ScaLAPACK 2.1  2.1 ScaLAPACK: Scalable Linear Algebra PACKage
psdtsv.f
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1  SUBROUTINE psdtsv( N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB,
2  \$ 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 IB, INFO, JA, LWORK, N, NRHS
13 * ..
14 * .. Array Arguments ..
15  INTEGER DESCA( * ), DESCB( * )
16  REAL B( * ), D( * ), DL( * ), DU( * ), WORK( * )
17 * ..
18 *
19 *
20 * Purpose
21 * =======
22 *
23 * PSDTSV 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 * tridiagonal diagonally dominant-like distributed
29 * matrix.
30 *
31 * Gaussian elimination without pivoting
32 * is used to factor a reordering
33 * of the matrix into L U.
34 *
35 * See PSDTTRF and PSDTTRS 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 * NRHS (global input) INTEGER
48 * The number of right hand sides, i.e., the number of columns
49 * of the distributed submatrix B(IB:IB+N-1, 1:NRHS).
50 * NRHS >= 0.
51 *
52 * DL (local input/local output) REAL pointer to local
53 * part of global vector storing the lower diagonal of the
54 * matrix. Globally, DL(1) is not referenced, and DL must be
55 * aligned with D.
56 * Must be of size >= DESCA( NB_ ).
57 * On exit, this array contains information containing the
58 * factors of the matrix.
59 *
60 * D (local input/local output) REAL pointer to local
61 * part of global vector storing the main diagonal of the
62 * matrix.
63 * On exit, this array contains information containing the
64 * factors of the matrix.
65 * Must be of size >= DESCA( NB_ ).
66 *
67 * DU (local input/local output) REAL pointer to local
68 * part of global vector storing the upper diagonal of the
69 * matrix. Globally, DU(n) is not referenced, and DU must be
70 * aligned with D.
71 * On exit, this array contains information containing the
72 * factors of the matrix.
73 * Must be of size >= DESCA( NB_ ).
74 *
75 * JA (global input) INTEGER
76 * The index in the global array A that points to the start of
77 * the matrix to be operated on (which may be either all of A
78 * or a submatrix of A).
79 *
80 * DESCA (global and local input) INTEGER array of dimension DLEN.
81 * if 1D type (DTYPE_A=501 or 502), DLEN >= 7;
82 * if 2D type (DTYPE_A=1), DLEN >= 9.
83 * The array descriptor for the distributed matrix A.
84 * Contains information of mapping of A to memory. Please
85 * see NOTES below for full description and options.
86 *
87 * B (local input/local output) REAL pointer into
88 * local memory to an array of local lead dimension lld_b>=NB.
89 * On entry, this array contains the
90 * the local pieces of the right hand sides
91 * B(IB:IB+N-1, 1:NRHS).
92 * On exit, this contains the local piece of the solutions
93 * distributed matrix X.
94 *
95 * IB (global input) INTEGER
96 * The row index in the global array B that points to the first
97 * row of the matrix to be operated on (which may be either
98 * all of B or a submatrix of B).
99 *
100 * DESCB (global and local input) INTEGER array of dimension DLEN.
101 * if 1D type (DTYPE_B=502), DLEN >=7;
102 * if 2D type (DTYPE_B=1), DLEN >= 9.
103 * The array descriptor for the distributed matrix B.
104 * Contains information of mapping of B to memory. Please
105 * see NOTES below for full description and options.
106 *
107 * WORK (local workspace/local output)
108 * REAL temporary workspace. This space may
109 * be overwritten in between calls to routines. WORK must be
110 * the size given in LWORK.
111 * On exit, WORK( 1 ) contains the minimal LWORK.
112 *
113 * LWORK (local input or global input) INTEGER
114 * Size of user-input workspace WORK.
115 * If LWORK is too small, the minimal acceptable size will be
116 * returned in WORK(1) and an error code is returned. LWORK>=
117 * (12*NPCOL+3*NB)
118 * +max(10*NPCOL+4*NRHS, 8*NPCOL)
119 *
120 * INFO (local output) INTEGER
121 * = 0: successful exit
122 * < 0: If the i-th argument is an array and the j-entry had
123 * an illegal value, then INFO = -(i*100+j), if the i-th
124 * argument is a scalar and had an illegal value, then
125 * INFO = -i.
126 * > 0: If INFO = K<=NPROCS, the submatrix stored on processor
127 * INFO and factored locally was not
128 * diagonally dominant-like, and
129 * the factorization was not completed.
130 * If INFO = K>NPROCS, the submatrix stored on processor
131 * INFO-NPROCS representing interactions with other
132 * processors was not
133 * stably factorable wo/interchanges,
134 * and the factorization was not completed.
135 *
136 * =====================================================================
137 *
138 *
139 * Restrictions
140 * ============
141 *
142 * The following are restrictions on the input parameters. Some of these
143 * are temporary and will be removed in future releases, while others
144 * may reflect fundamental technical limitations.
