1 SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
2 $ WORK, LWORK, RWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
11 DOUBLE PRECISION RCOND
12 * ..
13 * .. Array Arguments ..
14 INTEGER JPVT( * )
15 DOUBLE PRECISION RWORK( * )
16 COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * ZGELSY computes the minimum-norm solution to a complex linear least
23 * squares problem:
24 * minimize || A * X - B ||
25 * using a complete orthogonal factorization of A. A is an M-by-N
26 * matrix which may be rank-deficient.
27 *
28 * Several right hand side vectors b and solution vectors x can be
29 * handled in a single call; they are stored as the columns of the
30 * M-by-NRHS right hand side matrix B and the N-by-NRHS solution
31 * matrix X.
32 *
33 * The routine first computes a QR factorization with column pivoting:
34 * A * P = Q * [ R11 R12 ]
35 * [ 0 R22 ]
36 * with R11 defined as the largest leading submatrix whose estimated
37 * condition number is less than 1/RCOND. The order of R11, RANK,
38 * is the effective rank of A.
39 *
40 * Then, R22 is considered to be negligible, and R12 is annihilated
41 * by unitary transformations from the right, arriving at the
42 * complete orthogonal factorization:
43 * A * P = Q * [ T11 0 ] * Z
44 * [ 0 0 ]
45 * The minimum-norm solution is then
46 * X = P * Z**H [ inv(T11)*Q1**H*B ]
47 * [ 0 ]
48 * where Q1 consists of the first RANK columns of Q.
49 *
50 * This routine is basically identical to the original xGELSX except
51 * three differences:
52 * o The permutation of matrix B (the right hand side) is faster and
53 * more simple.
54 * o The call to the subroutine xGEQPF has been substituted by the
55 * the call to the subroutine xGEQP3. This subroutine is a Blas-3
56 * version of the QR factorization with column pivoting.
57 * o Matrix B (the right hand side) is updated with Blas-3.
58 *
59 * Arguments
60 * =========
61 *
62 * M (input) INTEGER
63 * The number of rows of the matrix A. M >= 0.
64 *
65 * N (input) INTEGER
66 * The number of columns of the matrix A. N >= 0.
67 *
68 * NRHS (input) INTEGER
69 * The number of right hand sides, i.e., the number of
70 * columns of matrices B and X. NRHS >= 0.
71 *
72 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
73 * On entry, the M-by-N matrix A.
74 * On exit, A has been overwritten by details of its
75 * complete orthogonal factorization.
76 *
77 * LDA (input) INTEGER
78 * The leading dimension of the array A. LDA >= max(1,M).
79 *
80 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
81 * On entry, the M-by-NRHS right hand side matrix B.
82 * On exit, the N-by-NRHS solution matrix X.
83 *
84 * LDB (input) INTEGER
85 * The leading dimension of the array B. LDB >= max(1,M,N).
86 *
87 * JPVT (input/output) INTEGER array, dimension (N)
88 * On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
89 * to the front of AP, otherwise column i is a free column.
90 * On exit, if JPVT(i) = k, then the i-th column of A*P
91 * was the k-th column of A.
92 *
93 * RCOND (input) DOUBLE PRECISION
94 * RCOND is used to determine the effective rank of A, which
95 * is defined as the order of the largest leading triangular
96 * submatrix R11 in the QR factorization with pivoting of A,
97 * whose estimated condition number < 1/RCOND.
98 *
99 * RANK (output) INTEGER
100 * The effective rank of A, i.e., the order of the submatrix
101 * R11. This is the same as the order of the submatrix T11
102 * in the complete orthogonal factorization of A.
103 *
104 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
105 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
106 *
107 * LWORK (input) INTEGER
108 * The dimension of the array WORK.
109 * The unblocked strategy requires that:
110 * LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
111 * where MN = min(M,N).
112 * The block algorithm requires that:
113 * LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
114 * where NB is an upper bound on the blocksize returned
115 * by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR,
116 * and ZUNMRZ.
117 *
118 * If LWORK = -1, then a workspace query is assumed; the routine
119 * only calculates the optimal size of the WORK array, returns
120 * this value as the first entry of the WORK array, and no error
121 * message related to LWORK is issued by XERBLA.
122 *
123 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
124 *
125 * INFO (output) INTEGER
126 * = 0: successful exit
127 * < 0: if INFO = -i, the i-th argument had an illegal value
128 *
129 * Further Details
130 * ===============
131 *
132 * Based on contributions by
133 * A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
134 * E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
135 * G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
136 *
137 * =====================================================================
138 *
139 * .. Parameters ..
