1: *> \brief <b> ZGELSX 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: *
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17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
22: * WORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER INFO, LDA, LDB, 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: *> This routine is deprecated and has been replaced by routine ZGELSY.
41: *>
42: *> ZGELSX computes the minimum-norm solution to a complex linear least
43: *> squares problem:
44: *> minimize || A * X - B ||
45: *> using a complete orthogonal factorization of A. A is an M-by-N
46: *> matrix which may be rank-deficient.
47: *>
48: *> Several right hand side vectors b and solution vectors x can be
49: *> handled in a single call; they are stored as the columns of the
50: *> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
51: *> matrix X.
52: *>
53: *> The routine first computes a QR factorization with column pivoting:
54: *> A * P = Q * [ R11 R12 ]
55: *> [ 0 R22 ]
56: *> with R11 defined as the largest leading submatrix whose estimated
57: *> condition number is less than 1/RCOND. The order of R11, RANK,
58: *> is the effective rank of A.
59: *>
60: *> Then, R22 is considered to be negligible, and R12 is annihilated
61: *> by unitary transformations from the right, arriving at the
62: *> complete orthogonal factorization:
63: *> A * P = Q * [ T11 0 ] * Z
64: *> [ 0 0 ]
65: *> The minimum-norm solution is then
66: *> X = P * Z**H [ inv(T11)*Q1**H*B ]
67: *> [ 0 ]
68: *> where Q1 consists of the first RANK columns of Q.
69: *> \endverbatim
70: *
71: * Arguments:
72: * ==========
73: *
74: *> \param[in] M
75: *> \verbatim
76: *> M is INTEGER
77: *> The number of rows of the matrix A. M >= 0.
78: *> \endverbatim
79: *>
80: *> \param[in] N
81: *> \verbatim
82: *> N is INTEGER
83: *> The number of columns of the matrix A. N >= 0.
84: *> \endverbatim
85: *>
86: *> \param[in] NRHS
87: *> \verbatim
88: *> NRHS is INTEGER
89: *> The number of right hand sides, i.e., the number of
90: *> columns of matrices B and X. NRHS >= 0.
91: *> \endverbatim
92: *>
93: *> \param[in,out] A
94: *> \verbatim
95: *> A is COMPLEX*16 array, dimension (LDA,N)
96: *> On entry, the M-by-N matrix A.
97: *> On exit, A has been overwritten by details of its
98: *> complete orthogonal factorization.
99: *> \endverbatim
100: *>
101: *> \param[in] LDA
102: *> \verbatim
103: *> LDA is INTEGER
104: *> The leading dimension of the array A. LDA >= max(1,M).
105: *> \endverbatim
106: *>
107: *> \param[in,out] B
108: *> \verbatim
109: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
110: *> On entry, the M-by-NRHS right hand side matrix B.
111: *> On exit, the N-by-NRHS solution matrix X.
112: *> If m >= n and RANK = n, the residual sum-of-squares for
113: *> the solution in the i-th column is given by the sum of
114: *> squares of elements N+1:M in that column.
115: *> \endverbatim
116: *>
117: *> \param[in] LDB
118: *> \verbatim
119: *> LDB is INTEGER
120: *> The leading dimension of the array B. LDB >= max(1,M,N).
121: *> \endverbatim
122: *>
123: *> \param[in,out] JPVT
124: *> \verbatim
125: *> JPVT is INTEGER array, dimension (N)
126: *> On entry, if JPVT(i) .ne. 0, the i-th column of A is an
127: *> initial column, otherwise it is a free column. Before
128: *> the QR factorization of A, all initial columns are
129: *> permuted to the leading positions; only the remaining
130: *> free columns are moved as a result of column pivoting
131: *> during the factorization.
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
156: *> (min(M,N) + max( N, 2*min(M,N)+NRHS )),
157: *> \endverbatim
158: *>
159: *> \param[out] RWORK
160: *> \verbatim
161: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
162: *> \endverbatim
163: *>
164: *> \param[out] INFO
165: *> \verbatim
166: *> INFO is INTEGER
167: *> = 0: successful exit
168: *> < 0: if INFO = -i, the i-th argument had an illegal value
169: *> \endverbatim
170: *
171: * Authors:
172: * ========
173: *
174: *> \author Univ. of Tennessee
175: *> \author Univ. of California Berkeley
176: *> \author Univ. of Colorado Denver
177: *> \author NAG Ltd.
