1: *> \brief <b> ZGELSY solves overdetermined or underdetermined systems for GE matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGELSY + dependencies
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
22: * WORK, LWORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
26: * DOUBLE PRECISION RCOND
27: * ..
28: * .. Array Arguments ..
29: * INTEGER JPVT( * )
30: * DOUBLE PRECISION RWORK( * )
31: * COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> ZGELSY computes the minimum-norm solution to a complex linear least
41: *> squares problem:
42: *> minimize || A * X - B ||
43: *> using a complete orthogonal factorization of A. A is an M-by-N
44: *> matrix which may be rank-deficient.
45: *>
46: *> Several right hand side vectors b and solution vectors x can be
47: *> handled in a single call; they are stored as the columns of the
48: *> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
49: *> matrix X.
50: *>
51: *> The routine first computes a QR factorization with column pivoting:
52: *> A * P = Q * [ R11 R12 ]
53: *> [ 0 R22 ]
54: *> with R11 defined as the largest leading submatrix whose estimated
55: *> condition number is less than 1/RCOND. The order of R11, RANK,
56: *> is the effective rank of A.
57: *>
58: *> Then, R22 is considered to be negligible, and R12 is annihilated
59: *> by unitary transformations from the right, arriving at the
60: *> complete orthogonal factorization:
61: *> A * P = Q * [ T11 0 ] * Z
62: *> [ 0 0 ]
63: *> The minimum-norm solution is then
64: *> X = P * Z**H [ inv(T11)*Q1**H*B ]
65: *> [ 0 ]
66: *> where Q1 consists of the first RANK columns of Q.
67: *>
68: *> This routine is basically identical to the original xGELSX except
69: *> three differences:
70: *> o The permutation of matrix B (the right hand side) is faster and
71: *> more simple.
72: *> o The call to the subroutine xGEQPF has been substituted by the
73: *> the call to the subroutine xGEQP3. This subroutine is a Blas-3
74: *> version of the QR factorization with column pivoting.
75: *> o Matrix B (the right hand side) is updated with Blas-3.
76: *> \endverbatim
77: *
78: * Arguments:
79: * ==========
80: *
81: *> \param[in] M
82: *> \verbatim
83: *> M is INTEGER
84: *> The number of rows of the matrix A. M >= 0.
85: *> \endverbatim
86: *>
87: *> \param[in] N
88: *> \verbatim
89: *> N is INTEGER
90: *> The number of columns of the matrix A. N >= 0.
91: *> \endverbatim
92: *>
93: *> \param[in] NRHS
94: *> \verbatim
95: *> NRHS is INTEGER
96: *> The number of right hand sides, i.e., the number of
97: *> columns of matrices B and X. NRHS >= 0.
98: *> \endverbatim
99: *>
100: *> \param[in,out] A
101: *> \verbatim
102: *> A is COMPLEX*16 array, dimension (LDA,N)
103: *> On entry, the M-by-N matrix A.
104: *> On exit, A has been overwritten by details of its
105: *> complete orthogonal factorization.
106: *> \endverbatim
107: *>
108: *> \param[in] LDA
109: *> \verbatim
110: *> LDA is INTEGER
111: *> The leading dimension of the array A. LDA >= max(1,M).
112: *> \endverbatim
113: *>
114: *> \param[in,out] B
115: *> \verbatim
116: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
117: *> On entry, the M-by-NRHS right hand side matrix B.
118: *> On exit, the N-by-NRHS solution matrix X.
119: *> \endverbatim
120: *>
121: *> \param[in] LDB
122: *> \verbatim
123: *> LDB is INTEGER
124: *> The leading dimension of the array B. LDB >= max(1,M,N).
125: *> \endverbatim
126: *>
127: *> \param[in,out] JPVT
128: *> \verbatim
129: *> JPVT is INTEGER array, dimension (N)
130: *> On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
131: *> to the front of AP, otherwise column i is a free column.
