ScaLAPACK 2.1  2.1
ScaLAPACK: Scalable Linear Algebra PACKage
psgbsv.f
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1  SUBROUTINE psgbsv( N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB,
2  $ DESCB, WORK, LWORK, INFO )
3 *
4 *
5 *
6 * -- ScaLAPACK routine (version 1.7) --
7 * University of Tennessee, Knoxville, Oak Ridge National Laboratory,
8 * and University of California, Berkeley.
9 * November 15, 1997
10 *
11 * .. Scalar Arguments ..
12  INTEGER BWL, BWU, IB, INFO, JA, LWORK, N, NRHS
13 * ..
14 * .. Array Arguments ..
15  INTEGER DESCA( * ), DESCB( * ), IPIV( * )
16  REAL A( * ), B( * ), WORK( * )
17 * ..
18 *
19 *
20 * Purpose
21 * =======
22 *
23 * PSGBSV solves a system of linear equations
24 *
25 * A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS)
26 *
27 * where A(1:N, JA:JA+N-1) is an N-by-N real
28 * banded distributed
29 * matrix with bandwidth BWL, BWU.
30 *
31 * Gaussian elimination with pivoting
32 * is used to factor a reordering
33 * of the matrix into P L U.
34 *
35 * See PSGBTRF and PSGBTRS for details.
36 *
37 * =====================================================================
38 *
39 * Arguments
40 * =========
41 *
42 *
43 * N (global input) INTEGER
44 * The number of rows and columns to be operated on, i.e. the
45 * order of the distributed submatrix A(1:N, JA:JA+N-1). N >= 0.
46 *
47 * BWL (global input) INTEGER
48 * Number of subdiagonals. 0 <= BWL <= N-1
49 *
50 * BWU (global input) INTEGER
51 * Number of superdiagonals. 0 <= BWU <= N-1
52 *
53 * NRHS (global input) INTEGER
54 * The number of right hand sides, i.e., the number of columns
55 * of the distributed submatrix B(IB:IB+N-1, 1:NRHS).
56 * NRHS >= 0.
57 *
58 * A (local input/local output) REAL pointer into
59 * local memory to an array with first dimension
60 * LLD_A >=(2*bwl+2*bwu+1) (stored in DESCA).
61 * On entry, this array contains the local pieces of the
62 * This local portion is stored in the packed banded format
63 * used in LAPACK. Please see the Notes below and the
64 * ScaLAPACK manual for more detail on the format of
65 * distributed matrices.
66 * On exit, this array contains information containing details
67 * of the factorization.
68 * Note that permutations are performed on the matrix, so that
69 * the factors returned are different from those returned
70 * by LAPACK.
71 *
72 * JA (global input) INTEGER
73 * The index in the global array A that points to the start of
74 * the matrix to be operated on (which may be either all of A
75 * or a submatrix of A).
76 *
77 * DESCA (global and local input) INTEGER array of dimension DLEN.
78 * if 1D type (DTYPE_A=501), DLEN >= 7;
79 * if 2D type (DTYPE_A=1), DLEN >= 9 .
80 * The array descriptor for the distributed matrix A.
81 * Contains information of mapping of A to memory. Please
82 * see NOTES below for full description and options.
83 *
84 * IPIV (local output) INTEGER array, dimension >= DESCA( NB ).
85 * Pivot indices for local factorizations.
86 * Users *should not* alter the contents between
87 * factorization and solve.
88 *
89 * B (local input/local output) REAL pointer into
90 * local memory to an array of local lead dimension lld_b>=NB.
91 * On entry, this array contains the
92 * the local pieces of the right hand sides
93 * B(IB:IB+N-1, 1:NRHS).
94 * On exit, this contains the local piece of the solutions
95 * distributed matrix X.
96 *
97 * IB (global input) INTEGER
98 * The row index in the global array B that points to the first
99 * row of the matrix to be operated on (which may be either
100 * all of B or a submatrix of B).
101 *
102 * DESCB (global and local input) INTEGER array of dimension DLEN.
103 * if 1D type (DTYPE_B=502), DLEN >=7;
104 * if 2D type (DTYPE_B=1), DLEN >= 9.
105 * The array descriptor for the distributed matrix B.
106 * Contains information of mapping of B to memory. Please
107 * see NOTES below for full description and options.
108 *
109 * WORK (local workspace/local output)
110 * REAL temporary workspace. This space may
111 * be overwritten in between calls to routines. WORK must be
112 * the size given in LWORK.
113 * On exit, WORK( 1 ) contains the minimal LWORK.
114 *
115 * LWORK (local input or global input) INTEGER
116 * Size of user-input workspace WORK.