145 *
146 * Non-cyclic restriction: VERY IMPORTANT!
147 * P*NB>= mod(JA-1,NB)+N.
148 * The mapping for matrices must be blocked, reflecting the nature
149 * of the divide and conquer algorithm as a task-parallel algorithm.
150 * This formula in words is: no processor may have more than one
151 * chunk of the matrix.
152 *
153 * Blocksize cannot be too small:
154 * If the matrix spans more than one processor, the following
155 * restriction on NB, the size of each block on each processor,
156 * must hold:
157 * NB >= 2
158 * The bulk of parallel computation is done on the matrix of size
159 * O(NB) on each processor. If this is too small, divide and conquer
160 * is a poor choice of algorithm.
161 *
162 * Submatrix reference:
163 * JA = IB
164 * Alignment restriction that prevents unnecessary communication.
165 *
166 *
167 * =====================================================================
168 *
169 *
170 * Notes
171 * =====
172 *
173 * If the factorization routine and the solve routine are to be called
174 * separately (to solve various sets of righthand sides using the same
175 * coefficient matrix), the auxiliary space AF *must not be altered*
176 * between calls to the factorization routine and the solve routine.
177 *
178 * The best algorithm for solving banded and tridiagonal linear systems
179 * depends on a variety of parameters, especially the bandwidth.
180 * Currently, only algorithms designed for the case N/P >> bw are
181 * implemented. These go by many names, including Divide and Conquer,
182 * Partitioning, domain decomposition-type, etc.
183 * For tridiagonal matrices, it is obvious: N/P >> bw(=1), and so D&C
184 * algorithms are the appropriate choice.
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 tridiagonal 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 ((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: tridiagonal 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 * However, for tridiagonal matrices, since the objects being
292 * distributed are the individual vectors storing the diagonals, we
293 * have adopted the convention that both the P-by-1 descriptor and
294 * the 1-by-P descriptor are allowed and are equivalent for
295 * tridiagonal matrices. Thus, for tridiagonal matrices,
296 * DTYPE_A = 501 or 502 can be used interchangeably
297 * without any other change.
298 * We require that the distributed vectors storing the diagonals of a
299 * tridiagonal matrix be aligned with each other. Because of this, a
300 * single descriptor, DESCA, serves to describe the distribution of
301 * of all diagonals simultaneously.
302 *
303 * IMPORTANT NOTE: the actual BLACS grid represented by the
304 * CTXT entry in the descriptor may be *either* P-by-1 or 1-by-P
305 * irrespective of which one-dimensional descriptor type
306 * (501 or 502) is input.
307 * This routine will interpret the grid properly either way.
308 * ScaLAPACK routines *do not support intercontext operations* so that
309 * the grid passed to a single ScaLAPACK routine *must be the same*
310 * for all array descriptors passed to that routine.
311 *
312 * NOTE: In all cases where 1D descriptors are used, 2D descriptors
313 * may also be used, since a one-dimensional array is a special case
314 * of a two-dimensional array with one dimension of size unity.
315 * The two-dimensional array used in this case *must* be of the
316 * proper orientation:
317 * If the appropriate one-dimensional descriptor is DTYPEA=501
318 * (1 by P type), then the two dimensional descriptor must
319 * have a CTXT value that refers to a 1 by P BLACS grid;
320 * If the appropriate one-dimensional descriptor is DTYPEA=502
321 * (P by 1 type), then the two dimensional descriptor must
322 * have a CTXT value that refers to a P by 1 BLACS grid.
323 *
324 *
325 * Summary of allowed descriptors, types, and BLACS grids:
326 * DTYPE 501 502 1 1
327 * BLACS grid 1xP or Px1 1xP or Px1 1xP Px1
328 * -----------------------------------------------------
329 * A OK OK OK NO
330 * B NO OK NO OK
331 *
332 * Note that a consequence of this chart is that it is not possible
333 * for *both* DTYPE_A and DTYPE_B to be 2D_type(1), as these lead
334 * to opposite requirements for the orientation of the BLACS grid,
335 * and as noted before, the *same* BLACS context must be used in
336 * all descriptors in a single ScaLAPACK subroutine call.
337 *
338 * Let A be a generic term for any 1D block cyclicly distributed array.
339 * Such a global array has an associated description vector DESCA.
340 * In the following comments, the character _ should be read as
341 * "of the global array".
342 *
343 * NOTATION STORED IN EXPLANATION
344 * --------------- ---------- ------------------------------------------
345 * DTYPE_A(global) DESCA( 1 ) The descriptor type. For 1D grids,
346 * TYPE_A = 501: 1-by-P grid.
347 * TYPE_A = 502: P-by-1 grid.
