ScaLAPACK 2.1  2.1
ScaLAPACK: Scalable Linear Algebra PACKage
pcptsv.f
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1  SUBROUTINE pcptsv( UPLO, N, NRHS, D, E, 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  CHARACTER UPLO
13  INTEGER IB, INFO, JA, LWORK, N, NRHS
14 * ..
15 * .. Array Arguments ..
16  INTEGER DESCA( * ), DESCB( * )
17  COMPLEX B( * ), E( * ), WORK( * )
18  REAL D( * )
19 * ..
20 *
21 *
22 * Purpose
23 * =======
24 *
25 * PCPTSV solves a system of linear equations
26 *
27 * A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS)
28 *
29 * where A(1:N, JA:JA+N-1) is an N-by-N complex
30 * tridiagonal symmetric positive definite distributed
31 * matrix.
32 *
33 * Cholesky factorization is used to factor a reordering of
34 * the matrix into L L'.
35 *
36 * See PCPTTRF and PCPTTRS for details.
37 *
38 * =====================================================================
39 *
40 * Arguments
41 * =========
42 *
43 * UPLO (global input) CHARACTER
44 * = 'U': Upper triangle of A(1:N, JA:JA+N-1) is stored;
45 * = 'L': Lower triangle of A(1:N, JA:JA+N-1) is stored.
46 *
47 * N (global input) INTEGER
48 * The number of rows and columns to be operated on, i.e. the
49 * order of the distributed submatrix A(1:N, JA:JA+N-1). N >= 0.
50 *
51 * NRHS (global input) INTEGER
52 * The number of right hand sides, i.e., the number of columns
53 * of the distributed submatrix B(IB:IB+N-1, 1:NRHS).
54 * NRHS >= 0.
55 *
56 * D (local input/local output) COMPLEX pointer to local
57 * part of global vector storing the main diagonal of the
58 * matrix.
59 * On exit, this array contains information containing the
60 * factors of the matrix.
61 * Must be of size >= DESCA( NB_ ).
62 *
63 * E (local input/local output) COMPLEX pointer to local
64 * part of global vector storing the upper diagonal of the
65 * matrix. Globally, DU(n) is not referenced, and DU must be
66 * aligned with D.
67 * On exit, this array contains information containing the
68 * factors of the matrix.
69 * Must be of size >= DESCA( NB_ ).
70 *
71 * JA (global input) INTEGER
72 * The index in the global array A that points to the start of
73 * the matrix to be operated on (which may be either all of A
74 * or a submatrix of A).
75 *
76 * DESCA (global and local input) INTEGER array of dimension DLEN.
77 * if 1D type (DTYPE_A=501 or 502), DLEN >= 7;
78 * if 2D type (DTYPE_A=1), DLEN >= 9.
79 * The array descriptor for the distributed matrix A.
80 * Contains information of mapping of A to memory. Please
81 * see NOTES below for full description and options.
82 *
83 * B (local input/local output) COMPLEX pointer into
84 * local memory to an array of local lead dimension lld_b>=NB.
85 * On entry, this array contains the
86 * the local pieces of the right hand sides
87 * B(IB:IB+N-1, 1:NRHS).
88 * On exit, this contains the local piece of the solutions
89 * distributed matrix X.
90 *
91 * IB (global input) INTEGER
92 * The row index in the global array B that points to the first
93 * row of the matrix to be operated on (which may be either
94 * all of B or a submatrix of B).
95 *
96 * DESCB (global and local input) INTEGER array of dimension DLEN.
97 * if 1D type (DTYPE_B=502), DLEN >=7;
98 * if 2D type (DTYPE_B=1), DLEN >= 9.
99 * The array descriptor for the distributed matrix B.
100 * Contains information of mapping of B to memory. Please
101 * see NOTES below for full description and options.
102 *
103 * WORK (local workspace/local output)
104 * COMPLEX temporary workspace. This space may
105 * be overwritten in between calls to routines. WORK must be
106 * the size given in LWORK.
