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 This routine is deprecated and has been replaced by routine ZGELSY.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: *> \ingroup complex16GEsolve
180: *
181: * =====================================================================
182: SUBROUTINE ZGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
183: $ WORK, RWORK, INFO )
184: *
185: * -- LAPACK driver routine --
186: * -- LAPACK is a software package provided by Univ. of Tennessee, --
187: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
188: *
189: * .. Scalar Arguments ..
190: INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
191: DOUBLE PRECISION RCOND
192: * ..
193: * .. Array Arguments ..
194: INTEGER JPVT( * )
195: DOUBLE PRECISION RWORK( * )
196: COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
197: * ..
198: *
199: * =====================================================================
200: *
201: * .. Parameters ..
202: INTEGER IMAX, IMIN
203: PARAMETER ( IMAX = 1, IMIN = 2 )
204: DOUBLE PRECISION ZERO, ONE, DONE, NTDONE
205: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, DONE = ZERO,
206: $ NTDONE = ONE )
207: COMPLEX*16 CZERO, CONE
208: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
209: $ CONE = ( 1.0D+0, 0.0D+0 ) )
210: * ..
211: * .. Local Scalars ..
212: INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
213: DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMAX, SMAXPR, SMIN, SMINPR,
214: $ SMLNUM
215: COMPLEX*16 C1, C2, S1, S2, T1, T2
216: * ..
217: * .. External Subroutines ..
218: EXTERNAL XERBLA, ZGEQPF, ZLAIC1, ZLASCL, ZLASET, ZLATZM,
219: $ ZTRSM, ZTZRQF, ZUNM2R
220: * ..
221: * .. External Functions ..
222: DOUBLE PRECISION DLAMCH, ZLANGE
223: EXTERNAL DLAMCH, ZLANGE
224: * ..
225: * .. Intrinsic Functions ..
226: INTRINSIC ABS, DCONJG, MAX, MIN
227: * ..
228: * .. Executable Statements ..
229: *
230: MN = MIN( M, N )
231: ISMIN = MN + 1
232: ISMAX = 2*MN + 1
233: *
234: * Test the input arguments.
235: *
236: INFO = 0
237: IF( M.LT.0 ) THEN
238: INFO = -1
239: ELSE IF( N.LT.0 ) THEN
240: INFO = -2
241: ELSE IF( NRHS.LT.0 ) THEN
242: INFO = -3
243: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
244: INFO = -5
245: ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
246: INFO = -7
247: END IF
248: *
249: IF( INFO.NE.0 ) THEN
250: CALL XERBLA( 'ZGELSX', -INFO )
251: RETURN
252: END IF
253: *
254: * Quick return if possible
255: *
256: IF( MIN( M, N, NRHS ).EQ.0 ) THEN
257: RANK = 0
258: RETURN
259: END IF
260: *
261: * Get machine parameters
262: *
263: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
264: BIGNUM = ONE / SMLNUM
265: CALL DLABAD( SMLNUM, BIGNUM )
266: *
267: * Scale A, B if max elements outside range [SMLNUM,BIGNUM]
268: *
269: ANRM = ZLANGE( 'M', M, N, A, LDA, RWORK )
270: IASCL = 0
271: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
272: *
273: * Scale matrix norm up to SMLNUM
274: *
275: CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
276: IASCL = 1
277: ELSE IF( ANRM.GT.BIGNUM ) THEN
278: *
279: * Scale matrix norm down to BIGNUM
280: *
281: CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
282: IASCL = 2
283: ELSE IF( ANRM.EQ.ZERO ) THEN
284: *
285: * Matrix all zero. Return zero solution.
286: *
287: CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
288: RANK = 0
289: GO TO 100
290: END IF
291: *
292: BNRM = ZLANGE( 'M', M, NRHS, B, LDB, RWORK )
293: IBSCL = 0
294: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
295: *
296: * Scale matrix norm up to SMLNUM
297: *
298: CALL ZLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
299: IBSCL = 1
300: ELSE IF( BNRM.GT.BIGNUM ) THEN
301: *
302: * Scale matrix norm down to BIGNUM
303: *
304: CALL ZLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
305: IBSCL = 2
306: END IF
307: *
308: * Compute QR factorization with column pivoting of A:
309: * A * P = Q * R
310: *
311: CALL ZGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), RWORK,
312: $ INFO )
313: *
314: * complex workspace MN+N. Real workspace 2*N. Details of Householder
315: * rotations stored in WORK(1:MN).
