1 SUBROUTINE PSPTTRF( N, D, E, JA, DESCA, AF, LAF, WORK, LWORK, 2 $ INFO ) 3* 4* -- ScaLAPACK routine (version 1.7) -- 5* University of Tennessee, Knoxville, Oak Ridge National Laboratory, 6* and University of California, Berkeley. 7* April 3, 2000 8* 9* .. Scalar Arguments .. 10 INTEGER INFO, JA, LAF, LWORK, N 11* .. 12* .. Array Arguments .. 13 INTEGER DESCA( * ) 14 REAL AF( * ), D( * ), E( * ), WORK( * ) 15* .. 16* 17* 18* Purpose 19* ======= 20* 21* PSPTTRF computes a Cholesky factorization 22* of an N-by-N real tridiagonal 23* symmetric positive definite distributed matrix 24* A(1:N, JA:JA+N-1). 25* Reordering is used to increase parallelism in the factorization. 26* This reordering results in factors that are DIFFERENT from those 27* produced by equivalent sequential codes. These factors cannot 28* be used directly by users; however, they can be used in 29* subsequent calls to PSPTTRS to solve linear systems. 30* 31* The factorization has the form 32* 33* P A(1:N, JA:JA+N-1) P^T = U' D U or 34* 35* P A(1:N, JA:JA+N-1) P^T = L D L', 36* 37* where U is a tridiagonal upper triangular matrix and L is tridiagonal 38* lower triangular, and P is a permutation matrix. 39* 40* ===================================================================== 41* 42* Arguments 43* ========= 44* 45* 46* N (global input) INTEGER 47* The number of rows and columns to be operated on, i.e. the 48* order of the distributed submatrix A(1:N, JA:JA+N-1). N >= 0. 49* 50* D (local input/local output) REAL pointer to local 51* part of global vector storing the main diagonal of the 52* matrix. 53* On exit, this array contains information containing the 54* factors of the matrix. 55* Must be of size >= DESCA( NB_ ). 56* 57* E (local input/local output) REAL pointer to local 58* part of global vector storing the upper diagonal of the 59* matrix. Globally, DU(n) is not referenced, and DU must be 60* aligned with D. 61* On exit, this array contains information containing the 62* factors of the matrix. 63* Must be of size >= DESCA( NB_ ). 64* 65* JA (global input) INTEGER 66* The index in the global array A that points to the start of 67* the matrix to be operated on (which may be either all of A 68* or a submatrix of A). 69* 70* DESCA (global and local input) INTEGER array of dimension DLEN. 71* if 1D type (DTYPE_A=501 or 502), DLEN >= 7; 72* if 2D type (DTYPE_A=1), DLEN >= 9. 73* The array descriptor for the distributed matrix A. 74* Contains information of mapping of A to memory. Please 75* see NOTES below for full description and options. 76* 77* AF (local output) REAL array, dimension LAF. 78* Auxiliary Fillin Space. 79* Fillin is created during the factorization routine 80* PSPTTRF and this is stored in AF. If a linear system 81* is to be solved using PSPTTRS after the factorization 82* routine, AF *must not be altered* after the factorization. 83* 84* LAF (local input) INTEGER 85* Size of user-input Auxiliary Fillin space AF. Must be >= 86* (NB+2) 87* If LAF is not large enough, an error code will be returned 88* and the minimum acceptable size will be returned in AF( 1 ) 89* 90* WORK (local workspace/local output) 91* REAL temporary workspace. This space may 92* be overwritten in between calls to routines. WORK must be 93* the size given in LWORK. 94* On exit, WORK( 1 ) contains the minimal LWORK. 