107 * On exit, WORK( 1 ) contains the minimal LWORK.
108 *
109 * LWORK (local input or global input) INTEGER
110 * Size of user-input workspace WORK.
111 * If LWORK is too small, the minimal acceptable size will be
112 * returned in WORK(1) and an error code is returned. LWORK>=
113 * (12*NPCOL + 3*NB)
114 * +max((10+2*min(100,NRHS))*NPCOL+4*NRHS, 8*NPCOL)
115 *
116 * INFO (local output) INTEGER
117 * = 0: successful exit
118 * < 0: If the i-th argument is an array and the j-entry had
119 * an illegal value, then INFO = -(i*100+j), if the i-th
120 * argument is a scalar and had an illegal value, then
121 * INFO = -i.
122 * > 0: If INFO = K<=NPROCS, the submatrix stored on processor
123 * INFO and factored locally was not
124 * positive definite, and
125 * the factorization was not completed.
126 * If INFO = K>NPROCS, the submatrix stored on processor
127 * INFO-NPROCS representing interactions with other
128 * processors was not
129 * positive definite,
130 * and the factorization was not completed.
131 *
132 * =====================================================================
133 *
134 *
135 * Restrictions
136 * ============
137 *
138 * The following are restrictions on the input parameters. Some of these
139 * are temporary and will be removed in future releases, while others
140 * may reflect fundamental technical limitations.
141 *
142 * Non-cyclic restriction: VERY IMPORTANT!
143 * P*NB>= mod(JA-1,NB)+N.
144 * The mapping for matrices must be blocked, reflecting the nature
145 * of the divide and conquer algorithm as a task-parallel algorithm.
146 * This formula in words is: no processor may have more than one
147 * chunk of the matrix.
148 *
149 * Blocksize cannot be too small:
150 * If the matrix spans more than one processor, the following
151 * restriction on NB, the size of each block on each processor,
152 * must hold:
153 * NB >= 2
154 * The bulk of parallel computation is done on the matrix of size
155 * O(NB) on each processor. If this is too small, divide and conquer
156 * is a poor choice of algorithm.
157 *
158 * Submatrix reference:
159 * JA = IB
160 * Alignment restriction that prevents unnecessary communication.
161 *
162 *
163 * =====================================================================
164 *
165 *
166 * Notes
167 * =====
168 *
169 * If the factorization routine and the solve routine are to be called
170 * separately (to solve various sets of righthand sides using the same
171 * coefficient matrix), the auxiliary space AF *must not be altered*
172 * between calls to the factorization routine and the solve routine.
173 *
174 * The best algorithm for solving banded and tridiagonal linear systems
175 * depends on a variety of parameters, especially the bandwidth.
176 * Currently, only algorithms designed for the case N/P >> bw are
177 * implemented. These go by many names, including Divide and Conquer,
178 * Partitioning, domain decomposition-type, etc.
179 * For tridiagonal matrices, it is obvious: N/P >> bw(=1), and so D&C
180 * algorithms are the appropriate choice.
181 *
182 * Algorithm description: Divide and Conquer
183 *
184 * The Divide and Conqer algorithm assumes the matrix is narrowly
185 * banded compared with the number of equations. In this situation,
186 * it is best to distribute the input matrix A one-dimensionally,
187 * with columns atomic and rows divided amongst the processes.
188 * The basic algorithm divides the tridiagonal matrix up into
189 * P pieces with one stored on each processor,
190 * and then proceeds in 2 phases for the factorization or 3 for the
191 * solution of a linear system.
192 * 1) Local Phase:
193 * The individual pieces are factored independently and in
194 * parallel. These factors are applied to the matrix creating
195 * fillin, which is stored in a non-inspectable way in auxiliary
196 * space AF. Mathematically, this is equivalent to reordering
197 * the matrix A as P A P^T and then factoring the principal
198 * leading submatrix of size equal to the sum of the sizes of
199 * the matrices factored on each processor. The factors of
200 * these submatrices overwrite the corresponding parts of A
201 * in memory.
