1: SUBROUTINE ZGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK,
2: $ INFO )
3: *
4: * -- LAPACK driver routine (version 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * February 2007
8: *
9: * .. Scalar Arguments ..
10: INTEGER INFO, LDA, LDB, LWORK, M, N, P
11: * ..
12: * .. Array Arguments ..
13: COMPLEX*16 A( LDA, * ), B( LDB, * ), C( * ), D( * ),
14: $ WORK( * ), X( * )
15: * ..
16: *
17: * Purpose
18: * =======
19: *
20: * ZGGLSE solves the linear equality-constrained least squares (LSE)
21: * problem:
22: *
23: * minimize || c - A*x ||_2 subject to B*x = d
24: *
25: * where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
26: * M-vector, and d is a given P-vector. It is assumed that
27: * P <= N <= M+P, and
28: *
29: * rank(B) = P and rank( ( A ) ) = N.
30: * ( ( B ) )
31: *
32: * These conditions ensure that the LSE problem has a unique solution,
33: * which is obtained using a generalized RQ factorization of the
34: * matrices (B, A) given by
35: *
36: * B = (0 R)*Q, A = Z*T*Q.
37: *
38: * Arguments
39: * =========
40: *
41: * M (input) INTEGER
42: * The number of rows of the matrix A. M >= 0.
43: *
44: * N (input) INTEGER
45: * The number of columns of the matrices A and B. N >= 0.
46: *
47: * P (input) INTEGER
48: * The number of rows of the matrix B. 0 <= P <= N <= M+P.
49: *
50: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
51: * On entry, the M-by-N matrix A.
52: * On exit, the elements on and above the diagonal of the array
53: * contain the min(M,N)-by-N upper trapezoidal matrix T.
54: *
55: * LDA (input) INTEGER
56: * The leading dimension of the array A. LDA >= max(1,M).
57: *
58: * B (input/output) COMPLEX*16 array, dimension (LDB,N)
59: * On entry, the P-by-N matrix B.
60: * On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
61: * contains the P-by-P upper triangular matrix R.
62: *
63: * LDB (input) INTEGER
64: * The leading dimension of the array B. LDB >= max(1,P).
65: *
66: * C (input/output) COMPLEX*16 array, dimension (M)
67: * On entry, C contains the right hand side vector for the
68: * least squares part of the LSE problem.
69: * On exit, the residual sum of squares for the solution
70: * is given by the sum of squares of elements N-P+1 to M of
71: * vector C.
72: *
73: * D (input/output) COMPLEX*16 array, dimension (P)
74: * On entry, D contains the right hand side vector for the
75: * constrained equation.
76: * On exit, D is destroyed.
77: *
78: * X (output) COMPLEX*16 array, dimension (N)
79: * On exit, X is the solution of the LSE problem.
80: *
81: * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
82: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
83: *
84: * LWORK (input) INTEGER
85: * The dimension of the array WORK. LWORK >= max(1,M+N+P).
86: * For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
87: * where NB is an upper bound for the optimal blocksizes for
88: * ZGEQRF, CGERQF, ZUNMQR and CUNMRQ.
89: *
90: * If LWORK = -1, then a workspace query is assumed; the routine
91: * only calculates the optimal size of the WORK array, returns
92: * this value as the first entry of the WORK array, and no error
93: * message related to LWORK is issued by XERBLA.
94: *
95: * INFO (output) INTEGER
96: * = 0: successful exit.
97: * < 0: if INFO = -i, the i-th argument had an illegal value.
98: * = 1: the upper triangular factor R associated with B in the
99: * generalized RQ factorization of the pair (B, A) is
100: * singular, so that rank(B) < P; the least squares
101: * solution could not be computed.
102: * = 2: the (N-P) by (N-P) part of the upper trapezoidal factor
103: * T associated with A in the generalized RQ factorization
104: * of the pair (B, A) is singular, so that
105: * rank( (A) ) < N; the least squares solution could not
106: * ( (B) )
107: * be computed.
108: *
109: * =====================================================================
110: *
111: * .. Parameters ..
112: COMPLEX*16 CONE
113: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
114: * ..
115: * .. Local Scalars ..
116: LOGICAL LQUERY
117: INTEGER LOPT, LWKMIN, LWKOPT, MN, NB, NB1, NB2, NB3,
118: $ NB4, NR
119: * ..
120: * .. External Subroutines ..
121: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGGRQF, ZTRMV,
122: $ ZTRTRS, ZUNMQR, ZUNMRQ
123: * ..
124: * .. External Functions ..
125: INTEGER ILAENV
126: EXTERNAL ILAENV
127: * ..
128: * .. Intrinsic Functions ..
129: INTRINSIC INT, MAX, MIN
130: * ..
131: * .. Executable Statements ..