140 INTEGER IMAX, IMIN
141 PARAMETER ( IMAX = 1, IMIN = 2 )
142 DOUBLE PRECISION ZERO, ONE
143 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
144 COMPLEX*16 CZERO, CONE
145 PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
146 $ CONE = ( 1.0D+0, 0.0D+0 ) )
147 * ..
148 * .. Local Scalars ..
149 LOGICAL LQUERY
150 INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKOPT, MN,
151 $ NB, NB1, NB2, NB3, NB4
152 DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMAX, SMAXPR, SMIN, SMINPR,
153 $ SMLNUM, WSIZE
154 COMPLEX*16 C1, C2, S1, S2
155 * ..
156 * .. External Subroutines ..
157 EXTERNAL DLABAD, XERBLA, ZCOPY, ZGEQP3, ZLAIC1, ZLASCL,
158 $ ZLASET, ZTRSM, ZTZRZF, ZUNMQR, ZUNMRZ
159 * ..
160 * .. External Functions ..
161 INTEGER ILAENV
162 DOUBLE PRECISION DLAMCH, ZLANGE
163 EXTERNAL ILAENV, DLAMCH, ZLANGE
164 * ..
165 * .. Intrinsic Functions ..
166 INTRINSIC ABS, DBLE, DCMPLX, MAX, MIN
167 * ..
168 * .. Executable Statements ..
169 *
170 MN = MIN( M, N )
171 ISMIN = MN + 1
172 ISMAX = 2*MN + 1
173 *
174 * Test the input arguments.
175 *
176 INFO = 0
177 NB1 = ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
178 NB2 = ILAENV( 1, 'ZGERQF', ' ', M, N, -1, -1 )
179 NB3 = ILAENV( 1, 'ZUNMQR', ' ', M, N, NRHS, -1 )
180 NB4 = ILAENV( 1, 'ZUNMRQ', ' ', M, N, NRHS, -1 )
181 NB = MAX( NB1, NB2, NB3, NB4 )
182 LWKOPT = MAX( 1, MN+2*N+NB*( N+1 ), 2*MN+NB*NRHS )
183 WORK( 1 ) = DCMPLX( LWKOPT )
184 LQUERY = ( LWORK.EQ.-1 )
185 IF( M.LT.0 ) THEN
186 INFO = -1
187 ELSE IF( N.LT.0 ) THEN
188 INFO = -2
189 ELSE IF( NRHS.LT.0 ) THEN
190 INFO = -3
191 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
192 INFO = -5
193 ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
194 INFO = -7
195 ELSE IF( LWORK.LT.( MN+MAX( 2*MN, N+1, MN+NRHS ) ) .AND. .NOT.
196 $ LQUERY ) THEN
197 INFO = -12
198 END IF
199 *
200 IF( INFO.NE.0 ) THEN
201 CALL XERBLA( 'ZGELSY', -INFO )
202 RETURN
203 ELSE IF( LQUERY ) THEN
204 RETURN
205 END IF
206 *
207 * Quick return if possible
208 *
209 IF( MIN( M, N, NRHS ).EQ.0 ) THEN
210 RANK = 0
211 RETURN
212 END IF
213 *
214 * Get machine parameters
215 *
216 SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
217 BIGNUM = ONE / SMLNUM
218 CALL DLABAD( SMLNUM, BIGNUM )
219 *
220 * Scale A, B if max entries outside range [SMLNUM,BIGNUM]
221 *
222 ANRM = ZLANGE( 'M', M, N, A, LDA, RWORK )
223 IASCL = 0
224 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
225 *
226 * Scale matrix norm up to SMLNUM
227 *
228 CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
229 IASCL = 1
230 ELSE IF( ANRM.GT.BIGNUM ) THEN
231 *
232 * Scale matrix norm down to BIGNUM
233 *
234 CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
235 IASCL = 2
236 ELSE IF( ANRM.EQ.ZERO ) THEN
237 *
238 * Matrix all zero. Return zero solution.