95* 96* LWORK (local input or global input) INTEGER 97* Size of user-input workspace WORK. 98* If LWORK is too small, the minimal acceptable size will be 99* returned in WORK(1) and an error code is returned. LWORK>= 100* 8*NPCOL 101* 102* INFO (local output) INTEGER 103* = 0: successful exit 104* < 0: If the i-th argument is an array and the j-entry had 105* an illegal value, then INFO = -(i*100+j), if the i-th 106* argument is a scalar and had an illegal value, then 107* INFO = -i. 108* > 0: If INFO = K<=NPROCS, the submatrix stored on processor 109* INFO and factored locally was not 110* positive definite, and 111* the factorization was not completed. 112* If INFO = K>NPROCS, the submatrix stored on processor 113* INFO-NPROCS representing interactions with other 114* processors was not 115* positive definite, 116* and the factorization was not completed. 117* 118* ===================================================================== 119* 120* 121* Restrictions 122* ============ 123* 124* The following are restrictions on the input parameters. Some of these 125* are temporary and will be removed in future releases, while others 126* may reflect fundamental technical limitations. 127* 128* Non-cyclic restriction: VERY IMPORTANT! 129* P*NB>= mod(JA-1,NB)+N. 130* The mapping for matrices must be blocked, reflecting the nature 131* of the divide and conquer algorithm as a task-parallel algorithm. 132* This formula in words is: no processor may have more than one 133* chunk of the matrix. 134* 135* Blocksize cannot be too small: 136* If the matrix spans more than one processor, the following 137* restriction on NB, the size of each block on each processor, 138* must hold: 139* NB >= 2 140* The bulk of parallel computation is done on the matrix of size 141* O(NB) on each processor. If this is too small, divide and conquer 142* is a poor choice of algorithm. 143* 144* Submatrix reference: 145* JA = IB 146* Alignment restriction that prevents unnecessary communication. 147* 148* 149* ===================================================================== 150* 151* 152* Notes 153* ===== 154* 155* If the factorization routine and the solve routine are to be called 156* separately (to solve various sets of righthand sides using the same 157* coefficient matrix), the auxiliary space AF *must not be altered* 158* between calls to the factorization routine and the solve routine. 159* 160* The best algorithm for solving banded and tridiagonal linear systems 161* depends on a variety of parameters, especially the bandwidth. 162* Currently, only algorithms designed for the case N/P >> bw are 163* implemented. These go by many names, including Divide and Conquer, 164* Partitioning, domain decomposition-type, etc. 165* For tridiagonal matrices, it is obvious: N/P >> bw(=1), and so D&C 166* algorithms are the appropriate choice. 167* 168* Algorithm description: Divide and Conquer 169* 170* The Divide and Conqer algorithm assumes the matrix is narrowly 171* banded compared with the number of equations. In this situation, 172* it is best to distribute the input matrix A one-dimensionally, 173* with columns atomic and rows divided amongst the processes. 174* The basic algorithm divides the tridiagonal matrix up into 175* P pieces with one stored on each processor, 176* and then proceeds in 2 phases for the factorization or 3 for the 177* solution of a linear system. 178* 1) Local Phase: 179* The individual pieces are factored independently and in 180* parallel. These factors are applied to the matrix creating 181* fillin, which is stored in a non-inspectable way in auxiliary 182* space AF. Mathematically, this is equivalent to reordering 183* the matrix A as P A P^T and then factoring the principal 184* leading submatrix of size equal to the sum of the sizes of 185* the matrices factored on each processor. The factors of 186* these submatrices overwrite the corresponding parts of A 187* in memory. 