LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
sgbtrf.f
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1 *> \brief \b SGBTRF
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGBTRF( 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 * REAL AB( LDAB, * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> SGBTRF computes an LU factorization of a real 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 REAL 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 realGBcomputational
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 sgbtrf( 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  REAL ab( ldab, * )
158 * ..
159 *
160 * =====================================================================
161 *
162 * .. Parameters ..
163  REAL one, zero
164  parameter( one = 1.0e+0, zero = 0.0e+0 )
165  INTEGER nbmax, ldwork
166  parameter( nbmax = 64, ldwork = nbmax+1 )
167 * ..
168 * .. Local Scalars ..
169  INTEGER i, i2, i3, ii, ip, j, j2, j3, jb, jj, jm, jp,
170  \$ ju, k2, km, kv, nb, nw
171  REAL temp
172 * ..
173 * .. Local Arrays ..
174  REAL work13( ldwork, nbmax ),
175  \$ work31( ldwork, nbmax )
176 * ..
177 * .. External Functions ..
178  INTEGER ilaenv, isamax
179  EXTERNAL ilaenv, isamax
180 * ..
181 * .. External Subroutines ..
182  EXTERNAL scopy, sgbtf2, sgemm, sger, slaswp, sscal,
183  \$ sswap, strsm, xerbla
184 * ..
185 * .. Intrinsic Functions ..
186  INTRINSIC max, min
187 * ..
188 * .. Executable Statements ..
189 *
190 * KV is the number of superdiagonals in the factor U, allowing for
191 * fill-in
192 *
193  kv = ku + kl
194 *
195 * Test the input parameters.
196 *
197  info = 0
198  IF( m.LT.0 ) THEN
199  info = -1
200  ELSE IF( n.LT.0 ) THEN
201  info = -2
202  ELSE IF( kl.LT.0 ) THEN
203  info = -3
204  ELSE IF( ku.LT.0 ) THEN
205  info = -4
206  ELSE IF( ldab.LT.kl+kv+1 ) THEN
207  info = -6
208  END IF
209  IF( info.NE.0 ) THEN
210  CALL xerbla( 'SGBTRF', -info )
211  return
212  END IF
213 *
214 * Quick return if possible
215 *
216  IF( m.EQ.0 .OR. n.EQ.0 )
217  \$ return
218 *
219 * Determine the block size for this environment
220 *
221  nb = ilaenv( 1, 'SGBTRF', ' ', m, n, kl, ku )
222 *
223 * The block size must not exceed the limit set by the size of the
224 * local arrays WORK13 and WORK31.
225 *
226  nb = min( nb, nbmax )
227 *
228  IF( nb.LE.1 .OR. nb.GT.kl ) THEN
229 *
230 * Use unblocked code
231 *
232  CALL sgbtf2( m, n, kl, ku, ab, ldab, ipiv, info )
233  ELSE
234 *
235 * Use blocked code
236 *
237 * Zero the superdiagonal elements of the work array WORK13
238 *
239  DO 20 j = 1, nb
240  DO 10 i = 1, j - 1
241  work13( i, j ) = zero
242  10 continue
243  20 continue
244 *
245 * Zero the subdiagonal elements of the work array WORK31
246 *
247  DO 40 j = 1, nb
248  DO 30 i = j + 1, nb
249  work31( i, j ) = zero
250  30 continue
251  40 continue
252 *
253 * Gaussian elimination with partial pivoting
254 *
255 * Set fill-in elements in columns KU+2 to KV to zero
256 *
257  DO 60 j = ku + 2, min( kv, n )
258  DO 50 i = kv - j + 2, kl
259  ab( i, j ) = zero
260  50 continue
261  60 continue
262 *
263 * JU is the index of the last column affected by the current
264 * stage of the factorization
265 *
266  ju = 1
267 *
268  DO 180 j = 1, min( m, n ), nb
269  jb = min( nb, min( m, n )-j+1 )
270 *
271 * The active part of the matrix is partitioned
272 *
273 * A11 A12 A13
274 * A21 A22 A23
275 * A31 A32 A33
276 *
277 * Here A11, A21 and A31 denote the current block of JB columns
278 * which is about to be factorized. The number of rows in the
279 * partitioning are JB, I2, I3 respectively, and the numbers
280 * of columns are JB, J2, J3. The superdiagonal elements of A13
281 * and the subdiagonal elements of A31 lie outside the band.
