WORK( M+NP+1 ), LWORK-M-NP, INFO )
199 LOPT = WORK( M+NP+1 )
200 *
201 * Update left-hand-side vector d = Q**H*d = ( d1 ) M
202 * ( d2 ) N-M
203 *
204 CALL ZUNMQR( 'Left', 'Conjugate transpose', N, 1, M, A, LDA, WORK,
205 $ D, MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
206 LOPT = MAX( LOPT, INT( WORK( M+NP+1 ) ) )
207 *
208 * Solve T22*y2 = d2 for y2
209 *
210 IF( N.GT.M ) THEN
211 CALL ZTRTRS( 'Upper', 'No transpose', 'Non unit', N-M, 1,
212 $ B( M+1, M+P-N+1 ), LDB, D( M+1 ), N-M, INFO )
213 *
214 IF( INFO.GT.0 ) THEN
215 INFO = 1
216 RETURN
217 END IF
218 *
219 CALL ZCOPY( N-M, D( M+1 ), 1, Y( M+P-N+1 ), 1 )
220 END IF
221 *
222 * Set y1 = 0
223 *
224 DO 10 I = 1, M + P - N
225 Y( I ) = CZERO
226 10 CONTINUE
227 *
228 * Update d1 = d1 - T12*y2
229 *
230 CALL ZGEMV( 'No transpose', M, N-M, -CONE, B( 1, M+P-N+1 ), LDB,
231 $ Y( M+P-N+1 ), 1, CONE, D, 1 )
232 *
233 * Solve triangular system: R11*x = d1
234 *
235 IF( M.GT.0 ) THEN
236 CALL ZTRTRS( 'Upper', 'No Transpose', 'Non unit', M, 1, A, LDA,
237 $ D, M, INFO )
238 *
239 IF( INFO.GT.0 ) THEN
240 INFO = 2
241 RETURN
242 END IF
243 *
244 * Copy D to X
245 *
246 CALL ZCOPY( M, D, 1, X, 1 )
247 END IF
248 *
249 * Backward transformation y = Z**H *y
250 *
251 CALL ZUNMRQ( 'Left', 'Conjugate transpose', P, 1, NP,
252 $ B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y,
253 $ MAX( 1, P ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
254 WORK( 1 ) = M + NP + MAX( LOPT, INT( WORK( M+NP+1 ) ) )
255 *
256 RETURN
257 *
258 * End of ZGGGLM
259 *
260 END