239 *
240 CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
241 RANK = 0
242 GO TO 70
243 END IF
244 *
245 BNRM = ZLANGE( 'M', M, NRHS, B, LDB, RWORK )
246 IBSCL = 0
247 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
248 *
249 * Scale matrix norm up to SMLNUM
250 *
251 CALL ZLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
252 IBSCL = 1
253 ELSE IF( BNRM.GT.BIGNUM ) THEN
254 *
255 * Scale matrix norm down to BIGNUM
256 *
257 CALL ZLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
258 IBSCL = 2
259 END IF
260 *
261 * Compute QR factorization with column pivoting of A:
262 * A * P = Q * R
263 *
264 CALL ZGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),
265 $ LWORK-MN, RWORK, INFO )
266 WSIZE = MN + DBLE( WORK( MN+1 ) )
267 *
268 * complex workspace: MN+NB*(N+1). real workspace 2*N.
269 * Details of Householder rotations stored in WORK(1:MN).
270 *
271 * Determine RANK using incremental condition estimation
272 *
273 WORK( ISMIN ) = CONE
274 WORK( ISMAX ) = CONE
275 SMAX = ABS( A( 1, 1 ) )
276 SMIN = SMAX
277 IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
278 RANK = 0
279 CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
280 GO TO 70
281 ELSE
282 RANK = 1
283 END IF
284 *
285 10 CONTINUE
286 IF( RANK.LT.MN ) THEN
287 I = RANK + 1
288 CALL ZLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
289 $ A( I, I ), SMINPR, S1, C1 )
290 CALL ZLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
291 $ A( I, I ), SMAXPR, S2, C2 )
292 *
293 IF( SMAXPR*RCOND.LE.SMINPR ) THEN
294 DO 20 I = 1, RANK
295 WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
296 WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
297 20 CONTINUE
298 WORK( ISMIN+RANK ) = C1
299 WORK( ISMAX+RANK ) = C2
300 SMIN = SMINPR
301 SMAX = SMAXPR
302 RANK = RANK + 1
303 GO TO 10
304 END IF
305 END IF
306 *
307 * complex workspace: 3*MN.
308 *
309 * Logically partition R = [ R11 R12 ]
310 * [ 0 R22 ]
311 * where R11 = R(1:RANK,1:RANK)
312 *
313 * [R11,R12] = [ T11, 0 ] * Y
314 *
315 IF( RANK.LT.N )
316 $ CALL ZTZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ),
317 $ LWORK-2*MN, INFO )
318 *
319 * complex workspace: 2*MN.
320 * Details of Householder rotations stored in WORK(MN+1:2*MN)
321 *
322 * B(1:M,1:NRHS) := Q**H * B(1:M,1:NRHS)
323 *
324 CALL ZUNMQR( 'Left', 'Conjugate transpose', M, NRHS, MN, A, LDA,
325 $ WORK( 1 ), B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO )
326 WSIZE = MAX( WSIZE, 2*MN+DBLE( WORK( 2*MN+1 ) ) )
327 *
328 * complex workspace: 2*MN+NB*NRHS.
329 *
330 * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
331 *
332 CALL ZTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
333 $ NRHS, CONE, A, LDA, B, LDB )
334 *
335 DO 40 J = 1, NRHS
336 DO 30 I = RANK + 1, N
337 B( I, J ) = CZERO
338 30 CONTINUE
339 40 CONTINUE
340 *
341 * B(1:N,1:NRHS) := Y**H * B(1:N,1:NRHS)
342 *
343 IF( RANK.LT.N ) THEN
344 CALL ZUNMRZ( 'Left', 'Conjugate transpose', N, NRHS, RANK,
345 $ N-RANK, A, LDA, WORK( MN+1 ), B, LDB,
346 $ WORK( 2*MN+1 ), LWORK-2*MN, INFO )
347 END IF
348 *
349 * complex workspace: 2*MN+NRHS.
350 *
351 * B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
352 *
353 DO 60 J = 1, NRHS
354 DO 50 I = 1, N
355 WORK( JPVT( I ) ) = B( I, J )
356 50 CONTINUE
357 CALL ZCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )
358 60 CONTINUE
359 *
360 * complex workspace: N.
361 *
362 * Undo scaling
363 *
364 IF( IASCL.EQ.1 ) THEN
365 CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
366 CALL ZLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
367 $ INFO )
368 ELSE IF( IASCL.EQ.2 ) THEN
369 CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
370 CALL ZLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
371 $ INFO )
372 END IF
373 IF( IBSCL.EQ.1 ) THEN
374 CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
375 ELSE IF( IBSCL.EQ.2 ) THEN
376 CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
377 END IF
378 *
379 70 CONTINUE
380 WORK( 1 ) = DCMPLX( LWKOPT )
381 *
382 RETURN
383 *
384 * End of ZGELSY
385 *
386 END
2 $ WORK, LWORK, RWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
11 DOUBLE PRECISION RCOND
12 * ..