132: *> On exit, if JPVT(i) = k, then the i-th column of A*P
133: *> was the k-th column of A.
134: *> \endverbatim
135: *>
136: *> \param[in] RCOND
137: *> \verbatim
138: *> RCOND is DOUBLE PRECISION
139: *> RCOND is used to determine the effective rank of A, which
140: *> is defined as the order of the largest leading triangular
141: *> submatrix R11 in the QR factorization with pivoting of A,
142: *> whose estimated condition number < 1/RCOND.
143: *> \endverbatim
144: *>
145: *> \param[out] RANK
146: *> \verbatim
147: *> RANK is INTEGER
148: *> The effective rank of A, i.e., the order of the submatrix
149: *> R11. This is the same as the order of the submatrix T11
150: *> in the complete orthogonal factorization of A.
151: *> \endverbatim
152: *>
153: *> \param[out] WORK
154: *> \verbatim
155: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
156: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
157: *> \endverbatim
158: *>
159: *> \param[in] LWORK
160: *> \verbatim
161: *> LWORK is INTEGER
162: *> The dimension of the array WORK.
163: *> The unblocked strategy requires that:
164: *> LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
165: *> where MN = min(M,N).
166: *> The block algorithm requires that:
167: *> LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
168: *> where NB is an upper bound on the blocksize returned
169: *> by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR,
170: *> and ZUNMRZ.
171: *>
172: *> If LWORK = -1, then a workspace query is assumed; the routine
173: *> only calculates the optimal size of the WORK array, returns
174: *> this value as the first entry of the WORK array, and no error
175: *> message related to LWORK is issued by XERBLA.
176: *> \endverbatim
177: *>
178: *> \param[out] RWORK
179: *> \verbatim
180: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
181: *> \endverbatim
182: *>
183: *> \param[out] INFO
184: *> \verbatim
185: *> INFO is INTEGER
186: *> = 0: successful exit
187: *> < 0: if INFO = -i, the i-th argument had an illegal value
188: *> \endverbatim
189: *
190: * Authors:
191: * ========
192: *
193: *> \author Univ. of Tennessee
194: *> \author Univ. of California Berkeley
195: *> \author Univ. of Colorado Denver
196: *> \author NAG Ltd.
197: *
198: *> \date November 2011
199: *
200: *> \ingroup complex16GEsolve
201: *
202: *> \par Contributors:
203: * ==================
204: *>
205: *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA \n
206: *> E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
207: *> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
208: *>
209: * =====================================================================
210: SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
211: $ WORK, LWORK, RWORK, INFO )
212: *
213: * -- LAPACK driver routine (version 3.4.0) --
214: * -- LAPACK is a software package provided by Univ. of Tennessee, --
215: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216: * November 2011
217: *
218: * .. Scalar Arguments ..
219: INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
220: DOUBLE PRECISION RCOND
221: * ..
222: * .. Array Arguments ..
223: INTEGER JPVT( * )
224: DOUBLE PRECISION RWORK( * )
225: COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
226: * ..
227: *
228: * =====================================================================
229: *
230: * .. Parameters ..
231: INTEGER IMAX, IMIN
232: PARAMETER ( IMAX = 1, IMIN = 2 )
233: DOUBLE PRECISION ZERO, ONE
234: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
235: COMPLEX*16 CZERO, CONE
236: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
237: $ CONE = ( 1.0D+0, 0.0D+0 ) )
238: * ..
239: * .. Local Scalars ..
240: LOGICAL LQUERY
241: INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKOPT, MN,
242: $ NB, NB1, NB2, NB3, NB4
243: DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMAX, SMAXPR, SMIN, SMINPR,
244: $ SMLNUM, WSIZE
245: COMPLEX*16 C1, C2, S1, S2
246: * ..
247: * .. External Subroutines ..
248: EXTERNAL DLABAD, XERBLA, ZCOPY, ZGEQP3, ZLAIC1, ZLASCL,
249: $ ZLASET, ZTRSM, ZTZRZF, ZUNMQR, ZUNMRZ
250: * ..