202 * 2) Reduced System Phase:
203 * A small ((P-1)) system is formed representing
204 * interaction of the larger blocks, and is stored (as are its
205 * factors) in the space AF. A parallel Block Cyclic Reduction
206 * algorithm is used. For a linear system, a parallel front solve
207 * followed by an analagous backsolve, both using the structure
208 * of the factored matrix, are performed.
209 * 3) Backsubsitution Phase:
210 * For a linear system, a local backsubstitution is performed on
211 * each processor in parallel.
212 *
213 *
214 * Descriptors
215 * ===========
216 *
217 * Descriptors now have *types* and differ from ScaLAPACK 1.0.
218 *
219 * Note: tridiagonal codes can use either the old two dimensional
220 * or new one-dimensional descriptors, though the processor grid in
221 * both cases *must be one-dimensional*. We describe both types below.
222 *
223 * Each global data object is described by an associated description
224 * vector. This vector stores the information required to establish
225 * the mapping between an object element and its corresponding process
226 * and memory location.
227 *
228 * Let A be a generic term for any 2D block cyclicly distributed array.
229 * Such a global array has an associated description vector DESCA.
230 * In the following comments, the character _ should be read as
231 * "of the global array".
232 *
233 * NOTATION STORED IN EXPLANATION
234 * --------------- -------------- --------------------------------------
235 * DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
236 * DTYPE_A = 1.
237 * CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
238 * the BLACS process grid A is distribu-
239 * ted over. The context itself is glo-
240 * bal, but the handle (the integer
241 * value) may vary.
242 * M_A (global) DESCA( M_ ) The number of rows in the global
243 * array A.
244 * N_A (global) DESCA( N_ ) The number of columns in the global
245 * array A.
246 * MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
247 * the rows of the array.
248 * NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
249 * the columns of the array.
250 * RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
251 * row of the array A is distributed.
252 * CSRC_A (global) DESCA( CSRC_ ) The process column over which the
253 * first column of the array A is
254 * distributed.
255 * LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
256 * array. LLD_A >= MAX(1,LOCr(M_A)).
257 *
258 * Let K be the number of rows or columns of a distributed matrix,
259 * and assume that its process grid has dimension p x q.
260 * LOCr( K ) denotes the number of elements of K that a process
261 * would receive if K were distributed over the p processes of its
262 * process column.
263 * Similarly, LOCc( K ) denotes the number of elements of K that a
264 * process would receive if K were distributed over the q processes of
265 * its process row.
266 * The values of LOCr() and LOCc() may be determined via a call to the
267 * ScaLAPACK tool function, NUMROC:
268 * LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
269 * LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
270 * An upper bound for these quantities may be computed by:
271 * LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
272 * LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
273 *
274 *
275 * One-dimensional descriptors:
276 *
277 * One-dimensional descriptors are a new addition to ScaLAPACK since
278 * version 1.0. They simplify and shorten the descriptor for 1D
279 * arrays.
280 *
281 * Since ScaLAPACK supports two-dimensional arrays as the fundamental
282 * object, we allow 1D arrays to be distributed either over the
283 * first dimension of the array (as if the grid were P-by-1) or the
284 * 2nd dimension (as if the grid were 1-by-P). This choice is
285 * indicated by the descriptor type (501 or 502)
286 * as described below.
287 * However, for tridiagonal matrices, since the objects being
288 * distributed are the individual vectors storing the diagonals, we
289 * have adopted the convention that both the P-by-1 descriptor and
290 * the 1-by-P descriptor are allowed and are equivalent for
291 * tridiagonal matrices. Thus, for tridiagonal matrices,
292 * DTYPE_A = 501 or 502 can be used interchangeably
293 * without any other change.
