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zgbtrf.f
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1 *> \brief \b ZGBTRF
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, KL, KU, LDAB, M, N
25 * ..
26 * .. Array Arguments ..
27 * INTEGER IPIV( * )
28 * COMPLEX*16 AB( LDAB, * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> ZGBTRF computes an LU factorization of a complex m-by-n band matrix A
38 *> using partial pivoting with row interchanges.
39 *>
40 *> This is the blocked version of the algorithm, calling Level 3 BLAS.
41 *> \endverbatim
42 *
43 * Arguments:
44 * ==========
45 *
46 *> \param[in] M
47 *> \verbatim
48 *> M is INTEGER
49 *> The number of rows of the matrix A. M >= 0.
50 *> \endverbatim
51 *>
52 *> \param[in] N
53 *> \verbatim
54 *> N is INTEGER
55 *> The number of columns of the matrix A. N >= 0.
56 *> \endverbatim
57 *>
58 *> \param[in] KL
59 *> \verbatim
60 *> KL is INTEGER
61 *> The number of subdiagonals within the band of A. KL >= 0.
62 *> \endverbatim
63 *>
64 *> \param[in] KU
65 *> \verbatim
66 *> KU is INTEGER
67 *> The number of superdiagonals within the band of A. KU >= 0.
68 *> \endverbatim
69 *>
70 *> \param[in,out] AB
71 *> \verbatim
72 *> AB is COMPLEX*16 array, dimension (LDAB,N)
73 *> On entry, the matrix A in band storage, in rows KL+1 to
74 *> 2*KL+KU+1; rows 1 to KL of the array need not be set.
75 *> The j-th column of A is stored in the j-th column of the
76 *> array AB as follows:
77 *> AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
78 *>
79 *> On exit, details of the factorization: U is stored as an
80 *> upper triangular band matrix with KL+KU superdiagonals in
81 *> rows 1 to KL+KU+1, and the multipliers used during the
82 *> factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
83 *> See below for further details.
84 *> \endverbatim
85 *>
86 *> \param[in] LDAB
87 *> \verbatim
88 *> LDAB is INTEGER
89 *> The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
90 *> \endverbatim
91 *>
92 *> \param[out] IPIV
93 *> \verbatim
94 *> IPIV is INTEGER array, dimension (min(M,N))
95 *> The pivot indices; for 1 <= i <= min(M,N), row i of the
96 *> matrix was interchanged with row IPIV(i).
97 *> \endverbatim
98 *>
99 *> \param[out] INFO
100 *> \verbatim
101 *> INFO is INTEGER
102 *> = 0: successful exit
103 *> < 0: if INFO = -i, the i-th argument had an illegal value
104 *> > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
105 *> has been completed, but the factor U is exactly
106 *> singular, and division by zero will occur if it is used
107 *> to solve a system of equations.
108 *> \endverbatim
109 *
110 * Authors:
111 * ========
112 *
113 *> \author Univ. of Tennessee
114 *> \author Univ. of California Berkeley
115 *> \author Univ. of Colorado Denver
116 *> \author NAG Ltd.
117 *
118 *> \date November 2011
119 *
120 *> \ingroup complex16GBcomputational
121 *
122 *> \par Further Details:
123 * =====================
124 *>
125 *> \verbatim
126 *>
127 *> The band storage scheme is illustrated by the following example, when
128 *> M = N = 6, KL = 2, KU = 1:
129 *>
130 *> On entry: On exit:
131 *>
132 *> * * * + + + * * * u14 u25 u36
133 *> * * + + + + * * u13 u24 u35 u46
134 *> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
135 *> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
136 *> a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
137 *> a31 a42 a53 a64 * * m31 m42 m53 m64 * *
138 *>
139 *> Array elements marked * are not used by the routine; elements marked
140 *> + need not be set on entry, but are required by the routine to store
141 *> elements of U because of fill-in resulting from the row interchanges.
142 *> \endverbatim
143 *>
144 * =====================================================================
145  SUBROUTINE zgbtrf( M, N, KL, KU, AB, LDAB, IPIV, INFO )
146 *
147 * -- LAPACK computational routine (version 3.4.0) --
148 * -- LAPACK is a software package provided by Univ. of Tennessee, --
149 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
150 * November 2011
151 *
152 * .. Scalar Arguments ..
153  INTEGER info, kl, ku, ldab, m, n
154 * ..