251: * .. External Functions ..
252: INTEGER ILAENV
253: DOUBLE PRECISION DLAMCH, ZLANGE
254: EXTERNAL ILAENV, DLAMCH, ZLANGE
255: * ..
256: * .. Intrinsic Functions ..
257: INTRINSIC ABS, DBLE, DCMPLX, MAX, MIN
258: * ..
259: * .. Executable Statements ..
260: *
261: MN = MIN( M, N )
262: ISMIN = MN + 1
263: ISMAX = 2*MN + 1
264: *
265: * Test the input arguments.
266: *
267: INFO = 0
268: NB1 = ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
269: NB2 = ILAENV( 1, 'ZGERQF', ' ', M, N, -1, -1 )
270: NB3 = ILAENV( 1, 'ZUNMQR', ' ', M, N, NRHS, -1 )
271: NB4 = ILAENV( 1, 'ZUNMRQ', ' ', M, N, NRHS, -1 )
272: NB = MAX( NB1, NB2, NB3, NB4 )
273: LWKOPT = MAX( 1, MN+2*N+NB*( N+1 ), 2*MN+NB*NRHS )
274: WORK( 1 ) = DCMPLX( LWKOPT )
275: LQUERY = ( LWORK.EQ.-1 )
276: IF( M.LT.0 ) THEN
277: INFO = -1
278: ELSE IF( N.LT.0 ) THEN
279: INFO = -2
280: ELSE IF( NRHS.LT.0 ) THEN
281: INFO = -3
282: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
283: INFO = -5
284: ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
285: INFO = -7
286: ELSE IF( LWORK.LT.( MN+MAX( 2*MN, N+1, MN+NRHS ) ) .AND. .NOT.
287: $ LQUERY ) THEN
288: INFO = -12
289: END IF
290: *
291: IF( INFO.NE.0 ) THEN
292: CALL XERBLA( 'ZGELSY', -INFO )
293: RETURN
294: ELSE IF( LQUERY ) THEN
295: RETURN
296: END IF
297: *
298: * Quick return if possible
299: *
300: IF( MIN( M, N, NRHS ).EQ.0 ) THEN
301: RANK = 0
302: RETURN
303: END IF
304: *
305: * Get machine parameters
306: *
307: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
308: BIGNUM = ONE / SMLNUM
309: CALL DLABAD( SMLNUM, BIGNUM )
310: *
311: * Scale A, B if max entries outside range [SMLNUM,BIGNUM]
312: *
313: ANRM = ZLANGE( 'M', M, N, A, LDA, RWORK )
314: IASCL = 0
315: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
316: *
317: * Scale matrix norm up to SMLNUM
318: *
319: CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
320: IASCL = 1
321: ELSE IF( ANRM.GT.BIGNUM ) THEN
322: *
323: * Scale matrix norm down to BIGNUM
324: *
325: CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
326: IASCL = 2
327: ELSE IF( ANRM.EQ.ZERO ) THEN
328: *
329: * Matrix all zero. Return zero solution.
330: *
331: CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
332: RANK = 0
333: GO TO 70
334: END IF
335: *
336: BNRM = ZLANGE( 'M', M, NRHS, B, LDB, RWORK )
337: IBSCL = 0
338: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
339: *
340: * Scale matrix norm up to SMLNUM
341: *
342: CALL ZLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
343: IBSCL = 1
344: ELSE IF( BNRM.GT.BIGNUM ) THEN
345: *
346: * Scale matrix norm down to BIGNUM
347: *
348: CALL ZLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
349: IBSCL = 2
350: END IF
351: *
352: * Compute QR factorization with column pivoting of A:
353: * A * P = Q * R
354: *
355: CALL ZGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),
356: $ LWORK-MN, RWORK, INFO )
357: WSIZE = MN + DBLE( WORK( MN+1 ) )
358: *
359: * complex workspace: MN+NB*(N+1). real workspace 2*N.