282 *
283  i2 = min( kl-jb, m-j-jb+1 )
284  i3 = min( jb, m-j-kl+1 )
285 *
286 * J2 and J3 are computed after JU has been updated.
287 *
288 * Factorize the current block of JB columns
289 *
290  DO 80 jj = j, j + jb - 1
291 *
292 * Set fill-in elements in column JJ+KV to zero
293 *
294  IF( jj+kv.LE.n ) THEN
295  DO 70 i = 1, kl
296  ab( i, jj+kv ) = zero
297  70 continue
298  END IF
299 *
300 * Find pivot and test for singularity. KM is the number of
301 * subdiagonal elements in the current column.
302 *
303  km = min( kl, m-jj )
304  jp = isamax( km+1, ab( kv+1, jj ), 1 )
305  ipiv( jj ) = jp + jj - j
306  IF( ab( kv+jp, jj ).NE.zero ) THEN
307  ju = max( ju, min( jj+ku+jp-1, n ) )
308  IF( jp.NE.1 ) THEN
309 *
310 * Apply interchange to columns J to J+JB-1
311 *
312  IF( jp+jj-1.LT.j+kl ) THEN
313 *
314  CALL sswap( jb, ab( kv+1+jj-j, j ), ldab-1,
315  \$ ab( kv+jp+jj-j, j ), ldab-1 )
316  ELSE
317 *
318 * The interchange affects columns J to JJ-1 of A31
319 * which are stored in the work array WORK31
320 *
321  CALL sswap( jj-j, ab( kv+1+jj-j, j ), ldab-1,
322  \$ work31( jp+jj-j-kl, 1 ), ldwork )
323  CALL sswap( j+jb-jj, ab( kv+1, jj ), ldab-1,
324  \$ ab( kv+jp, jj ), ldab-1 )
325  END IF
326  END IF
327 *
328 * Compute multipliers
329 *
330  CALL sscal( km, one / ab( kv+1, jj ), ab( kv+2, jj ),
331  \$ 1 )
332 *
333 * Update trailing submatrix within the band and within
334 * the current block. JM is the index of the last column
335 * which needs to be updated.
336 *
337  jm = min( ju, j+jb-1 )
338  IF( jm.GT.jj )
339  \$ CALL sger( km, jm-jj, -one, ab( kv+2, jj ), 1,
340  \$ ab( kv, jj+1 ), ldab-1,
341  \$ ab( kv+1, jj+1 ), ldab-1 )
342  ELSE
343 *
344 * If pivot is zero, set INFO to the index of the pivot
345 * unless a zero pivot has already been found.
346 *
347  IF( info.EQ.0 )
348  \$ info = jj
349  END IF
350 *
351 * Copy current column of A31 into the work array WORK31
352 *
353  nw = min( jj-j+1, i3 )
354  IF( nw.GT.0 )
355  \$ CALL scopy( nw, ab( kv+kl+1-jj+j, jj ), 1,
356  \$ work31( 1, jj-j+1 ), 1 )
357  80 continue
358  IF( j+jb.LE.n ) THEN
359 *
360 * Apply the row interchanges to the other blocks.
361 *
362  j2 = min( ju-j+1, kv ) - jb
363  j3 = max( 0, ju-j-kv+1 )
364 *
365 * Use SLASWP to apply the row interchanges to A12, A22, and
366 * A32.
367 *
368  CALL slaswp( j2, ab( kv+1-jb, j+jb ), ldab-1, 1, jb,
369  \$ ipiv( j ), 1 )
370 *
371 * Adjust the pivot indices.
372 *
373  DO 90 i = j, j + jb - 1
374  ipiv( i ) = ipiv( i ) + j - 1
375  90 continue
376 *
377 * Apply the row interchanges to A13, A23, and A33
378 * columnwise.