13 * .. Array Arguments ..
14 INTEGER JPVT( * )
15 DOUBLE PRECISION RWORK( * )
16 COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * ZGELSY computes the minimum-norm solution to a complex linear least
23 * squares problem:
24 * minimize || A * X - B ||
25 * using a complete orthogonal factorization of A. A is an M-by-N
26 * matrix which may be rank-deficient.
27 *
28 * Several right hand side vectors b and solution vectors x can be
29 * handled in a single call; they are stored as the columns of the
30 * M-by-NRHS right hand side matrix B and the N-by-NRHS solution
31 * matrix X.
32 *
33 * The routine first computes a QR factorization with column pivoting:
34 * A * P = Q * [ R11 R12 ]
35 * [ 0 R22 ]
36 * with R11 defined as the largest leading submatrix whose estimated
37 * condition number is less than 1/RCOND. The order of R11, RANK,
38 * is the effective rank of A.
39 *
40 * Then, R22 is considered to be negligible, and R12 is annihilated
41 * by unitary transformations from the right, arriving at the
42 * complete orthogonal factorization:
43 * A * P = Q * [ T11 0 ] * Z
44 * [ 0 0 ]
45 * The minimum-norm solution is then
46 * X = P * Z**H [ inv(T11)*Q1**H*B ]
47 * [ 0 ]
48 * where Q1 consists of the first RANK columns of Q.
49 *
50 * This routine is basically identical to the original xGELSX except
51 * three differences:
52 * o The permutation of matrix B (the right hand side) is faster and
53 * more simple.
54 * o The call to the subroutine xGEQPF has been substituted by the
55 * the call to the subroutine xGEQP3. This subroutine is a Blas-3
56 * version of the QR factorization with column pivoting.
57 * o Matrix B (the right hand side) is updated with Blas-3.
58 *
59 * Arguments
60 * =========
61 *
62 * M (input) INTEGER
63 * The number of rows of the matrix A. M >= 0.
64 *
65 * N (input) INTEGER
66 * The number of columns of the matrix A. N >= 0.
67 *
68 * NRHS (input) INTEGER
69 * The number of right hand sides, i.e., the number of
70 * columns of matrices B and X. NRHS >= 0.
71 *
72 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
73 * On entry, the M-by-N matrix A.
74 * On exit, A has been overwritten by details of its
75 * complete orthogonal factorization.
76 *
77 * LDA (input) INTEGER
78 * The leading dimension of the array A. LDA >= max(1,M).
79 *
80 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
81 * On entry, the M-by-NRHS right hand side matrix B.
82 * On exit, the N-by-NRHS solution matrix X.
83 *
84 * LDB (input) INTEGER
85 * The leading dimension of the array B. LDB >= max(1,M,N).
86 *
87 * JPVT (input/output) INTEGER array, dimension (N)
88 * On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
89 * to the front of AP, otherwise column i is a free column.
90 * On exit, if JPVT(i) = k, then the i-th column of A*P
91 * was the k-th column of A.
92 *
93 * RCOND (input) DOUBLE PRECISION
94 * RCOND is used to determine the effective rank of A, which
95 * is defined as the order of the largest leading triangular
96 * submatrix R11 in the QR factorization with pivoting of A,
97 * whose estimated condition number < 1/RCOND.
98 *
99 * RANK (output) INTEGER
100 * The effective rank of A, i.e., the order of the submatrix
101 * R11. This is the same as the order of the submatrix T11
102 * in the complete orthogonal factorization of A.
103 *
104 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
105 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
106 *
107 * LWORK (input) INTEGER
108 * The dimension of the array WORK.
109 * The unblocked strategy requires that:
110 * LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
111 * where MN = min(M,N).
112 * The block algorithm requires that:
113 * LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
114 * where NB is an upper bound on the blocksize returned
115 * by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR,
116 * and ZUNMRZ.
117 *
118 * If LWORK = -1, then a workspace query is assumed; the routine
119 * only calculates the optimal size of the WORK array, returns
120 * this value as the first entry of the WORK array, and no error
121 * message related to LWORK is issued by XERBLA.
122 *
123 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
124 *
125 * INFO (output) INTEGER
126 * = 0: successful exit
127 * < 0: if INFO = -i, the i-th argument had an illegal value
128 *
129 * Further Details
130 * ===============
131 *
132 * Based on contributions by
133 * A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
134 * E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
135 * G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
136 *
137 * =====================================================================
138 *
139 * .. Parameters ..