294 * We require that the distributed vectors storing the diagonals of a
295 * tridiagonal matrix be aligned with each other. Because of this, a
296 * single descriptor, DESCA, serves to describe the distribution of
297 * of all diagonals simultaneously.
298 *
299 * IMPORTANT NOTE: the actual BLACS grid represented by the
300 * CTXT entry in the descriptor may be *either* P-by-1 or 1-by-P
301 * irrespective of which one-dimensional descriptor type
302 * (501 or 502) is input.
303 * This routine will interpret the grid properly either way.
304 * ScaLAPACK routines *do not support intercontext operations* so that
305 * the grid passed to a single ScaLAPACK routine *must be the same*
306 * for all array descriptors passed to that routine.
307 *
308 * NOTE: In all cases where 1D descriptors are used, 2D descriptors
309 * may also be used, since a one-dimensional array is a special case
310 * of a two-dimensional array with one dimension of size unity.
311 * The two-dimensional array used in this case *must* be of the
312 * proper orientation:
313 * If the appropriate one-dimensional descriptor is DTYPEA=501
314 * (1 by P type), then the two dimensional descriptor must
315 * have a CTXT value that refers to a 1 by P BLACS grid;
316 * If the appropriate one-dimensional descriptor is DTYPEA=502
317 * (P by 1 type), then the two dimensional descriptor must
318 * have a CTXT value that refers to a P by 1 BLACS grid.
319 *
320 *
321 * Summary of allowed descriptors, types, and BLACS grids:
322 * DTYPE 501 502 1 1
323 * BLACS grid 1xP or Px1 1xP or Px1 1xP Px1
324 * -----------------------------------------------------
325 * A OK OK OK NO
326 * B NO OK NO OK
327 *
328 * Note that a consequence of this chart is that it is not possible
329 * for *both* DTYPE_A and DTYPE_B to be 2D_type(1), as these lead
330 * to opposite requirements for the orientation of the BLACS grid,
331 * and as noted before, the *same* BLACS context must be used in
332 * all descriptors in a single ScaLAPACK subroutine call.
333 *
334 * Let A be a generic term for any 1D block cyclicly distributed array.
335 * Such a global array has an associated description vector DESCA.
336 * In the following comments, the character _ should be read as
337 * "of the global array".
338 *
339 * NOTATION STORED IN EXPLANATION
340 * --------------- ---------- ------------------------------------------
341 * DTYPE_A(global) DESCA( 1 ) The descriptor type. For 1D grids,
342 * TYPE_A = 501: 1-by-P grid.
343 * TYPE_A = 502: P-by-1 grid.
344 * CTXT_A (global) DESCA( 2 ) The BLACS context handle, indicating
345 * the BLACS process grid A is distribu-
346 * ted over. The context itself is glo-
347 * bal, but the handle (the integer
348 * value) may vary.
349 * N_A (global) DESCA( 3 ) The size of the array dimension being
350 * distributed.
351 * NB_A (global) DESCA( 4 ) The blocking factor used to distribute
352 * the distributed dimension of the array.
353 * SRC_A (global) DESCA( 5 ) The process row or column over which the
354 * first row or column of the array
355 * is distributed.
356 * Ignored DESCA( 6 ) Ignored for tridiagonal matrices.
357 * Reserved DESCA( 7 ) Reserved for future use.
358 *
359 *
360 *
361 * =====================================================================
362 *
363 * Code Developer: Andrew J. Cleary, University of Tennessee.
364 * Current address: Lawrence Livermore National Labs.
365 * This version released: August, 2001.
366 *
367 * =====================================================================
368 *
369 * ..
370 * .. Parameters ..