155 * .. Array Arguments ..
156  INTEGER ipiv( * )
157  COMPLEX*16 ab( ldab, * )
158 * ..
159 *
160 * =====================================================================
161 *
162 * .. Parameters ..
163  COMPLEX*16 one, zero
164  parameter( one = ( 1.0d+0, 0.0d+0 ),
165  $ zero = ( 0.0d+0, 0.0d+0 ) )
166  INTEGER nbmax, ldwork
167  parameter( nbmax = 64, ldwork = nbmax+1 )
168 * ..
169 * .. Local Scalars ..
170  INTEGER i, i2, i3, ii, ip, j, j2, j3, jb, jj, jm, jp,
171  $ ju, k2, km, kv, nb, nw
172  COMPLEX*16 temp
173 * ..
174 * .. Local Arrays ..
175  COMPLEX*16 work13( ldwork, nbmax ),
176  $ work31( ldwork, nbmax )
177 * ..
178 * .. External Functions ..
179  INTEGER ilaenv, izamax
180  EXTERNAL ilaenv, izamax
181 * ..
182 * .. External Subroutines ..
183  EXTERNAL xerbla, zcopy, zgbtf2, zgemm, zgeru, zlaswp,
184  $ zscal, zswap, ztrsm
185 * ..
186 * .. Intrinsic Functions ..
187  INTRINSIC max, min
188 * ..
189 * .. Executable Statements ..
190 *
191 * KV is the number of superdiagonals in the factor U, allowing for
192 * fill-in
193 *
194  kv = ku + kl
195 *
196 * Test the input parameters.
197 *
198  info = 0
199  IF( m.LT.0 ) THEN
200  info = -1
201  ELSE IF( n.LT.0 ) THEN
202  info = -2
203  ELSE IF( kl.LT.0 ) THEN
204  info = -3
205  ELSE IF( ku.LT.0 ) THEN
206  info = -4
207  ELSE IF( ldab.LT.kl+kv+1 ) THEN
208  info = -6
209  END IF
210  IF( info.NE.0 ) THEN
211  CALL xerbla( 'ZGBTRF', -info )
212  return
213  END IF
214 *
215 * Quick return if possible
216 *
217  IF( m.EQ.0 .OR. n.EQ.0 )
218  $ return
219 *
220 * Determine the block size for this environment
221 *
222  nb = ilaenv( 1, 'ZGBTRF', ' ', m, n, kl, ku )
223 *
224 * The block size must not exceed the limit set by the size of the
225 * local arrays WORK13 and WORK31.
226 *
227  nb = min( nb, nbmax )
228 *
229  IF( nb.LE.1 .OR. nb.GT.kl ) THEN
230 *
231 * Use unblocked code
232 *
233  CALL zgbtf2( m, n, kl, ku, ab, ldab, ipiv, info )
234  ELSE
235 *
236 * Use blocked code
237 *
238 * Zero the superdiagonal elements of the work array WORK13
239 *
240  DO 20 j = 1, nb
241  DO 10 i = 1, j - 1
242  work13( i, j ) = zero
243  10 continue
244  20 continue
245 *
246 * Zero the subdiagonal elements of the work array WORK31
247 *
248  DO 40 j = 1, nb
249  DO 30 i = j + 1, nb
250  work31( i, j ) = zero
251  30 continue
252  40 continue
253 *
254 * Gaussian elimination with partial pivoting
255 *
256 * Set fill-in elements in columns KU+2 to KV to zero
257 *
258  DO 60 j = ku + 2, min( kv, n )
259  DO 50 i = kv - j + 2, kl
260  ab( i, j ) = zero
261  50 continue
262  60 continue
263 *
264 * JU is the index of the last column affected by the current
265 * stage of the factorization
266 *
267  ju = 1
268 *
269  DO 180 j = 1, min( m, n ), nb
270  jb = min( nb, min( m, n )-j+1 )
271 *
272 * The active part of the matrix is partitioned
273 *
274 * A11 A12 A13
275 * A21 A22 A23
276 * A31 A32 A33
277 *
278 * Here A11, A21 and A31 denote the current block of JB columns
279 * which is about to be factorized. The number of rows in the
280 * partitioning are JB, I2, I3 respectively, and the numbers
281 * of columns are JB, J2, J3. The superdiagonal elements of A13
282 * and the subdiagonal elements of A31 lie outside the band.