188* 2) Reduced System Phase: 189* A small ((P-1)) system is formed representing 190* interaction of the larger blocks, and is stored (as are its 191* factors) in the space AF. A parallel Block Cyclic Reduction 192* algorithm is used. For a linear system, a parallel front solve 193* followed by an analagous backsolve, both using the structure 194* of the factored matrix, are performed. 195* 3) Backsubsitution Phase: 196* For a linear system, a local backsubstitution is performed on 197* each processor in parallel. 198* 199* 200* Descriptors 201* =========== 202* 203* Descriptors now have *types* and differ from ScaLAPACK 1.0. 204* 205* Note: tridiagonal codes can use either the old two dimensional 206* or new one-dimensional descriptors, though the processor grid in 207* both cases *must be one-dimensional*. We describe both types below. 208* 209* Each global data object is described by an associated description 210* vector. This vector stores the information required to establish 211* the mapping between an object element and its corresponding process 212* and memory location. 213* 214* Let A be a generic term for any 2D block cyclicly distributed array. 215* Such a global array has an associated description vector DESCA. 216* In the following comments, the character _ should be read as 217* "of the global array". 218* 219* NOTATION STORED IN EXPLANATION 220* --------------- -------------- -------------------------------------- 221* DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, 222* DTYPE_A = 1. 223* CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating 224* the BLACS process grid A is distribu- 225* ted over. The context itself is glo- 226* bal, but the handle (the integer 227* value) may vary. 228* M_A (global) DESCA( M_ ) The number of rows in the global 229* array A. 230* N_A (global) DESCA( N_ ) The number of columns in the global 231* array A. 232* MB_A (global) DESCA( MB_ ) The blocking factor used to distribute 233* the rows of the array. 234* NB_A (global) DESCA( NB_ ) The blocking factor used to distribute 235* the columns of the array. 236* RSRC_A (global) DESCA( RSRC_ ) The process row over which the first 237* row of the array A is distributed. 238* CSRC_A (global) DESCA( CSRC_ ) The process column over which the 239* first column of the array A is 240* distributed. 241* LLD_A (local) DESCA( LLD_ ) The leading dimension of the local 242* array. LLD_A >= MAX(1,LOCr(M_A)). 243* 244* Let K be the number of rows or columns of a distributed matrix, 245* and assume that its process grid has dimension p x q. 246* LOCr( K ) denotes the number of elements of K that a process 247* would receive if K were distributed over the p processes of its 248* process column. 249* Similarly, LOCc( K ) denotes the number of elements of K that a 250* process would receive if K were distributed over the q processes of 251* its process row. 252* The values of LOCr() and LOCc() may be determined via a call to the 253* ScaLAPACK tool function, NUMROC: 254* LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), 255* LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). 256* An upper bound for these quantities may be computed by: 257* LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A 258* LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A 259* 260* 261* One-dimensional descriptors: 262* 263* One-dimensional descriptors are a new addition to ScaLAPACK since 264* version 1.0. They simplify and shorten the descriptor for 1D 265* arrays. 