316: *
317: * Determine RANK using incremental condition estimation
318: *
319: WORK( ISMIN ) = CONE
320: WORK( ISMAX ) = CONE
321: SMAX = ABS( A( 1, 1 ) )
322: SMIN = SMAX
323: IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
324: RANK = 0
325: CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
326: GO TO 100
327: ELSE
328: RANK = 1
329: END IF
330: *
331: 10 CONTINUE
332: IF( RANK.LT.MN ) THEN
333: I = RANK + 1
334: CALL ZLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
335: $ A( I, I ), SMINPR, S1, C1 )
336: CALL ZLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
337: $ A( I, I ), SMAXPR, S2, C2 )
338: *
339: IF( SMAXPR*RCOND.LE.SMINPR ) THEN
340: DO 20 I = 1, RANK
341: WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
342: WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
343: 20 CONTINUE
344: WORK( ISMIN+RANK ) = C1
345: WORK( ISMAX+RANK ) = C2
346: SMIN = SMINPR
347: SMAX = SMAXPR
348: RANK = RANK + 1
349: GO TO 10
350: END IF
351: END IF
352: *
353: * Logically partition R = [ R11 R12 ]
354: * [ 0 R22 ]
355: * where R11 = R(1:RANK,1:RANK)
356: *
357: * [R11,R12] = [ T11, 0 ] * Y
358: *
359: IF( RANK.LT.N )
360: $ CALL ZTZRQF( RANK, N, A, LDA, WORK( MN+1 ), INFO )
361: *
362: * Details of Householder rotations stored in WORK(MN+1:2*MN)
363: *
364: * B(1:M,1:NRHS) := Q**H * B(1:M,1:NRHS)
365: *
366: CALL ZUNM2R( 'Left', 'Conjugate transpose', M, NRHS, MN, A, LDA,
367: $ WORK( 1 ), B, LDB, WORK( 2*MN+1 ), INFO )
368: *
369: * workspace NRHS
370: *
371: * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
372: *
373: CALL ZTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
374: $ NRHS, CONE, A, LDA, B, LDB )
375: *
376: DO 40 I = RANK + 1, N
377: DO 30 J = 1, NRHS
378: B( I, J ) = CZERO
379: 30 CONTINUE
380: 40 CONTINUE
381: *
382: * B(1:N,1:NRHS) := Y**H * B(1:N,1:NRHS)
383: *
384: IF( RANK.LT.N ) THEN
385: DO 50 I = 1, RANK
386: CALL ZLATZM( 'Left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA,
387: $ DCONJG( WORK( MN+I ) ), B( I, 1 ),
388: $ B( RANK+1, 1 ), LDB, WORK( 2*MN+1 ) )
389: 50 CONTINUE
390: END IF
391: *
392: * workspace NRHS
393: *
394: * B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
395: *
396: DO 90 J = 1, NRHS
397: DO 60 I = 1, N
398: WORK( 2*MN+I ) = NTDONE
399: 60 CONTINUE
400: DO 80 I = 1, N
401: IF( WORK( 2*MN+I ).EQ.NTDONE ) THEN
402: IF( JPVT( I ).NE.I ) THEN
403: K = I
404: T1 = B( K, J )
405: T2 = B( JPVT( K ), J )
406: 70 CONTINUE
407: B( JPVT( K ), J ) = T1
408: WORK( 2*MN+K ) = DONE
409: T1 = T2
410: K = JPVT( K )
411: T2 = B( JPVT( K ), J )
412: IF( JPVT( K ).NE.I )
413: $ GO TO 70
414: B( I, J ) = T1
415: WORK( 2*MN+K ) = DONE
416: END IF
417: END IF
418: 80 CONTINUE
419: 90 CONTINUE
420: *
421: * Undo scaling
422: *
423: IF( IASCL.EQ.1 ) THEN
424: CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
425: CALL ZLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
426: $ INFO )
427: ELSE IF( IASCL.EQ.2 ) THEN
428: CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
429: CALL ZLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
430: $ INFO )
431: END IF
432: IF( IBSCL.EQ.1 ) THEN
433: CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
434: ELSE IF( IBSCL.EQ.2 ) THEN
435: CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
436: END IF
437: *
438: 100 CONTINUE
439: *
440: RETURN
441: *
442: * End of ZGELSX
443: *
444: END
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