360: * Details of Householder rotations stored in WORK(1:MN).
361: *
362: * Determine RANK using incremental condition estimation
363: *
364: WORK( ISMIN ) = CONE
365: WORK( ISMAX ) = CONE
366: SMAX = ABS( A( 1, 1 ) )
367: SMIN = SMAX
368: IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
369: RANK = 0
370: CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
371: GO TO 70
372: ELSE
373: RANK = 1
374: END IF
375: *
376: 10 CONTINUE
377: IF( RANK.LT.MN ) THEN
378: I = RANK + 1
379: CALL ZLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
380: $ A( I, I ), SMINPR, S1, C1 )
381: CALL ZLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
382: $ A( I, I ), SMAXPR, S2, C2 )
383: *
384: IF( SMAXPR*RCOND.LE.SMINPR ) THEN
385: DO 20 I = 1, RANK
386: WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
387: WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
388: 20 CONTINUE
389: WORK( ISMIN+RANK ) = C1
390: WORK( ISMAX+RANK ) = C2
391: SMIN = SMINPR
392: SMAX = SMAXPR
393: RANK = RANK + 1
394: GO TO 10
395: END IF
396: END IF
397: *
398: * complex workspace: 3*MN.
399: *
400: * Logically partition R = [ R11 R12 ]
401: * [ 0 R22 ]
402: * where R11 = R(1:RANK,1:RANK)
403: *
404: * [R11,R12] = [ T11, 0 ] * Y
405: *
406: IF( RANK.LT.N )
407: $ CALL ZTZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ),
408: $ LWORK-2*MN, INFO )
409: *
410: * complex workspace: 2*MN.
411: * Details of Householder rotations stored in WORK(MN+1:2*MN)
412: *
413: * B(1:M,1:NRHS) := Q**H * B(1:M,1:NRHS)
414: *
415: CALL ZUNMQR( 'Left', 'Conjugate transpose', M, NRHS, MN, A, LDA,
416: $ WORK( 1 ), B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO )
417: WSIZE = MAX( WSIZE, 2*MN+DBLE( WORK( 2*MN+1 ) ) )
418: *
419: * complex workspace: 2*MN+NB*NRHS.
420: *
421: * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
422: *
423: CALL ZTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
424: $ NRHS, CONE, A, LDA, B, LDB )
425: *
426: DO 40 J = 1, NRHS
427: DO 30 I = RANK + 1, N
428: B( I, J ) = CZERO
429: 30 CONTINUE
430: 40 CONTINUE
431: *
432: * B(1:N,1:NRHS) := Y**H * B(1:N,1:NRHS)
433: *
434: IF( RANK.LT.N ) THEN
435: CALL ZUNMRZ( 'Left', 'Conjugate transpose', N, NRHS, RANK,
436: $ N-RANK, A, LDA, WORK( MN+1 ), B, LDB,
437: $ WORK( 2*MN+1 ), LWORK-2*MN, INFO )
438: END IF
439: *
440: * complex workspace: 2*MN+NRHS.
441: *
442: * B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
443: *
444: DO 60 J = 1, NRHS
445: DO 50 I = 1, N
446: WORK( JPVT( I ) ) = B( I, J )
447: 50 CONTINUE
448: CALL ZCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )
449: 60 CONTINUE
450: *
451: * complex workspace: N.
452: *
453: * Undo scaling
454: *
455: IF( IASCL.EQ.1 ) THEN
456: CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
457: CALL ZLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
458: $ INFO )
459: ELSE IF( IASCL.EQ.2 ) THEN
460: CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
461: CALL ZLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
462: $ INFO )
463: END IF
464: IF( IBSCL.EQ.1 ) THEN
465: CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
466: ELSE IF( IBSCL.EQ.2 ) THEN
467: CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
468: END IF
469: *
470: 70 CONTINUE
471: WORK( 1 ) = DCMPLX( LWKOPT )
472: *
473: RETURN
474: *
475: * End of ZGELSY
476: *
477: END
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