283 *
284  i2 = min( kl-jb, m-j-jb+1 )
285  i3 = min( jb, m-j-kl+1 )
286 *
287 * J2 and J3 are computed after JU has been updated.
288 *
289 * Factorize the current block of JB columns
290 *
291  DO 80 jj = j, j + jb - 1
292 *
293 * Set fill-in elements in column JJ+KV to zero
294 *
295  IF( jj+kv.LE.n ) THEN
296  DO 70 i = 1, kl
297  ab( i, jj+kv ) = zero
298  70 continue
299  END IF
300 *
301 * Find pivot and test for singularity. KM is the number of
302 * subdiagonal elements in the current column.
303 *
304  km = min( kl, m-jj )
305  jp = izamax( km+1, ab( kv+1, jj ), 1 )
306  ipiv( jj ) = jp + jj - j
307  IF( ab( kv+jp, jj ).NE.zero ) THEN
308  ju = max( ju, min( jj+ku+jp-1, n ) )
309  IF( jp.NE.1 ) THEN
310 *
311 * Apply interchange to columns J to J+JB-1
312 *
313  IF( jp+jj-1.LT.j+kl ) THEN
314 *
315  CALL zswap( jb, ab( kv+1+jj-j, j ), ldab-1,
316  $ ab( kv+jp+jj-j, j ), ldab-1 )
317  ELSE
318 *
319 * The interchange affects columns J to JJ-1 of A31
320 * which are stored in the work array WORK31
321 *
322  CALL zswap( jj-j, ab( kv+1+jj-j, j ), ldab-1,
323  $ work31( jp+jj-j-kl, 1 ), ldwork )
324  CALL zswap( j+jb-jj, ab( kv+1, jj ), ldab-1,
325  $ ab( kv+jp, jj ), ldab-1 )
326  END IF
327  END IF
328 *
329 * Compute multipliers
330 *
331  CALL zscal( km, one / ab( kv+1, jj ), ab( kv+2, jj ),
332  $ 1 )
333 *
334 * Update trailing submatrix within the band and within
335 * the current block. JM is the index of the last column
336 * which needs to be updated.
337 *
338  jm = min( ju, j+jb-1 )
339  IF( jm.GT.jj )
340  $ CALL zgeru( km, jm-jj, -one, ab( kv+2, jj ), 1,
341  $ ab( kv, jj+1 ), ldab-1,
342  $ ab( kv+1, jj+1 ), ldab-1 )
343  ELSE
344 *
345 * If pivot is zero, set INFO to the index of the pivot
346 * unless a zero pivot has already been found.
347 *
348  IF( info.EQ.0 )
349  $ info = jj
350  END IF
351 *
352 * Copy current column of A31 into the work array WORK31
353 *
354  nw = min( jj-j+1, i3 )
355  IF( nw.GT.0 )
356  $ CALL zcopy( nw, ab( kv+kl+1-jj+j, jj ), 1,
357  $ work31( 1, jj-j+1 ), 1 )
358  80 continue
359  IF( j+jb.LE.n ) THEN
360 *
361 * Apply the row interchanges to the other blocks.
362 *
363  j2 = min( ju-j+1, kv ) - jb
364  j3 = max( 0, ju-j-kv+1 )
365 *
366 * Use ZLASWP to apply the row interchanges to A12, A22, and
367 * A32.
368 *
369  CALL zlaswp( j2, ab( kv+1-jb, j+jb ), ldab-1, 1, jb,
370  $ ipiv( j ), 1 )
371 *
372 * Adjust the pivot indices.
373 *
374  DO 90 i = j, j + jb - 1
375  ipiv( i ) = ipiv( i ) + j - 1
376  90 continue
377 *
378 * Apply the row interchanges to A13, A23, and A33
379 * columnwise.