371  REAL ONE, ZERO
372  parameter( one = 1.0e+0 )
373  parameter( zero = 0.0e+0 )
374  COMPLEX CONE, CZERO
375  parameter( cone = ( 1.0e+0, 0.0e+0 ) )
376  parameter( czero = ( 0.0e+0, 0.0e+0 ) )
377  INTEGER INT_ONE
378  parameter( int_one = 1 )
379  INTEGER DESCMULT, BIGNUM
380  parameter(descmult = 100, bignum = descmult * descmult)
381  INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
382  $ lld_, mb_, m_, nb_, n_, rsrc_
383  parameter( block_cyclic_2d = 1, dlen_ = 9, dtype_ = 1,
384  $ ctxt_ = 2, m_ = 3, n_ = 4, mb_ = 5, nb_ = 6,
385  $ rsrc_ = 7, csrc_ = 8, lld_ = 9 )
386 * ..
387 * .. Local Scalars ..
388  INTEGER ICTXT, MYCOL, MYROW, NB, NPCOL, NPROW,
389  $ ws_factor
390 * ..
391 * .. External Subroutines ..
392  EXTERNAL pcpttrf, pcpttrs, pxerbla
393 * ..
394 * .. Executable Statements ..
395 *
396 * Note: to avoid duplication, most error checking is not performed
397 * in this routine and is left to routines
398 * PCPTTRF and PCPTTRS.
399 *
400 * Begin main code
401 *
402  info = 0
403 *
404 * Get block size to calculate workspace requirements
405 *
406  IF( desca( dtype_ ) .EQ. block_cyclic_2d ) THEN
407  nb = desca( nb_ )
408  ictxt = desca( ctxt_ )
409  ELSEIF( desca( dtype_ ) .EQ. 501 ) THEN
410  nb = desca( 4 )
411  ictxt = desca( 2 )
412  ELSEIF( desca( dtype_ ) .EQ. 502 ) THEN
413  nb = desca( 4 )
414  ictxt = desca( 2 )
415  ELSE
416  info = -( 5*100 + dtype_ )
417  CALL pxerbla( ictxt,
418  $ 'PCPTSV',
419  $ -info )
420  RETURN
421  ENDIF
422 *
423  CALL blacs_gridinfo( ictxt, nprow, npcol, myrow, mycol )
424 *
425 *
426 * Size needed for AF in factorization
427 *
428  ws_factor = (12*npcol + 3*nb)
429 *
430 * Factor the matrix
431 *
432  CALL pcpttrf( n, d, e, ja, desca, work, min( lwork, ws_factor ),
433  $ work( 1+ws_factor ), lwork-ws_factor, info )
434 *
435 * Check info for error conditions
436 *
437  IF( info.NE.0 ) THEN
438  IF( info .LT. 0 ) THEN
439  CALL pxerbla( ictxt, 'PCPTSV', -info )
440  ENDIF
441  RETURN
442  END IF
443 *
444 * Solve the system using the factorization
445 *
446  CALL pcpttrs( uplo, n, nrhs, d, e, ja, desca, b, ib, descb, work,
447  $ min( lwork, ws_factor ), work( 1+ws_factor),
448  $ lwork-ws_factor, info )
449 *
450 * Check info for error conditions
451 *
452  IF( info.NE.0 ) THEN
453  CALL pxerbla( ictxt, 'PCPTSV', -info )
454  RETURN
455  END IF
456 *
457  RETURN
458 *
459 * End of PCPTSV
460 *
461  END
pcpttrs
subroutine pcpttrs(UPLO, N, NRHS, D, E, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO)
Definition: pcpttrs.f:3
pcptsv
subroutine pcptsv(UPLO, N, NRHS, D, E, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO)
Definition: pcptsv.f:3
pcpttrf
subroutine pcpttrf(N, D, E, JA, DESCA, AF, LAF, WORK, LWORK, INFO)
Definition: pcpttrf.f:3
pxerbla
subroutine pxerbla(ICTXT, SRNAME, INFO)
Definition: pxerbla.f:2
min
#define min(A, B)
Definition: pcgemr.c:181