117 * If LWORK is too small, the minimal acceptable size will be
118 * returned in WORK(1) and an error code is returned. LWORK>=
119 * (NB+bwu)*(bwl+bwu)+6*(bwl+bwu)*(bwl+2*bwu)
120 * +max(NRHS*(NB+2*bwl+4*bwu), 1)
121 *
122 * INFO (global output) INTEGER
123 * = 0: successful exit
124 * < 0: If the i-th argument is an array and the j-entry had
125 * an illegal value, then INFO = -(i*100+j), if the i-th
126 * argument is a scalar and had an illegal value, then
127 * INFO = -i.
128 * > 0: If INFO = K<=NPROCS, the submatrix stored on processor
129 * INFO and factored locally was not
130 * nonsingular, and
131 * the factorization was not completed.
132 * If INFO = K>NPROCS, the submatrix stored on processor
133 * INFO-NPROCS representing interactions with other
134 * processors was not
135 * nonsingular,
136 * and the factorization was not completed.
137 *
138 * =====================================================================
139 *
140 *
141 * Restrictions
142 * ============
143 *
144 * The following are restrictions on the input parameters. Some of these
145 * are temporary and will be removed in future releases, while others
146 * may reflect fundamental technical limitations.
147 *
148 * Non-cyclic restriction: VERY IMPORTANT!
149 * P*NB>= mod(JA-1,NB)+N.
150 * The mapping for matrices must be blocked, reflecting the nature
151 * of the divide and conquer algorithm as a task-parallel algorithm.
152 * This formula in words is: no processor may have more than one
153 * chunk of the matrix.
154 *
155 * Blocksize cannot be too small:
156 * If the matrix spans more than one processor, the following
157 * restriction on NB, the size of each block on each processor,
158 * must hold:
159 * NB >= (BWL+BWU)+1
160 * The bulk of parallel computation is done on the matrix of size
161 * O(NB) on each processor. If this is too small, divide and conquer
162 * is a poor choice of algorithm.
163 *
164 * Submatrix reference:
165 * JA = IB
166 * Alignment restriction that prevents unnecessary communication.
167 *
168 *
169 * =====================================================================
170 *
171 *
172 * Notes
173 * =====
174 *
175 * If the factorization routine and the solve routine are to be called
176 * separately (to solve various sets of righthand sides using the same
177 * coefficient matrix), the auxiliary space AF *must not be altered*
178 * between calls to the factorization routine and the solve routine.
179 *
180 * The best algorithm for solving banded and tridiagonal linear systems
181 * depends on a variety of parameters, especially the bandwidth.
182 * Currently, only algorithms designed for the case N/P >> bw are
183 * implemented. These go by many names, including Divide and Conquer,
184 * Partitioning, domain decomposition-type, etc.
185 *
186 * Algorithm description: Divide and Conquer
187 *
188 * The Divide and Conqer algorithm assumes the matrix is narrowly
189 * banded compared with the number of equations. In this situation,
190 * it is best to distribute the input matrix A one-dimensionally,
191 * with columns atomic and rows divided amongst the processes.
192 * The basic algorithm divides the banded matrix up into
193 * P pieces with one stored on each processor,
194 * and then proceeds in 2 phases for the factorization or 3 for the
195 * solution of a linear system.
196 * 1) Local Phase:
197 * The individual pieces are factored independently and in
198 * parallel. These factors are applied to the matrix creating
199 * fillin, which is stored in a non-inspectable way in auxiliary
200 * space AF. Mathematically, this is equivalent to reordering
201 * the matrix A as P A P^T and then factoring the principal
202 * leading submatrix of size equal to the sum of the sizes of
203 * the matrices factored on each processor. The factors of
204 * these submatrices overwrite the corresponding parts of A
205 * in memory.
206 * 2) Reduced System Phase:
207 * A small (max(bwl,bwu)* (P-1)) system is formed representing
208 * interaction of the larger blocks, and is stored (as are its
209 * factors) in the space AF. A parallel Block Cyclic Reduction
210 * algorithm is used. For a linear system, a parallel front solve
211 * followed by an analagous backsolve, both using the structure
212 * of the factored matrix, are performed.
213 * 3) Backsubsitution Phase:
214 * For a linear system, a local backsubstitution is performed on
215 * each processor in parallel.
216 *
217 *
218 * Descriptors
219 * ===========
220 *
221 * Descriptors now have *types* and differ from ScaLAPACK 1.0.
222 *
223 * Note: banded codes can use either the old two dimensional
224 * or new one-dimensional descriptors, though the processor grid in
225 * both cases *must be one-dimensional*. We describe both types below.
226 *
227 * Each global data object is described by an associated description
228 * vector. This vector stores the information required to establish
229 * the mapping between an object element and its corresponding process
230 * and memory location.
231 *
232 * Let A be a generic term for any 2D block cyclicly distributed array.
233 * Such a global array has an associated description vector DESCA.
234 * In the following comments, the character _ should be read as
235 * "of the global array".