266* 267* Since ScaLAPACK supports two-dimensional arrays as the fundamental 268* object, we allow 1D arrays to be distributed either over the 269* first dimension of the array (as if the grid were P-by-1) or the 270* 2nd dimension (as if the grid were 1-by-P). This choice is 271* indicated by the descriptor type (501 or 502) 272* as described below. 273* However, for tridiagonal matrices, since the objects being 274* distributed are the individual vectors storing the diagonals, we 275* have adopted the convention that both the P-by-1 descriptor and 276* the 1-by-P descriptor are allowed and are equivalent for 277* tridiagonal matrices. Thus, for tridiagonal matrices, 278* DTYPE_A = 501 or 502 can be used interchangeably 279* without any other change. 280* We require that the distributed vectors storing the diagonals of a 281* tridiagonal matrix be aligned with each other. Because of this, a 282* single descriptor, DESCA, serves to describe the distribution of 283* of all diagonals simultaneously. 284* 285* IMPORTANT NOTE: the actual BLACS grid represented by the 286* CTXT entry in the descriptor may be *either* P-by-1 or 1-by-P 287* irrespective of which one-dimensional descriptor type 288* (501 or 502) is input. 289* This routine will interpret the grid properly either way. 290* ScaLAPACK routines *do not support intercontext operations* so that 291* the grid passed to a single ScaLAPACK routine *must be the same* 292* for all array descriptors passed to that routine. 293* 294* NOTE: In all cases where 1D descriptors are used, 2D descriptors 295* may also be used, since a one-dimensional array is a special case 296* of a two-dimensional array with one dimension of size unity. 297* The two-dimensional array used in this case *must* be of the 298* proper orientation: 299* If the appropriate one-dimensional descriptor is DTYPEA=501 300* (1 by P type), then the two dimensional descriptor must 301* have a CTXT value that refers to a 1 by P BLACS grid; 302* If the appropriate one-dimensional descriptor is DTYPEA=502 303* (P by 1 type), then the two dimensional descriptor must 304* have a CTXT value that refers to a P by 1 BLACS grid. 305* 306* 307* Summary of allowed descriptors, types, and BLACS grids: 308* DTYPE 501 502 1 1 309* BLACS grid 1xP or Px1 1xP or Px1 1xP Px1 310* ----------------------------------------------------- 311* A OK OK OK NO 312* B NO OK NO OK 313* 314* Let A be a generic term for any 1D block cyclicly distributed array. 315* Such a global array has an associated description vector DESCA. 316* In the following comments, the character _ should be read as 317* "of the global array". 318* 319* NOTATION STORED IN EXPLANATION 320* --------------- ---------- ------------------------------------------ 321* DTYPE_A(global) DESCA( 1 ) The descriptor type. For 1D grids, 322* TYPE_A = 501: 1-by-P grid. 323* TYPE_A = 502: P-by-1 grid. 324* CTXT_A (global) DESCA( 2 ) The BLACS context handle, indicating 325* the BLACS process grid A is distribu- 326* ted over. The context itself is glo- 327* bal, but the handle (the integer 328* value) may vary. 329* N_A (global) DESCA( 3 ) The size of the array dimension being 330* distributed. 331* NB_A (global) DESCA( 4 ) The blocking factor used to distribute 332* the distributed dimension of the array. 333* SRC_A (global) DESCA( 5 ) The process row or column over which the 334* first row or column of the array 335* is distributed. 336* Ignored DESCA( 6 ) Ignored for tridiagonal matrices. 