140 INTEGER IMAX, IMIN
141 PARAMETER ( IMAX = 1, IMIN = 2 )
142 DOUBLE PRECISION ZERO, ONE
143 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
144 COMPLEX*16 CZERO, CONE
145 PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
146 $ CONE = ( 1.0D+0, 0.0D+0 ) )
147 * ..
148 * .. Local Scalars ..
149 LOGICAL LQUERY
150 INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKOPT, MN,
151 $ NB, NB1, NB2, NB3, NB4
152 DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMAX, SMAXPR, SMIN, SMINPR,
153 $ SMLNUM, WSIZE
154 COMPLEX*16 C1, C2, S1, S2
155 * ..
156 * .. External Subroutines ..
157 EXTERNAL DLABAD, XERBLA, ZCOPY, ZGEQP3, ZLAIC1, ZLASCL,
158 $ ZLASET, ZTRSM, ZTZRZF, ZUNMQR, ZUNMRZ
159 * ..
160 * .. External Functions ..
161 INTEGER ILAENV
162 DOUBLE PRECISION DLAMCH, ZLANGE
163 EXTERNAL ILAENV, DLAMCH, ZLANGE
164 * ..
165 * .. Intrinsic Functions ..
166 INTRINSIC ABS, DBLE, DCMPLX, MAX, MIN
167 * ..
168 * .. Executable Statements ..
169 *
170 MN = MIN( M, N )
171 ISMIN = MN + 1
172 ISMAX = 2*MN + 1
173 *
174 * Test the input arguments.
175 *
176 INFO = 0
177 NB1 = ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
178 NB2 = ILAENV( 1, 'ZGERQF', ' ', M, N, -1, -1 )
179 NB3 = ILAENV( 1, 'ZUNMQR', ' ', M, N, NRHS, -1 )
180 NB4 = ILAENV( 1, 'ZUNMRQ', ' ', M, N, NRHS, -1 )
181 NB = MAX( NB1, NB2, NB3, NB4 )
182 LWKOPT = MAX( 1, MN+2*N+NB*( N+1 ), 2*MN+NB*NRHS )
183 WORK( 1 ) = DCMPLX( LWKOPT )
184 LQUERY = ( LWORK.EQ.-1 )
185 IF( M.LT.0 ) THEN
186 INFO = -1
187 ELSE IF( N.LT.0 ) THEN
188 INFO = -2
189 ELSE IF( NRHS.LT.0 ) THEN
190 INFO = -3
191 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
192 INFO = -5
193 ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
194 INFO = -7
195 ELSE IF( LWORK.LT.( MN+MAX( 2*MN, N+1, MN+NRHS ) ) .AND. .NOT.
196 $ LQUERY ) THEN
197 INFO = -12
198 END IF
199 *
200 IF( INFO.NE.0 ) THEN
201 CALL XERBLA( 'ZGELSY', -INFO )
202 RETURN
203 ELSE IF( LQUERY ) THEN
204 RETURN
205 END IF
206 *
207 * Quick return if possible
208 *
209 IF( MIN( M, N, NRHS ).EQ.0 ) THEN
210 RANK = 0
211 RETURN
212 END IF
213 *
214 * Get machine parameters
215 *
216 SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
217 BIGNUM = ONE / SMLNUM
218 CALL DLABAD( SMLNUM, BIGNUM )
219 *
220 * Scale A, B if max entries outside range [SMLNUM,BIGNUM]
221 *
222 ANRM = ZLANGE( 'M', M, N, A, LDA, RWORK )
223 IASCL = 0
224 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
225 *
226 * Scale matrix norm up to SMLNUM
227 *
228 CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
229 IASCL = 1
230 ELSE IF( ANRM.GT.BIGNUM ) THEN
231 *
232 * Scale matrix norm down to BIGNUM
233 *
234 CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
235 IASCL = 2
236 ELSE IF( ANRM.EQ.ZERO ) THEN
237 *
238 * Matrix all zero. Return zero solution.