236 *
237 * NOTATION STORED IN EXPLANATION
238 * --------------- -------------- --------------------------------------
239 * DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
240 * DTYPE_A = 1.
241 * CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
242 * the BLACS process grid A is distribu-
243 * ted over. The context itself is glo-
244 * bal, but the handle (the integer
245 * value) may vary.
246 * M_A (global) DESCA( M_ ) The number of rows in the global
247 * array A.
248 * N_A (global) DESCA( N_ ) The number of columns in the global
249 * array A.
250 * MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
251 * the rows of the array.
252 * NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
253 * the columns of the array.
254 * RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
255 * row of the array A is distributed.
256 * CSRC_A (global) DESCA( CSRC_ ) The process column over which the
257 * first column of the array A is
258 * distributed.
259 * LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
260 * array. LLD_A >= MAX(1,LOCr(M_A)).
261 *
262 * Let K be the number of rows or columns of a distributed matrix,
263 * and assume that its process grid has dimension p x q.
264 * LOCr( K ) denotes the number of elements of K that a process
265 * would receive if K were distributed over the p processes of its
266 * process column.
267 * Similarly, LOCc( K ) denotes the number of elements of K that a
268 * process would receive if K were distributed over the q processes of
269 * its process row.
270 * The values of LOCr() and LOCc() may be determined via a call to the
271 * ScaLAPACK tool function, NUMROC:
272 * LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
273 * LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
274 * An upper bound for these quantities may be computed by:
275 * LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
276 * LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
277 *
278 *
279 * One-dimensional descriptors:
280 *
281 * One-dimensional descriptors are a new addition to ScaLAPACK since
282 * version 1.0. They simplify and shorten the descriptor for 1D
283 * arrays.
284 *
285 * Since ScaLAPACK supports two-dimensional arrays as the fundamental
286 * object, we allow 1D arrays to be distributed either over the
287 * first dimension of the array (as if the grid were P-by-1) or the
288 * 2nd dimension (as if the grid were 1-by-P). This choice is
289 * indicated by the descriptor type (501 or 502)
290 * as described below.
291 *
292 * IMPORTANT NOTE: the actual BLACS grid represented by the
293 * CTXT entry in the descriptor may be *either* P-by-1 or 1-by-P
294 * irrespective of which one-dimensional descriptor type
295 * (501 or 502) is input.
296 * This routine will interpret the grid properly either way.
297 * ScaLAPACK routines *do not support intercontext operations* so that
298 * the grid passed to a single ScaLAPACK routine *must be the same*
299 * for all array descriptors passed to that routine.
300 *
301 * NOTE: In all cases where 1D descriptors are used, 2D descriptors
302 * may also be used, since a one-dimensional array is a special case
303 * of a two-dimensional array with one dimension of size unity.
304 * The two-dimensional array used in this case *must* be of the
305 * proper orientation:
306 * If the appropriate one-dimensional descriptor is DTYPEA=501
307 * (1 by P type), then the two dimensional descriptor must
308 * have a CTXT value that refers to a 1 by P BLACS grid;
309 * If the appropriate one-dimensional descriptor is DTYPEA=502
310 * (P by 1 type), then the two dimensional descriptor must
311 * have a CTXT value that refers to a P by 1 BLACS grid.
312 *
313 *
314 * Summary of allowed descriptors, types, and BLACS grids:
315 * DTYPE 501 502 1 1
316 * BLACS grid 1xP or Px1 1xP or Px1 1xP Px1
317 * -----------------------------------------------------
318 * A OK NO OK NO
319 * B NO OK NO OK
320 *
321 * Note that a consequence of this chart is that it is not possible
322 * for *both* DTYPE_A and DTYPE_B to be 2D_type(1), as these lead
323 * to opposite requirements for the orientation of the BLACS grid,
324 * and as noted before, the *same* BLACS context must be used in
325 * all descriptors in a single ScaLAPACK subroutine call.
326 *
327 * Let A be a generic term for any 1D block cyclicly distributed array.
328 * Such a global array has an associated description vector DESCA.
329 * In the following comments, the character _ should be read as
330 * "of the global array".
331 *
332 * NOTATION STORED IN EXPLANATION
333 * --------------- ---------- ------------------------------------------
334 * DTYPE_A(global) DESCA( 1 ) The descriptor type. For 1D grids,
335 * TYPE_A = 501: 1-by-P grid.
336 * TYPE_A = 502: P-by-1 grid.
337 * CTXT_A (global) DESCA( 2 ) The BLACS context handle, indicating
338 * the BLACS process grid A is distribu-
339 * ted over. The context itself is glo-
340 * bal, but the handle (the integer
341 * value) may vary.
342 * N_A (global) DESCA( 3 ) The size of the array dimension being
343 * distributed.