337* Reserved DESCA( 7 ) Reserved for future use. 338* 339* 340* 341* ===================================================================== 342* 343* Code Developer: Andrew J. Cleary, University of Tennessee. 344* Current address: Lawrence Livermore National Labs. 345* 346* ===================================================================== 347* 348* .. Parameters .. 349 REAL ONE 350 PARAMETER ( ONE = 1.0E+0 ) 351 REAL ZERO 352 PARAMETER ( ZERO = 0.0E+0 ) 353 INTEGER INT_ONE 354 PARAMETER ( INT_ONE = 1 ) 355 INTEGER DESCMULT, BIGNUM 356 PARAMETER ( DESCMULT = 100, BIGNUM = DESCMULT*DESCMULT ) 357 INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_, 358 $ LLD_, MB_, M_, NB_, N_, RSRC_ 359 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, 360 $ CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, 361 $ RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) 362* .. 363* .. Local Scalars .. 364 INTEGER COMM_PROC, CSRC, FIRST_PROC, I, ICTXT, 365 $ ICTXT_NEW, ICTXT_SAVE, IDUM3, INT_TEMP, JA_NEW, 366 $ LAF_MIN, LEVEL_DIST, LLDA, MYCOL, MYROW, 367 $ MY_NUM_COLS, NB, NP, NPCOL, NPROW, NP_SAVE, 368 $ ODD_SIZE, PART_OFFSET, PART_SIZE, RETURN_CODE, 369 $ STORE_N_A, TEMP, WORK_SIZE_MIN 370* .. 371* .. Local Arrays .. 372 INTEGER DESCA_1XP( 7 ), PARAM_CHECK( 7, 3 ) 373* .. 374* .. External Subroutines .. 375 EXTERNAL BLACS_GRIDEXIT, BLACS_GRIDINFO, DESC_CONVERT, 376 $ GLOBCHK, IGAMX2D, IGEBR2D, IGEBS2D, PXERBLA, 377 $ RESHAPE, SGERV2D, SGESD2D, SPTTRF, SPTTRSV, 378 $ STRRV2D, STRSD2D 379* .. 380* .. External Functions .. 381 INTEGER NUMROC 382 EXTERNAL NUMROC 383* .. 384* .. Intrinsic Functions .. 385 INTRINSIC MOD, REAL 386* .. 387* .. Executable Statements .. 388* 389* Test the input parameters 390* 391 INFO = 0 392* 393* Convert descriptor into standard form for easy access to 394* parameters, check that grid is of right shape. 395* 396 DESCA_1XP( 1 ) = 501 397* 398 TEMP = DESCA( DTYPE_ ) 399 IF( TEMP.EQ.502 ) THEN 400* Temporarily set the descriptor type to 1xP type 401 DESCA( DTYPE_ ) = 501 402 END IF 403* 404 CALL DESC_CONVERT( DESCA, DESCA_1XP, RETURN_CODE ) 405* 406 DESCA( DTYPE_ ) = TEMP 407* 408 IF( RETURN_CODE.NE.0 ) THEN 409 INFO = -( 5*100+2 ) 410 END IF 411* 412* Get values out of descriptor for use in code. 413* 414 ICTXT = DESCA_1XP( 2 ) 415 CSRC = DESCA_1XP( 5 ) 416 NB = DESCA_1XP( 4 ) 417 LLDA = DESCA_1XP( 6 ) 418 STORE_N_A = DESCA_1XP( 3 ) 419* 420* Get grid parameters 421* 422* 423 CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) 424 NP = NPROW*NPCOL 425* 426* 427* 428 IF( LWORK.LT.-1 ) THEN 429 INFO = -9 430 ELSE IF( LWORK.EQ.-1 ) THEN 431 IDUM3 = -1 432 ELSE 433 IDUM3 = 1 434 END IF 435* 436 IF( N.LT.0 ) THEN 437 INFO = -1 438 END IF 439* 440 IF( N+JA-1.GT.STORE_N_A ) THEN 441 INFO = -( 5*100+6 ) 442 END IF 443* 444* Argument checking that is specific to Divide & Conquer routine 445* 446 IF( NPROW.NE.1 ) THEN 447 INFO = -( 5*100+2 ) 448 END IF 449* 450 IF( N.GT.NP*NB-MOD( JA-1, NB ) ) THEN 451 INFO = -( 1 ) 452 CALL PXERBLA( ICTXT, 'PSPTTRF, D&C alg.: only 1 block per proc' 453 $ , -INFO ) 454 RETURN 455 END IF 456* 457 IF( ( JA+N-1.GT.NB ) .AND. ( NB.LT.2*INT_ONE ) ) THEN 458 INFO = -( 5*100+4 ) 459 CALL PXERBLA( ICTXT, 'PSPTTRF, D&C alg.: NB too small', -INFO ) 460 RETURN 461 END IF 462* 463* 464* Check auxiliary storage size 465* 466 LAF_MIN = ( 12*NPCOL+3*NB ) 467* 468 IF( LAF.LT.LAF_MIN ) THEN 469 INFO = -7 470* put minimum value of laf into AF( 1 ) 471 AF( 1 ) = LAF_MIN 472 CALL PXERBLA( ICTXT, 'PSPTTRF: auxiliary storage error ', 473 $ -INFO ) 474 RETURN 475 END IF 476* 477* Check worksize 478* 479 WORK_SIZE_MIN = 8*NPCOL 480* 481 WORK( 1 ) = WORK_SIZE_MIN 482* 483 IF( LWORK.LT.WORK_SIZE_MIN ) THEN 484 IF( LWORK.NE.