239 *
240 CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
241 RANK = 0
242 GO TO 70
243 END IF
244 *
245 BNRM = ZLANGE( 'M', M, NRHS, B, LDB, RWORK )
246 IBSCL = 0
247 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
248 *
249 * Scale matrix norm up to SMLNUM
250 *
251 CALL ZLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
252 IBSCL = 1
253 ELSE IF( BNRM.GT.BIGNUM ) THEN
254 *
255 * Scale matrix norm down to BIGNUM
256 *
257 CALL ZLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
258 IBSCL = 2
259 END IF
260 *
261 * Compute QR factorization with column pivoting of A:
262 * A * P = Q * R
263 *
264 CALL ZGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),
265 $ LWORK-MN, RWORK, INFO )
266 WSIZE = MN + DBLE( WORK( MN+1 ) )
267 *
268 * complex workspace: MN+NB*(N+1). real workspace 2*N.
269 * Details of Householder rotations stored in WORK(1:MN).
270 *
271 * Determine RANK using incremental condition estimation
272 *
273 WORK( ISMIN ) = CONE
274 WORK( ISMAX ) = CONE
275 SMAX = ABS( A( 1, 1 ) )
276 SMIN = SMAX
277 IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
278 RANK = 0
279 CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
280 GO TO 70
281 ELSE
282 RANK = 1
283 END IF
284 *
285 10 CONTINUE
286 IF( RANK.LT.MN ) THEN
287 I = RANK + 1
288 CALL ZLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
289 $ A( I, I ), SMINPR, S1, C1 )
290 CALL ZLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
291 $ A( I, I ), SMAXPR, S2, C2 )
292 *
293 IF( SMAXPR*RCOND.LE.SMINPR ) THEN
294 DO 20 I = 1, RANK
295 WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
296 WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
297 20 CONTINUE
298 WORK( ISMIN+RANK ) = C1
299 WORK( ISMAX+RANK ) = C2
300 SMIN = SMINPR
301 SMAX = SMAXPR
302 RANK = RANK + 1
303 GO TO 10
304 END IF
305 END IF
306 *
307 * complex workspace: 3*MN.
308 *
309 * Logically partition R = [ R11 R12 ]
310 * [ 0 R22 ]
311 * where R11 = R(1:RANK,1:RANK)
312 *
313 * [R11,R12] = [ T11, 0 ] * Y
314 *
315 IF( RANK.LT.N )
316 $ CALL ZTZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ),
317 $ LWORK-2*MN, INFO )
318 *
319 * complex workspace: 2*MN.
320 * Details of Householder rotations stored in WORK(MN+1:2*MN)
321 *
322 * B(1:M,1:NRHS) := Q**H * B(1:M,1:NRHS)
323 *
324 CALL ZUNMQR( 'Left', 'Conjugate transpose', M, NRHS, MN, A, LDA,
325 $ WORK( 1 ), B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO )
326 WSIZE = MAX( WSIZE, 2*MN+DBLE( WORK( 2*MN+1 ) ) )
327 *
328 * complex workspace: 2*MN+NB*NRHS.
329 *
330 * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
331 *
332 CALL ZTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
333 $ NRHS, CONE, A, LDA, B, LDB )
334 *
335 DO 40 J = 1, NRHS
336 DO 30 I = RANK + 1, N
337 B( I, J ) = CZERO
338 30 CONTINUE
339 40 CONTINUE
340 *
341 * B(1:N,1:NRHS) := Y**H * B(1:N,1:NRHS)
342 *
343 IF( RANK.LT.N ) THEN
344 CALL ZUNMRZ( 'Left', 'Conjugate transpose', N, NRHS, RANK,
345 $ N-RANK, A, LDA, WORK( MN+1 ), B, LDB,
346 $ WORK( 2*MN+1 ), LWORK-2*MN, INFO )
347 END IF
348 *
349 * complex workspace: 2*MN+NRHS.
350 *
351 * B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
352 *
353 DO 60 J = 1, NRHS
354 DO 50 I = 1, N
355 WORK( JPVT( I ) ) = B( I, J )
356 50 CONTINUE
357 CALL ZCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )
358 60 CONTINUE
359 *
360 * complex workspace: N.
361 *
362 * Undo scaling
363 *
364 IF( IASCL.EQ.1 ) THEN
365 CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
366 CALL ZLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
367 $ INFO )
368 ELSE IF( IASCL.EQ.2 ) THEN
369 CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
370 CALL ZLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
371 $ INFO )
372 END IF
373 IF( IBSCL.EQ.1 ) THEN
374 CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
375 ELSE IF( IBSCL.EQ.2 ) THEN
376 CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
377 END IF
378 *
379 70 CONTINUE
380 WORK( 1 ) = DCMPLX( LWKOPT )
381 *
382 RETURN
383 *
384 * End of ZGELSY
385 *
386 END