344 * NB_A (global) DESCA( 4 ) The blocking factor used to distribute
345 * the distributed dimension of the array.
346 * SRC_A (global) DESCA( 5 ) The process row or column over which the
347 * first row or column of the array
348 * is distributed.
349 * LLD_A (local) DESCA( 6 ) The leading dimension of the local array
350 * storing the local blocks of the distri-
351 * buted array A. Minimum value of LLD_A
352 * depends on TYPE_A.
353 * TYPE_A = 501: LLD_A >=
354 * size of undistributed dimension, 1.
355 * TYPE_A = 502: LLD_A >=NB_A, 1.
356 * Reserved DESCA( 7 ) Reserved for future use.
357 *
358 *
359 *
360 * =====================================================================
361 *
362 * Code Developer: Andrew J. Cleary, University of Tennessee.
363 * Current address: Lawrence Livermore National Labs.
364 * This version released: August, 2001.
365 *
366 * =====================================================================
367 *
368 * ..
369 * .. Parameters ..
370  REAL ONE, ZERO
371  parameter( one = 1.0e+0 )
372  parameter( zero = 0.0e+0 )
373  INTEGER INT_ONE
374  parameter( int_one = 1 )
375  INTEGER DESCMULT, BIGNUM
376  parameter(descmult = 100, bignum = descmult * descmult)
377  INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
378  $ lld_, mb_, m_, nb_, n_, rsrc_
379  parameter( block_cyclic_2d = 1, dlen_ = 9, dtype_ = 1,
380  $ ctxt_ = 2, m_ = 3, n_ = 4, mb_ = 5, nb_ = 6,
381  $ rsrc_ = 7, csrc_ = 8, lld_ = 9 )
382 * ..
383 * .. Local Scalars ..
384  INTEGER ICTXT, MYCOL, MYROW, NB, NPCOL, NPROW,
385  $ ws_factor
386 * ..
387 * .. External Subroutines ..
388  EXTERNAL psgbtrf, psgbtrs, pxerbla
389 * ..
390 * .. Executable Statements ..
391 *
392 * Note: to avoid duplication, most error checking is not performed
393 * in this routine and is left to routines
394 * PSGBTRF and PSGBTRS.
395 *
396 * Begin main code
397 *
398  info = 0
399 *
400 * Get block size to calculate workspace requirements
401 *
402  IF( desca( dtype_ ) .EQ. block_cyclic_2d ) THEN
403  nb = desca( nb_ )
404  ictxt = desca( ctxt_ )
405  ELSEIF( desca( dtype_ ) .EQ. 501 ) THEN
406  nb = desca( 4 )
407  ictxt = desca( 2 )
408  ELSE
409  info = -( 6*100 + dtype_ )
410  CALL pxerbla( ictxt,
411  $ 'PSGBSV',
412  $ -info )
413  RETURN
414  ENDIF
415 *
416  CALL blacs_gridinfo( ictxt, nprow, npcol, myrow, mycol )
417 *
418 *
419 * Size needed for AF in factorization
420 *
421  ws_factor = (nb+bwu)*(bwl+bwu)+6*(bwl+bwu)*(bwl+2*bwu)
422 *
423 * Factor the matrix
424 *
425  CALL psgbtrf( n, bwl, bwu, a, ja, desca, ipiv, work,
426  $ min( lwork, ws_factor ), work( 1+ws_factor ),
427  $ lwork-ws_factor, info )
428 *
429 * Check info for error conditions
430 *
431  IF( info.NE.0 ) THEN
432  IF( info .LT. 0 ) THEN
433  CALL pxerbla( ictxt, 'PSGBSV', -info )
434  ENDIF
435  RETURN
436  END IF
437 *
438 * Solve the system using the factorization
439 *
440  CALL psgbtrs( 'N', n, bwl, bwu, nrhs, a, ja, desca, ipiv, b, ib,
441  $ descb, work, min( lwork, ws_factor ),
442  $ work( 1+ws_factor), lwork-ws_factor, info )
443 *
444 * Check info for error conditions
445 *
446  IF( info.NE.0 ) THEN
447  CALL pxerbla( ictxt, 'PSGBSV', -info )
448  RETURN
449  END IF
450 *
451  RETURN
452 *
453 * End of PSGBSV
454 *
455  END
psgbtrf
subroutine psgbtrf(N, BWL, BWU, A, JA, DESCA, IPIV, AF, LAF, WORK, LWORK, INFO)
Definition: psgbtrf.f:3
psgbsv
subroutine psgbsv(N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, WORK, LWORK, INFO)
Definition: psgbsv.f:3
pxerbla
subroutine pxerbla(ICTXT, SRNAME, INFO)
Definition: pxerbla.f:2
psgbtrs
subroutine psgbtrs(TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO)
Definition: psgbtrs.f:3
min
#define min(A, B)
Definition: pcgemr.c:181