-1 ) THEN 485 INFO = -9 486 CALL PXERBLA( ICTXT, 'PSPTTRF: worksize error ', -INFO ) 487 END IF 488 RETURN 489 END IF 490* 491* Pack params and positions into arrays for global consistency check 492* 493 PARAM_CHECK( 7, 1 ) = DESCA( 5 ) 494 PARAM_CHECK( 6, 1 ) = DESCA( 4 ) 495 PARAM_CHECK( 5, 1 ) = DESCA( 3 ) 496 PARAM_CHECK( 4, 1 ) = DESCA( 1 ) 497 PARAM_CHECK( 3, 1 ) = JA 498 PARAM_CHECK( 2, 1 ) = N 499 PARAM_CHECK( 1, 1 ) = IDUM3 500* 501 PARAM_CHECK( 7, 2 ) = 505 502 PARAM_CHECK( 6, 2 ) = 504 503 PARAM_CHECK( 5, 2 ) = 503 504 PARAM_CHECK( 4, 2 ) = 501 505 PARAM_CHECK( 3, 2 ) = 4 506 PARAM_CHECK( 2, 2 ) = 1 507 PARAM_CHECK( 1, 2 ) = 9 508* 509* Want to find errors with MIN( ), so if no error, set it to a big 510* number. If there already is an error, multiply by the the 511* descriptor multiplier. 512* 513 IF( INFO.GE.0 ) THEN 514 INFO = BIGNUM 515 ELSE IF( INFO.LT.-DESCMULT ) THEN 516 INFO = -INFO 517 ELSE 518 INFO = -INFO*DESCMULT 519 END IF 520* 521* Check consistency across processors 522* 523 CALL GLOBCHK( ICTXT, 7, PARAM_CHECK, 7, PARAM_CHECK( 1, 3 ), 524 $ INFO ) 525* 526* Prepare output: set info = 0 if no error, and divide by DESCMULT 527* if error is not in a descriptor entry. 528* 529 IF( INFO.EQ.BIGNUM ) THEN 530 INFO = 0 531 ELSE IF( MOD( INFO, DESCMULT ).EQ.0 ) THEN 532 INFO = -INFO / DESCMULT 533 ELSE 534 INFO = -INFO 535 END IF 536* 537 IF( INFO.LT.0 ) THEN 538 CALL PXERBLA( ICTXT, 'PSPTTRF', -INFO ) 539 RETURN 540 END IF 541* 542* Quick return if possible 543* 544 IF( N.EQ.0 ) 545 $ RETURN 546* 547* 548* Adjust addressing into matrix space to properly get into 549* the beginning part of the relevant data 550* 551 PART_OFFSET = NB*( ( JA-1 ) / ( NPCOL*NB ) ) 552* 553 IF( ( MYCOL-CSRC ).LT.( JA-PART_OFFSET-1 ) / NB ) THEN 554 PART_OFFSET = PART_OFFSET + NB 555 END IF 556* 557 IF( MYCOL.LT.CSRC ) THEN 558 PART_OFFSET = PART_OFFSET - NB 559 END IF 560* 561* Form a new BLACS grid (the "standard form" grid) with only procs 562* holding part of the matrix, of size 1xNP where NP is adjusted, 563* starting at csrc=0, with JA modified to reflect dropped procs. 564* 565* First processor to hold part of the matrix: 566* 567 FIRST_PROC = MOD( ( JA-1 ) / NB+CSRC, NPCOL ) 568* 569* Calculate new JA one while dropping off unused processors. 570* 571 JA_NEW = MOD( JA-1, NB ) + 1 572* 573* Save and compute new value of NP 574* 575 NP_SAVE = NP 576 NP = ( JA_NEW+N-2 ) / NB + 1 577* 578* Call utility routine that forms "standard-form" grid 579* 580 CALL RESHAPE( ICTXT, INT_ONE, ICTXT_NEW, INT_ONE, FIRST_PROC, 581 $ INT_ONE, NP ) 582* 583* Use new context from standard grid as context. 584* 585 ICTXT_SAVE = ICTXT 586 ICTXT = ICTXT_NEW 587 DESCA_1XP( 2 ) = ICTXT_NEW 588* 589* Get information about new grid. 590* 591 CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) 592* 593* Drop out processors that do not have part of the matrix. 594* 595 IF( MYROW.LT.0 ) THEN 596 GO TO 90 597 END IF 598* 599* ******************************** 600* Values reused throughout routine 601* 602* User-input value of partition size 603* 604 PART_SIZE = NB 605* 606* Number of columns in each processor 607* 608 MY_NUM_COLS = NUMROC( N, PART_SIZE, MYCOL, 0, NPCOL ) 609* 610* Offset in columns to beginning of main partition in each proc 611* 612 IF( MYCOL.EQ.0 ) THEN 613 PART_OFFSET = PART_OFFSET + MOD( JA_NEW-1, PART_SIZE ) 614 MY_NUM_COLS = MY_NUM_COLS - MOD( JA_NEW-1, PART_SIZE ) 615 END IF 616* 617* Size of main (or odd) partition in each processor 618* 619 ODD_SIZE = MY_NUM_COLS 620 IF( MYCOL.LT.NP-1 ) THEN 621 ODD_SIZE = ODD_SIZE - INT_ONE 622 END IF 623* 624* 625* Zero out space for fillin 626* 627 DO 10 I = 1, LAF_MIN 628 AF( I ) = ZERO 629 10 CONTINUE 630* 631* Begin main code 632* 633* 634******************************************************************** 635* PHASE 1: Local computation phase. 