178: *
179: *> \date December 2016
180: *
181: *> \ingroup complex16GEsolve
182: *
183: * =====================================================================
184: SUBROUTINE ZGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
185: $ WORK, RWORK, INFO )
186: *
187: * -- LAPACK driver routine (version 3.7.0) --
188: * -- LAPACK is a software package provided by Univ. of Tennessee, --
189: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
190: * December 2016
191: *
192: * .. Scalar Arguments ..
193: INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
194: DOUBLE PRECISION RCOND
195: * ..
196: * .. Array Arguments ..
197: INTEGER JPVT( * )
198: DOUBLE PRECISION RWORK( * )
199: COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
200: * ..
201: *
202: * =====================================================================
203: *
204: * .. Parameters ..
205: INTEGER IMAX, IMIN
206: PARAMETER ( IMAX = 1, IMIN = 2 )
207: DOUBLE PRECISION ZERO, ONE, DONE, NTDONE
208: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, DONE = ZERO,
209: $ NTDONE = ONE )
210: COMPLEX*16 CZERO, CONE
211: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
212: $ CONE = ( 1.0D+0, 0.0D+0 ) )
213: * ..
214: * .. Local Scalars ..
215: INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
216: DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMAX, SMAXPR, SMIN, SMINPR,
217: $ SMLNUM
218: COMPLEX*16 C1, C2, S1, S2, T1, T2
219: * ..
220: * .. External Subroutines ..
221: EXTERNAL XERBLA, ZGEQPF, ZLAIC1, ZLASCL, ZLASET, ZLATZM,
222: $ ZTRSM, ZTZRQF, ZUNM2R
223: * ..
224: * .. External Functions ..
225: DOUBLE PRECISION DLAMCH, ZLANGE
226: EXTERNAL DLAMCH, ZLANGE
227: * ..
228: * .. Intrinsic Functions ..
229: INTRINSIC ABS, DCONJG, MAX, MIN
230: * ..
231: * .. Executable Statements ..
232: *
233: MN = MIN( M, N )
234: ISMIN = MN + 1
235: ISMAX = 2*MN + 1
236: *
237: * Test the input arguments.
238: *
239: INFO = 0
240: IF( M.LT.0 ) THEN
241: INFO = -1
242: ELSE IF( N.LT.0 ) THEN
243: INFO = -2
244: ELSE IF( NRHS.LT.0 ) THEN
245: INFO = -3
246: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
247: INFO = -5
248: ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
249: INFO = -7
250: END IF
251: *
252: IF( INFO.NE.0 ) THEN
253: CALL XERBLA( 'ZGELSX', -INFO )
254: RETURN
255: END IF
256: *
257: * Quick return if possible
258: *
259: IF( MIN( M, N, NRHS ).EQ.0 ) THEN
260: RANK = 0
261: RETURN
262: END IF
263: *
264: * Get machine parameters
265: *
266: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
267: BIGNUM = ONE / SMLNUM
268: CALL DLABAD( SMLNUM, BIGNUM )
269: *
270: * Scale A, B if max elements outside range [SMLNUM,BIGNUM]
271: *
272: ANRM = ZLANGE( 'M', M, N, A, LDA, RWORK )
273: IASCL = 0
274: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
275: *
276: * Scale matrix norm up to SMLNUM
277: *
278: CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
279: IASCL = 1
280: ELSE IF( ANRM.GT.BIGNUM ) THEN
281: *
282: * Scale matrix norm down to BIGNUM
283: *
284: CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
285: IASCL = 2
286: ELSE IF( ANRM.EQ.ZERO ) THEN
287: *
288: * Matrix all zero. Return zero solution.
289: *
290: CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
291: RANK = 0
292: GO TO 100
293: END IF
294: *
295: BNRM = ZLANGE( 'M', M, NRHS, B, LDB, RWORK )
296: IBSCL = 0
297: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
298: *
299: * Scale matrix norm up to SMLNUM
300: *
301: CALL ZLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
302: IBSCL = 1
303: ELSE IF( BNRM.GT.BIGNUM ) THEN
304: *
305: * Scale matrix norm down to BIGNUM
306: *
307: CALL ZLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
308: IBSCL = 2
309: END IF
310: *
311: * Compute QR factorization with column pivoting of A:
312: * A * P = Q * R
313: *
314: CALL ZGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), RWORK,
315: $ INFO )
316: *
317: * complex workspace MN+N. Real workspace 2*N. Details of Householder
318: * rotations stored in WORK(1:MN).