380 *
381  k2 = j - 1 + jb + j2
382  DO 110 i = 1, j3
383  jj = k2 + i
384  DO 100 ii = j + i - 1, j + jb - 1
385  ip = ipiv( ii )
386  IF( ip.NE.ii ) THEN
387  temp = ab( kv+1+ii-jj, jj )
388  ab( kv+1+ii-jj, jj ) = ab( kv+1+ip-jj, jj )
389  ab( kv+1+ip-jj, jj ) = temp
390  END IF
391  100 continue
392  110 continue
393 *
394 * Update the relevant part of the trailing submatrix
395 *
396  IF( j2.GT.0 ) THEN
397 *
398 * Update A12
399 *
400  CALL ztrsm( 'Left', 'Lower', 'No transpose', 'Unit',
401  $ jb, j2, one, ab( kv+1, j ), ldab-1,
402  $ ab( kv+1-jb, j+jb ), ldab-1 )
403 *
404  IF( i2.GT.0 ) THEN
405 *
406 * Update A22
407 *
408  CALL zgemm( 'No transpose', 'No transpose', i2, j2,
409  $ jb, -one, ab( kv+1+jb, j ), ldab-1,
410  $ ab( kv+1-jb, j+jb ), ldab-1, one,
411  $ ab( kv+1, j+jb ), ldab-1 )
412  END IF
413 *
414  IF( i3.GT.0 ) THEN
415 *
416 * Update A32
417 *
418  CALL zgemm( 'No transpose', 'No transpose', i3, j2,
419  $ jb, -one, work31, ldwork,
420  $ ab( kv+1-jb, j+jb ), ldab-1, one,
421  $ ab( kv+kl+1-jb, j+jb ), ldab-1 )
422  END IF
423  END IF
424 *
425  IF( j3.GT.0 ) THEN
426 *
427 * Copy the lower triangle of A13 into the work array
428 * WORK13
429 *
430  DO 130 jj = 1, j3
431  DO 120 ii = jj, jb
432  work13( ii, jj ) = ab( ii-jj+1, jj+j+kv-1 )
433  120 continue
434  130 continue
435 *
436 * Update A13 in the work array
437 *
438  CALL ztrsm( 'Left', 'Lower', 'No transpose', 'Unit',
439  $ jb, j3, one, ab( kv+1, j ), ldab-1,
440  $ work13, ldwork )
441 *
442  IF( i2.GT.0 ) THEN
443 *
444 * Update A23
445 *
446  CALL zgemm( 'No transpose', 'No transpose', i2, j3,
447  $ jb, -one, ab( kv+1+jb, j ), ldab-1,
448  $ work13, ldwork, one, ab( 1+jb, j+kv ),
449  $ ldab-1 )
450  END IF
451 *
452  IF( i3.GT.0 ) THEN
453 *
454 * Update A33
455 *
456  CALL zgemm( 'No transpose', 'No transpose', i3, j3,
457  $ jb, -one, work31, ldwork, work13,
458  $ ldwork, one, ab( 1+kl, j+kv ), ldab-1 )
459  END IF
460 *
461 * Copy the lower triangle of A13 back into place
462 *
463  DO 150 jj = 1, j3
464  DO 140 ii = jj, jb
465  ab( ii-jj+1, jj+j+kv-1 ) = work13( ii, jj )
466  140 continue
467  150 continue
468  END IF
469  ELSE
470 *
471 * Adjust the pivot indices.
472 *
473  DO 160 i = j, j + jb - 1
474  ipiv( i ) = ipiv( i ) + j - 1
475  160 continue
476  END IF
477 *
478 * Partially undo the interchanges in the current block to
479 * restore the upper triangular form of A31 and copy the upper
480 * triangle of A31 back into place
481 *
482  DO 170 jj = j + jb - 1, j, -1
483  jp = ipiv( jj ) - jj + 1
484  IF( jp.NE.1 ) THEN
485 *
486 * Apply interchange to columns J to JJ-1
487 *
488  IF( jp+jj-1.LT.j+kl ) THEN
489 *
490 * The interchange does not affect A31
491 *
492  CALL zswap( jj-j, ab( kv+1+jj-j, j ), ldab-1,
493  $ ab( kv+jp+jj-j, j ), ldab-1 )
494  ELSE
495 *
496 * The interchange does affect A31
497 *
498  CALL zswap( jj-j, ab( kv+1+jj-j, j ), ldab-1,
499  $ work31( jp+jj-j-kl, 1 ), ldwork )
500  END IF
501  END IF
502 *
503 * Copy the current column of A31 back into place
504 *
505  nw = min( i3, jj-j+1 )
506  IF( nw.GT.0 )
507  $ CALL zcopy( nw, work31( 1, jj-j+1 ), 1,
508  $ ab( kv+kl+1-jj+j, jj ), 1 )
509  170 continue
510  180 continue
511  END IF
512 *
513  return
514 *
515 * End of ZGBTRF
516 *
517  END