636******************************************************************** 637* 638* 639 IF( MYCOL.LT.NP-1 ) THEN 640* Transfer last triangle D_i of local matrix to next processor 641* which needs it to calculate fillin due to factorization of 642* its main (odd) block A_i. 643* Overlap the send with the factorization of A_i. 644* 645 CALL STRSD2D( ICTXT, 'U', 'N', 1, 1, 646 $ E( PART_OFFSET+ODD_SIZE+1 ), LLDA-1, 0, MYCOL+1 ) 647* 648 END IF 649* 650* 651* Factor main partition A_i = L_i {L_i}^T in each processor 652* Or A_i = {U_i}^T {U_i} if E is the upper superdiagonal 653* 654 CALL SPTTRF( ODD_SIZE, D( PART_OFFSET+1 ), E( PART_OFFSET+1 ), 655 $ INFO ) 656* 657 IF( INFO.NE.0 ) THEN 658 INFO = MYCOL + 1 659 GO TO 20 660 END IF 661* 662* 663 IF( MYCOL.LT.NP-1 ) THEN 664* Apply factorization to odd-even connection block B_i 665* 666* 667* Perform the triangular system solve {L_i}{{B'}_i}^T = {B_i}^T 668* by dividing B_i by diagonal element 669* 670 E( PART_OFFSET+ODD_SIZE ) = E( PART_OFFSET+ODD_SIZE ) / 671 $ D( PART_OFFSET+ODD_SIZE ) 672* 673* 674* 675* Compute contribution to diagonal block(s) of reduced system. 676* {C'}_i = {C_i}-{{B'}_i}{{B'}_i}^T 677* 678 D( PART_OFFSET+ODD_SIZE+1 ) = D( PART_OFFSET+ODD_SIZE+1 ) - 679 $ D( PART_OFFSET+ODD_SIZE )* 680 $ ( E( PART_OFFSET+ODD_SIZE )* 681 $ ( E( PART_OFFSET+ODD_SIZE ) ) ) 682* 683 END IF 684* End of "if ( MYCOL .lt. NP-1 )..." loop 685* 686* 687 20 CONTINUE 688* If the processor could not locally factor, it jumps here. 689* 690 IF( MYCOL.NE.0 ) THEN 691* 692* Receive previously transmitted matrix section, which forms 693* the right-hand-side for the triangular solve that calculates 694* the "spike" fillin. 695* 696* 697 CALL STRRV2D( ICTXT, 'U', 'N', 1, 1, AF( 1 ), ODD_SIZE, 0, 698 $ MYCOL-1 ) 699* 700 IF( INFO.EQ.0 ) THEN 701* 702* Calculate the "spike" fillin, ${L_i} {{G}_i}^T = {D_i}$ . 703* 704 CALL SPTTRSV( 'N', ODD_SIZE, INT_ONE, D( PART_OFFSET+1 ), 705 $ E( PART_OFFSET+1 ), AF( 1 ), ODD_SIZE, INFO ) 706* 707* Divide by D 708* 709 DO 30 I = 1, ODD_SIZE 710 AF( I ) = AF( I ) / D( PART_OFFSET+I ) 711 30 CONTINUE 712* 713* 714* Calculate the update block for previous proc, E_i = G_i{G_i}^T 715* 716* 717* Since there is no element-by-element vector multiplication in 718* the BLAS, this loop must be hardwired in without a BLAS call 719* 720 INT_TEMP = ODD_SIZE*INT_ONE + 2 + 1 721 AF( INT_TEMP ) = 0 722* 723 DO 40 I = 1, ODD_SIZE 724 AF( INT_TEMP ) = AF( INT_TEMP ) - 725 $ D( PART_OFFSET+I )*( AF( I )* 726 $ ( AF( I ) ) ) 727 40 CONTINUE 728* 729* 730* Initiate send of E_i to previous processor to overlap 731* with next computation. 732* 733 CALL SGESD2D( ICTXT, INT_ONE, INT_ONE, AF( ODD_SIZE+3 ), 734 $ INT_ONE, 0, MYCOL-1 ) 735* 736* 737 IF( MYCOL.LT.NP-1 ) THEN 738* 739* Calculate off-diagonal block(s) of reduced system. 740* Note: for ease of use in solution of reduced system, store 741* L's off-diagonal block in transpose form. 742* {F_i}^T = {H_i}{{B'}_i}^T 743* 744 AF( ODD_SIZE+1 ) = -D( PART_OFFSET+ODD_SIZE )* 745 $ ( E( PART_OFFSET+ODD_SIZE )* 746 $ AF( ODD_SIZE ) ) 747* 748* 749 END IF 750* 751 END IF 752* End of "if ( MYCOL .ne. 0 )..." 