348 * CTXT_A (global) DESCA( 2 ) The BLACS context handle, indicating
349 * the BLACS process grid A is distribu-
350 * ted over. The context itself is glo-
351 * bal, but the handle (the integer
352 * value) may vary.
353 * N_A (global) DESCA( 3 ) The size of the array dimension being
354 * distributed.
355 * NB_A (global) DESCA( 4 ) The blocking factor used to distribute
356 * the distributed dimension of the array.
357 * SRC_A (global) DESCA( 5 ) The process row or column over which the
358 * first row or column of the array
359 * is distributed.
360 * Ignored DESCA( 6 ) Ignored for tridiagonal matrices.
361 * Reserved DESCA( 7 ) Reserved for future use.
362 *
363 *
364 *
365 * =====================================================================
366 *
367 * Code Developer: Andrew J. Cleary, University of Tennessee.
368 * Current address: Lawrence Livermore National Labs.
369 * This version released: August, 2001.
370 *
371 * =====================================================================
372 *
373 * ..
374 * .. Parameters ..
375  REAL ONE, ZERO
376  parameter( one = 1.0e+0 )
377  parameter( zero = 0.0e+0 )
378  INTEGER INT_ONE
379  parameter( int_one = 1 )
380  INTEGER DESCMULT, BIGNUM
381  parameter(descmult = 100, bignum = descmult * descmult)
382  INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
383  \$ lld_, mb_, m_, nb_, n_, rsrc_
384  parameter( block_cyclic_2d = 1, dlen_ = 9, dtype_ = 1,
385  \$ ctxt_ = 2, m_ = 3, n_ = 4, mb_ = 5, nb_ = 6,
386  \$ rsrc_ = 7, csrc_ = 8, lld_ = 9 )
387 * ..
388 * .. Local Scalars ..
389  INTEGER ICTXT, MYCOL, MYROW, NB, NPCOL, NPROW,
390  \$ ws_factor
391 * ..
392 * .. External Subroutines ..
393  EXTERNAL psdttrf, psdttrs, pxerbla
394 * ..
395 * .. Executable Statements ..
396 *
397 * Note: to avoid duplication, most error checking is not performed
398 * in this routine and is left to routines
399 * PSDTTRF and PSDTTRS.
400 *
401 * Begin main code
402 *
403  info = 0
404 *
405 * Get block size to calculate workspace requirements
406 *
407  IF( desca( dtype_ ) .EQ. block_cyclic_2d ) THEN
408  nb = desca( nb_ )
409  ictxt = desca( ctxt_ )
410  ELSEIF( desca( dtype_ ) .EQ. 501 ) THEN
411  nb = desca( 4 )
412  ictxt = desca( 2 )
413  ELSEIF( desca( dtype_ ) .EQ. 502 ) THEN
414  nb = desca( 4 )
415  ictxt = desca( 2 )
416  ELSE
417  info = -( 6*100 + dtype_ )
418  CALL pxerbla( ictxt,
419  \$ 'PSDTSV',
420  \$ -info )
421  RETURN
422  ENDIF
423 *
424  CALL blacs_gridinfo( ictxt, nprow, npcol, myrow, mycol )
425 *
426 *
427 * Size needed for AF in factorization
428 *
429  ws_factor = (12*npcol+3*nb)
430 *
431 * Factor the matrix
432 *
433  CALL psdttrf( n, dl, d, du, ja, desca, work,
434  \$ min( lwork, ws_factor ), work( 1+ws_factor ),
435  \$ lwork-ws_factor, info )
436 *
437 * Check info for error conditions
438 *
439  IF( info.NE.0 ) THEN
440  IF( info .LT. 0 ) THEN
441  CALL pxerbla( ictxt, 'PSDTSV', -info )
442  ENDIF
443  RETURN
444  END IF
445 *
446 * Solve the system using the factorization
447 *
448  CALL psdttrs( 'N', n, nrhs, dl, d, du, ja, desca, b, ib, descb,
449  \$ work, min( lwork, ws_factor ), work( 1+ws_factor),
450  \$ lwork-ws_factor, info )
451 *
452 * Check info for error conditions
453 *
454  IF( info.NE.0 ) THEN
455  CALL pxerbla( ictxt, 'PSDTSV', -info )
456  RETURN
457  END IF
458 *
459  RETURN
460 *
461 * End of PSDTSV
462 *
463  END
psdtsv
subroutine psdtsv(N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO)
Definition: psdtsv.f:3
pxerbla
subroutine pxerbla(ICTXT, SRNAME, INFO)
Definition: pxerbla.f:2
psdttrf
subroutine psdttrf(N, DL, D, DU, JA, DESCA, AF, LAF, WORK, LWORK, INFO)
Definition: psdttrf.f:3
min
#define min(A, B)
Definition: pcgemr.c:181
psdttrs
subroutine psdttrs(TRANS, N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO)
Definition: psdttrs.f:3