1: *> \brief \b ZGGGLM
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGGGLM + dependencies
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13: *> [ZIP]</a>
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
22: * INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER INFO, LDA, LDB, LWORK, M, N, P
26: * ..
27: * .. Array Arguments ..
28: * COMPLEX*16 A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
29: * $ X( * ), Y( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZGGGLM solves a general Gauss-Markov linear model (GLM) problem:
39: *>
40: *> minimize || y ||_2 subject to d = A*x + B*y
41: *> x
42: *>
43: *> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
44: *> given N-vector. It is assumed that M <= N <= M+P, and
45: *>
46: *> rank(A) = M and rank( A B ) = N.
47: *>
48: *> Under these assumptions, the constrained equation is always
49: *> consistent, and there is a unique solution x and a minimal 2-norm
50: *> solution y, which is obtained using a generalized QR factorization
51: *> of the matrices (A, B) given by
52: *>
53: *> A = Q*(R), B = Q*T*Z.
54: *> (0)
55: *>
56: *> In particular, if matrix B is square nonsingular, then the problem
57: *> GLM is equivalent to the following weighted linear least squares
58: *> problem
59: *>
60: *> minimize || inv(B)*(d-A*x) ||_2
61: *> x
62: *>
63: *> where inv(B) denotes the inverse of B.
64: *> \endverbatim
65: *
66: * Arguments:
67: * ==========
68: *
69: *> \param[in] N
70: *> \verbatim
71: *> N is INTEGER
72: *> The number of rows of the matrices A and B. N >= 0.
73: *> \endverbatim
74: *>
75: *> \param[in] M
76: *> \verbatim
77: *> M is INTEGER
78: *> The number of columns of the matrix A. 0 <= M <= N.
79: *> \endverbatim
80: *>
81: *> \param[in] P
82: *> \verbatim
83: *> P is INTEGER
84: *> The number of columns of the matrix B. P >= N-M.
85: *> \endverbatim
86: *>
87: *> \param[in,out] A
88: *> \verbatim
89: *> A is COMPLEX*16 array, dimension (LDA,M)
90: *> On entry, the N-by-M matrix A.
91: *> On exit, the upper triangular part of the array A contains
92: *> the M-by-M upper triangular matrix R.
93: *> \endverbatim
94: *>
95: *> \param[in] LDA
96: *> \verbatim
97: *> LDA is INTEGER
98: *> The leading dimension of the array A. LDA >= max(1,N).
99: *> \endverbatim
100: *>
101: *> \param[in,out] B
102: *> \verbatim
103: *> B is COMPLEX*16 array, dimension (LDB,P)
104: *> On entry, the N-by-P matrix B.
105: *> On exit, if N <= P, the upper triangle of the subarray
106: *> B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
107: *> if N > P, the elements on and above the (N-P)th subdiagonal
108: *> contain the N-by-P upper trapezoidal matrix T.
109: *> \endverbatim
110: *>
111: *> \param[in] LDB
112: *> \verbatim
113: *> LDB is INTEGER
114: *> The leading dimension of the array B. LDB >= max(1,N).
115: *> \endverbatim
116: *>
117: *> \param[in,out] D
118: *> \verbatim
119: *> D is COMPLEX*16 array, dimension (N)
120: *> On entry, D is the left hand side of the GLM equation.
121: *> On exit, D is destroyed.
122: *> \endverbatim
123: *>
124: *> \param[out] X
125: *> \verbatim
126: *> X is COMPLEX*16 array, dimension (M)
127: *> \endverbatim
128: *>
129: *> \param[out] Y
130: *> \verbatim
131: *> Y is COMPLEX*16 array, dimension (P)
132: *>
133: *> On exit, X and Y are the solutions of the GLM problem.
134: *> \endverbatim
135: *>
136: *> \param[out] WORK
137: *> \verbatim
138: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
139: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
140: *> \endverbatim
141: *>
142: *> \param[in] LWORK
143: *> \verbatim
144: *> LWORK is INTEGER
145: *> The dimension of the array WORK. LWORK >= max(1,N+M+P).
146: *> For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
147: *> where NB is an upper bound for the optimal blocksizes for
148: *> ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.
149: *>
150: *> If LWORK = -1, then a workspace query is assumed; the routine
151: *> only calculates the optimal size of the WORK array, returns
152: *> this value as the first entry of the WORK array, and no error
153: *> message related to LWORK is issued by XERBLA.
154: *> \endverbatim
155: *>
156: *> \param[out] INFO
157: *> \verbatim
158: *> INFO is INTEGER
159: *> = 0: successful exit.
160: *> < 0: if INFO = -i, the i-th argument had an illegal value.
161: *> = 1: the upper triangular factor R associated with A in the
162: *> generalized QR factorization of the pair (A, B) is
163: *> singular, so that rank(A) < M; the least squares
164: *> solution could not be computed.
165: *> = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal
166: *> factor T associated with B in the generalized QR
167: *> factorization of the pair (A, B) is singular, so that
168: *> rank( A B ) < N; the least squares solution could not
169: *> be computed.
170: *> \endverbatim
171: *
172: * Authors:
173: * ========
174: *
175: *> \author Univ. of Tennessee
176: *> \author Univ. of California Berkeley
177: *> \author Univ. of Colorado Denver
178: *> \author NAG Ltd.
179: *
180: *> \date November 2015
181: *
182: *> \ingroup complex16OTHEReigen
183: *
184: * =====================================================================
185: SUBROUTINE ZGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
186: $ INFO )
187: *
188: * -- LAPACK driver routine (version 3.6.0) --
189: * -- LAPACK is a software package provided by Univ. of Tennessee, --
190: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
191: * November 2015
192: *
193: * .. Scalar Arguments ..