319: *
320: * Determine RANK using incremental condition estimation
321: *
322: WORK( ISMIN ) = CONE
323: WORK( ISMAX ) = CONE
324: SMAX = ABS( A( 1, 1 ) )
325: SMIN = SMAX
326: IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
327: RANK = 0
328: CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
329: GO TO 100
330: ELSE
331: RANK = 1
332: END IF
333: *
334: 10 CONTINUE
335: IF( RANK.LT.MN ) THEN
336: I = RANK + 1
337: CALL ZLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
338: $ A( I, I ), SMINPR, S1, C1 )
339: CALL ZLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
340: $ A( I, I ), SMAXPR, S2, C2 )
341: *
342: IF( SMAXPR*RCOND.LE.SMINPR ) THEN
343: DO 20 I = 1, RANK
344: WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
345: WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
346: 20 CONTINUE
347: WORK( ISMIN+RANK ) = C1
348: WORK( ISMAX+RANK ) = C2
349: SMIN = SMINPR
350: SMAX = SMAXPR
351: RANK = RANK + 1
352: GO TO 10
353: END IF
354: END IF
355: *
356: * Logically partition R = [ R11 R12 ]
357: * [ 0 R22 ]
358: * where R11 = R(1:RANK,1:RANK)
359: *
360: * [R11,R12] = [ T11, 0 ] * Y
361: *
362: IF( RANK.LT.N )
363: $ CALL ZTZRQF( RANK, N, A, LDA, WORK( MN+1 ), INFO )
364: *
365: * Details of Householder rotations stored in WORK(MN+1:2*MN)
366: *
367: * B(1:M,1:NRHS) := Q**H * B(1:M,1:NRHS)
368: *
369: CALL ZUNM2R( 'Left', 'Conjugate transpose', M, NRHS, MN, A, LDA,
370: $ WORK( 1 ), B, LDB, WORK( 2*MN+1 ), INFO )
371: *
372: * workspace NRHS
373: *
374: * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
375: *
376: CALL ZTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
377: $ NRHS, CONE, A, LDA, B, LDB )
378: *
379: DO 40 I = RANK + 1, N
380: DO 30 J = 1, NRHS
381: B( I, J ) = CZERO
382: 30 CONTINUE
383: 40 CONTINUE
384: *
385: * B(1:N,1:NRHS) := Y**H * B(1:N,1:NRHS)
386: *
387: IF( RANK.LT.N ) THEN
388: DO 50 I = 1, RANK
389: CALL ZLATZM( 'Left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA,
390: $ DCONJG( WORK( MN+I ) ), B( I, 1 ),
391: $ B( RANK+1, 1 ), LDB, WORK( 2*MN+1 ) )
392: 50 CONTINUE
393: END IF
394: *
395: * workspace NRHS
396: *
397: * B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
398: *
399: DO 90 J = 1, NRHS
400: DO 60 I = 1, N
401: WORK( 2*MN+I ) = NTDONE
402: 60 CONTINUE
403: DO 80 I = 1, N
404: IF( WORK( 2*MN+I ).EQ.NTDONE ) THEN
405: IF( JPVT( I ).NE.I ) THEN
406: K = I
407: T1 = B( K, J )
408: T2 = B( JPVT( K ), J )
409: 70 CONTINUE
410: B( JPVT( K ), J ) = T1
411: WORK( 2*MN+K ) = DONE
412: T1 = T2
413: K = JPVT( K )
414: T2 = B( JPVT( K ), J )
415: IF( JPVT( K ).NE.I )
416: $ GO TO 70
417: B( I, J ) = T1
418: WORK( 2*MN+K ) = DONE
419: END IF
420: END IF
421: 80 CONTINUE
422: 90 CONTINUE
423: *
424: * Undo scaling
425: *
426: IF( IASCL.EQ.1 ) THEN
427: CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
428: CALL ZLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
429: $ INFO )
430: ELSE IF( IASCL.EQ.2 ) THEN
431: CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
432: CALL ZLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
433: $ INFO )
434: END IF
435: IF( IBSCL.EQ.1 ) THEN
436: CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
437: ELSE IF( IBSCL.EQ.2 ) THEN
438: CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
439: END IF
440: *
441: 100 CONTINUE
442: *
443: RETURN
444: *
445: * End of ZGELSX
446: *
447: END
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