379 *
380  k2 = j - 1 + jb + j2
381  DO 110 i = 1, j3
382  jj = k2 + i
383  DO 100 ii = j + i - 1, j + jb - 1
384  ip = ipiv( ii )
385  IF( ip.NE.ii ) THEN
386  temp = ab( kv+1+ii-jj, jj )
387  ab( kv+1+ii-jj, jj ) = ab( kv+1+ip-jj, jj )
388  ab( kv+1+ip-jj, jj ) = temp
389  END IF
390  100 continue
391  110 continue
392 *
393 * Update the relevant part of the trailing submatrix
394 *
395  IF( j2.GT.0 ) THEN
396 *
397 * Update A12
398 *
399  CALL strsm( 'Left', 'Lower', 'No transpose', 'Unit',
400  \$ jb, j2, one, ab( kv+1, j ), ldab-1,
401  \$ ab( kv+1-jb, j+jb ), ldab-1 )
402 *
403  IF( i2.GT.0 ) THEN
404 *
405 * Update A22
406 *
407  CALL sgemm( 'No transpose', 'No transpose', i2, j2,
408  \$ jb, -one, ab( kv+1+jb, j ), ldab-1,
409  \$ ab( kv+1-jb, j+jb ), ldab-1, one,
410  \$ ab( kv+1, j+jb ), ldab-1 )
411  END IF
412 *
413  IF( i3.GT.0 ) THEN
414 *
415 * Update A32
416 *
417  CALL sgemm( 'No transpose', 'No transpose', i3, j2,
418  \$ jb, -one, work31, ldwork,
419  \$ ab( kv+1-jb, j+jb ), ldab-1, one,
420  \$ ab( kv+kl+1-jb, j+jb ), ldab-1 )
421  END IF
422  END IF
423 *
424  IF( j3.GT.0 ) THEN
425 *
426 * Copy the lower triangle of A13 into the work array
427 * WORK13
428 *
429  DO 130 jj = 1, j3
430  DO 120 ii = jj, jb
431  work13( ii, jj ) = ab( ii-jj+1, jj+j+kv-1 )
432  120 continue
433  130 continue
434 *
435 * Update A13 in the work array
436 *
437  CALL strsm( 'Left', 'Lower', 'No transpose', 'Unit',
438  \$ jb, j3, one, ab( kv+1, j ), ldab-1,
439  \$ work13, ldwork )
440 *
441  IF( i2.GT.0 ) THEN
442 *
443 * Update A23
444 *
445  CALL sgemm( 'No transpose', 'No transpose', i2, j3,
446  \$ jb, -one, ab( kv+1+jb, j ), ldab-1,
447  \$ work13, ldwork, one, ab( 1+jb, j+kv ),
448  \$ ldab-1 )
449  END IF
450 *
451  IF( i3.GT.0 ) THEN
452 *
453 * Update A33
454 *
455  CALL sgemm( 'No transpose', 'No transpose', i3, j3,
456  \$ jb, -one, work31, ldwork, work13,
457  \$ ldwork, one, ab( 1+kl, j+kv ), ldab-1 )
458  END IF
459 *
460 * Copy the lower triangle of A13 back into place
461 *
462  DO 150 jj = 1, j3
463  DO 140 ii = jj, jb
464  ab( ii-jj+1, jj+j+kv-1 ) = work13( ii, jj )
465  140 continue
466  150 continue
467  END IF
468  ELSE
469 *
470 * Adjust the pivot indices.
471 *
472  DO 160 i = j, j + jb - 1
473  ipiv( i ) = ipiv( i ) + j - 1
474  160 continue
475  END IF
476 *
477 * Partially undo the interchanges in the current block to
478 * restore the upper triangular form of A31 and copy the upper
479 * triangle of A31 back into place
480 *
481  DO 170 jj = j + jb - 1, j, -1
482  jp = ipiv( jj ) - jj + 1
483  IF( jp.NE.1 ) THEN
484 *
485 * Apply interchange to columns J to JJ-1
486 *
487  IF( jp+jj-1.LT.j+kl ) THEN
488 *
489 * The interchange does not affect A31
490 *
491  CALL sswap( jj-j, ab( kv+1+jj-j, j ), ldab-1,
492  \$ ab( kv+jp+jj-j, j ), ldab-1 )
493  ELSE
494 *
495 * The interchange does affect A31
496 *
497  CALL sswap( jj-j, ab( kv+1+jj-j, j ), ldab-1,
498  \$ work31( jp+jj-j-kl, 1 ), ldwork )
499  END IF
500  END IF
501 *
502 * Copy the current column of A31 back into place
503 *
504  nw = min( i3, jj-j+1 )
505  IF( nw.GT.0 )
506  \$ CALL scopy( nw, work31( 1, jj-j+1 ), 1,
507  \$ ab( kv+kl+1-jj+j, jj ), 1 )
508  170 continue
509  180 continue
510  END IF
511 *
512  return
513 *
514 * End of SGBTRF
515 *
516  END