194: INTEGER INFO, LDA, LDB, LWORK, M, N, P
195: * ..
196: * .. Array Arguments ..
197: COMPLEX*16 A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
198: $ X( * ), Y( * )
199: * ..
200: *
201: * ===================================================================
202: *
203: * .. Parameters ..
204: COMPLEX*16 CZERO, CONE
205: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
206: $ CONE = ( 1.0D+0, 0.0D+0 ) )
207: * ..
208: * .. Local Scalars ..
209: LOGICAL LQUERY
210: INTEGER I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3,
211: $ NB4, NP
212: * ..
213: * .. External Subroutines ..
214: EXTERNAL XERBLA, ZCOPY, ZGEMV, ZGGQRF, ZTRTRS, ZUNMQR,
215: $ ZUNMRQ
216: * ..
217: * .. External Functions ..
218: INTEGER ILAENV
219: EXTERNAL ILAENV
220: * ..
221: * .. Intrinsic Functions ..
222: INTRINSIC INT, MAX, MIN
223: * ..
224: * .. Executable Statements ..
225: *
226: * Test the input parameters
227: *
228: INFO = 0
229: NP = MIN( N, P )
230: LQUERY = ( LWORK.EQ.-1 )
231: IF( N.LT.0 ) THEN
232: INFO = -1
233: ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
234: INFO = -2
235: ELSE IF( P.LT.0 .OR. P.LT.N-M ) THEN
236: INFO = -3
237: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
238: INFO = -5
239: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
240: INFO = -7
241: END IF
242: *
243: * Calculate workspace
244: *
245: IF( INFO.EQ.0) THEN
246: IF( N.EQ.0 ) THEN
247: LWKMIN = 1
248: LWKOPT = 1
249: ELSE
250: NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, M, -1, -1 )
251: NB2 = ILAENV( 1, 'ZGERQF', ' ', N, M, -1, -1 )
252: NB3 = ILAENV( 1, 'ZUNMQR', ' ', N, M, P, -1 )
253: NB4 = ILAENV( 1, 'ZUNMRQ', ' ', N, M, P, -1 )
254: NB = MAX( NB1, NB2, NB3, NB4 )
255: LWKMIN = M + N + P
256: LWKOPT = M + NP + MAX( N, P )*NB
257: END IF
258: WORK( 1 ) = LWKOPT
259: *
260: IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
261: INFO = -12
262: END IF
263: END IF
264: *
265: IF( INFO.NE.0 ) THEN
266: CALL XERBLA( 'ZGGGLM', -INFO )
267: RETURN
268: ELSE IF( LQUERY ) THEN
269: RETURN
270: END IF
271: *
272: * Quick return if possible
273: *
274: IF( N.EQ.0 )
275: $ RETURN
276: *
277: * Compute the GQR factorization of matrices A and B:
278: *
279: * Q**H*A = ( R11 ) M, Q**H*B*Z**H = ( T11 T12 ) M
280: * ( 0 ) N-M ( 0 T22 ) N-M
281: * M M+P-N N-M
282: *
283: * where R11 and T22 are upper triangular, and Q and Z are
284: * unitary.
285: *
286: CALL ZGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ),
287: $ WORK( M+NP+1 ), LWORK-M-NP, INFO )
288: LOPT = WORK( M+NP+1 )
289: *
290: * Update left-hand-side vector d = Q**H*d = ( d1 ) M
291: * ( d2 ) N-M
292: *
293: CALL ZUNMQR( 'Left', 'Conjugate transpose', N, 1, M, A, LDA, WORK,
294: $ D, MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
295: LOPT = MAX( LOPT, INT( WORK( M+NP+1 ) ) )
296: *
297: * Solve T22*y2 = d2 for y2
298: *
299: IF( N.GT.M ) THEN
300: CALL ZTRTRS( 'Upper', 'No transpose', 'Non unit', N-M, 1,
301: $ B( M+1, M+P-N+1 ), LDB, D( M+1 ), N-M, INFO )
302: *
303: IF( INFO.GT.0 ) THEN
304: INFO = 1
305: RETURN
306: END IF
307: *
308: CALL ZCOPY( N-M, D( M+1 ), 1, Y( M+P-N+1 ), 1 )
309: END IF
310: *
311: * Set y1 = 0
312: *
313: DO 10 I = 1, M + P - N
314: Y( I ) = CZERO
315: 10 CONTINUE
316: *
317: * Update d1 = d1 - T12*y2
318: *
319: CALL ZGEMV( 'No transpose', M, N-M, -CONE, B( 1, M+P-N+1 ), LDB,
320: $ Y( M+P-N+1 ), 1, CONE, D, 1 )
321: *
322: * Solve triangular system: R11*x = d1
323: *
324: IF( M.GT.0 ) THEN
325: CALL ZTRTRS( 'Upper', 'No Transpose', 'Non unit', M, 1, A, LDA,
326: $ D, M, INFO )
327: *
328: IF( INFO.GT.0 ) THEN
329: INFO = 2
330: RETURN
331: END IF
332: *
333: * Copy D to X
334: *
335: CALL ZCOPY( M, D, 1, X, 1 )
336: END IF
337: *
338: * Backward transformation y = Z**H *y
339: *
340: CALL ZUNMRQ( 'Left', 'Conjugate transpose', P, 1, NP,
341: $ B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y,
342: $ MAX( 1, P ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
343: WORK( 1 ) = M + NP + MAX( LOPT, INT( WORK( M+NP+1 ) ) )
344: *
345: RETURN
346: *
347: * End of ZGGGLM
348: *
349: END
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