132: *
133: * Test the input parameters
134: *
135: INFO = 0
136: MN = MIN( M, N )
137: LQUERY = ( LWORK.EQ.-1 )
138: IF( M.LT.0 ) THEN
139: INFO = -1
140: ELSE IF( N.LT.0 ) THEN
141: INFO = -2
142: ELSE IF( P.LT.0 .OR. P.GT.N .OR. P.LT.N-M ) THEN
143: INFO = -3
144: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
145: INFO = -5
146: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
147: INFO = -7
148: END IF
149: *
150: * Calculate workspace
151: *
152: IF( INFO.EQ.0) THEN
153: IF( N.EQ.0 ) THEN
154: LWKMIN = 1
155: LWKOPT = 1
156: ELSE
157: NB1 = ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
158: NB2 = ILAENV( 1, 'ZGERQF', ' ', M, N, -1, -1 )
159: NB3 = ILAENV( 1, 'ZUNMQR', ' ', M, N, P, -1 )
160: NB4 = ILAENV( 1, 'ZUNMRQ', ' ', M, N, P, -1 )
161: NB = MAX( NB1, NB2, NB3, NB4 )
162: LWKMIN = M + N + P
163: LWKOPT = P + MN + MAX( M, N )*NB
164: END IF
165: WORK( 1 ) = LWKOPT
166: *
167: IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
168: INFO = -12
169: END IF
170: END IF
171: *
172: IF( INFO.NE.0 ) THEN
173: CALL XERBLA( 'ZGGLSE', -INFO )
174: RETURN
175: ELSE IF( LQUERY ) THEN
176: RETURN
177: END IF
178: *
179: * Quick return if possible
180: *
181: IF( N.EQ.0 )
182: $ RETURN
183: *
184: * Compute the GRQ factorization of matrices B and A:
185: *
186: * B*Q' = ( 0 T12 ) P Z'*A*Q' = ( R11 R12 ) N-P
187: * N-P P ( 0 R22 ) M+P-N
188: * N-P P
189: *
190: * where T12 and R11 are upper triangular, and Q and Z are
191: * unitary.
192: *
193: CALL ZGGRQF( P, M, N, B, LDB, WORK, A, LDA, WORK( P+1 ),
194: $ WORK( P+MN+1 ), LWORK-P-MN, INFO )
195: LOPT = WORK( P+MN+1 )
196: *
197: * Update c = Z'*c = ( c1 ) N-P
198: * ( c2 ) M+P-N
199: *
200: CALL ZUNMQR( 'Left', 'Conjugate Transpose', M, 1, MN, A, LDA,
201: $ WORK( P+1 ), C, MAX( 1, M ), WORK( P+MN+1 ),
202: $ LWORK-P-MN, INFO )
203: LOPT = MAX( LOPT, INT( WORK( P+MN+1 ) ) )
204: *
205: * Solve T12*x2 = d for x2
206: *
207: IF( P.GT.0 ) THEN
208: CALL ZTRTRS( 'Upper', 'No transpose', 'Non-unit', P, 1,
209: $ B( 1, N-P+1 ), LDB, D, P, INFO )
210: *
211: IF( INFO.GT.0 ) THEN
212: INFO = 1
213: RETURN
214: END IF
215: *
216: * Put the solution in X
217: *
218: CALL ZCOPY( P, D, 1, X( N-P+1 ), 1 )
219: *
220: * Update c1
221: *
222: CALL ZGEMV( 'No transpose', N-P, P, -CONE, A( 1, N-P+1 ), LDA,
223: $ D, 1, CONE, C, 1 )
224: END IF
225: *
226: * Solve R11*x1 = c1 for x1
227: *
228: IF( N.GT.P ) THEN
229: CALL ZTRTRS( 'Upper', 'No transpose', 'Non-unit', N-P, 1,
230: $ A, LDA, C, N-P, INFO )
231: *
232: IF( INFO.GT.0 ) THEN
233: INFO = 2
234: RETURN
235: END IF
236: *
237: * Put the solutions in X
238: *
239: CALL ZCOPY( N-P, C, 1, X, 1 )
240: END IF
241: *
242: * Compute the residual vector:
243: *
244: IF( M.LT.N ) THEN
245: NR = M + P - N
246: IF( NR.GT.0 )
247: $ CALL ZGEMV( 'No transpose', NR, N-M, -CONE, A( N-P+1, M+1 ),
248: $ LDA, D( NR+1 ), 1, CONE, C( N-P+1 ), 1 )
249: ELSE
250: NR = P
251: END IF
252: IF( NR.GT.0 ) THEN
253: CALL ZTRMV( 'Upper', 'No transpose', 'Non unit', NR,
254: $ A( N-P+1, N-P+1 ), LDA, D, 1 )
255: CALL ZAXPY( NR, -CONE, D, 1, C( N-P+1 ), 1 )
256: END IF
257: *
258: * Backward transformation x = Q'*x
259: *
260: CALL ZUNMRQ( 'Left', 'Conjugate Transpose', N, 1, P, B, LDB,
261: $ WORK( 1 ), X, N, WORK( P+MN+1 ), LWORK-P-MN, INFO )
262: WORK( 1 ) = P + MN + MAX( LOPT, INT( WORK( P+MN+1 ) ) )
263: *
264: RETURN
265: *
266: * End of ZGGLSE
267: *
268: END
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