753* 754 END IF 755* End of "if (info.eq.0) then" 756* 757* 758* Check to make sure no processors have found errors 759* 760 CALL IGAMX2D( ICTXT, 'A', ' ', 1, 1, INFO, 1, INFO, INFO, -1, 0, 761 $ 0 ) 762* 763 IF( MYCOL.EQ.0 ) THEN 764 CALL IGEBS2D( ICTXT, 'A', ' ', 1, 1, INFO, 1 ) 765 ELSE 766 CALL IGEBR2D( ICTXT, 'A', ' ', 1, 1, INFO, 1, 0, 0 ) 767 END IF 768* 769 IF( INFO.NE.0 ) THEN 770 GO TO 80 771 END IF 772* No errors found, continue 773* 774* 775******************************************************************** 776* PHASE 2: Formation and factorization of Reduced System. 777******************************************************************** 778* 779* Gather up local sections of reduced system 780* 781* 782* The last processor does not participate in the factorization of 783* the reduced system, having sent its E_i already. 784 IF( MYCOL.EQ.NPCOL-1 ) THEN 785 GO TO 70 786 END IF 787* 788* Initiate send of off-diag block(s) to overlap with next part. 789* Off-diagonal block needed on neighboring processor to start 790* algorithm. 791* 792 IF( ( MOD( MYCOL+1, 2 ).EQ.0 ) .AND. ( MYCOL.GT.0 ) ) THEN 793* 794 CALL SGESD2D( ICTXT, INT_ONE, INT_ONE, AF( ODD_SIZE+1 ), 795 $ INT_ONE, 0, MYCOL-1 ) 796* 797 END IF 798* 799* Copy last diagonal block into AF storage for subsequent 800* operations. 801* 802 AF( ODD_SIZE+2 ) = REAL( D( PART_OFFSET+ODD_SIZE+1 ) ) 803* 804* Receive cont. to diagonal block that is stored on this proc. 805* 806 IF( MYCOL.LT.NPCOL-1 ) THEN 807* 808 CALL SGERV2D( ICTXT, INT_ONE, INT_ONE, AF( ODD_SIZE+2+1 ), 809 $ INT_ONE, 0, MYCOL+1 ) 810* 811* Add contribution to diagonal block 812* 813 AF( ODD_SIZE+2 ) = AF( ODD_SIZE+2 ) + AF( ODD_SIZE+3 ) 814* 815 END IF 816* 817* 818* ************************************* 819* Modification Loop 820* 821* The distance for sending and receiving for each level starts 822* at 1 for the first level. 823 LEVEL_DIST = 1 824* 825* Do until this proc is needed to modify other procs' equations 826* 827 50 CONTINUE 828 IF( MOD( ( MYCOL+1 ) / LEVEL_DIST, 2 ).NE.0 ) 829 $ GO TO 60 830* 831* Receive and add contribution to diagonal block from the left 832* 833 IF( MYCOL-LEVEL_DIST.GE.0 ) THEN 834 CALL SGERV2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE, 0, 835 $ MYCOL-LEVEL_DIST ) 836* 837 AF( ODD_SIZE+2 ) = AF( ODD_SIZE+2 ) + WORK( 1 ) 838* 839 END IF 840* 841* Receive and add contribution to diagonal block from the right 842* 843 IF( MYCOL+LEVEL_DIST.LT.NPCOL-1 ) THEN 844 CALL SGERV2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE, 0, 845 $ MYCOL+LEVEL_DIST ) 846* 847 AF( ODD_SIZE+2 ) = AF( ODD_SIZE+2 ) + WORK( 1 ) 848* 849 END IF 850* 851 LEVEL_DIST = LEVEL_DIST*2 852* 853 GO TO 50 854 60 CONTINUE 855* [End of GOTO Loop] 856* 857* 858* ********************************* 859* Calculate and use this proc's blocks to modify other procs'... 860 IF( AF( ODD_SIZE+2 ).EQ.ZERO ) THEN 861 INFO = NPCOL + MYCOL 862 END IF 863* 864* **************************************************************** 865* Receive offdiagonal block from processor to right. 866* If this is the first group of processors, the receive comes 867* from a different processor than otherwise. 868* 869 IF( LEVEL_DIST.EQ.1 ) THEN 870 COMM_PROC = MYCOL + 1 871* 872* Move block into place that it will be expected to be for 873* calcs. 874* 875 AF( ODD_SIZE+3 ) = AF( ODD_SIZE+1 ) 876* 877 ELSE 878 COMM_PROC = MYCOL + LEVEL_DIST / 2 879 END IF 880* 881 IF( MYCOL / LEVEL_DIST.LE.( NPCOL-1 ) / LEVEL_DIST-2 ) THEN 882* 883 CALL SGERV2D( ICTXT, INT_ONE, INT_ONE, AF( ODD_SIZE+1 ), 884 $ INT_ONE, 0, COMM_PROC ) 885* 886 IF( INFO.EQ.0 ) THEN 887* 888* 889* Modify upper off_diagonal block with diagonal block 890* 891* 892 AF( ODD_SIZE+1 ) = AF( ODD_SIZE+1 ) / AF( ODD_SIZE+2 ) 893* 894 END IF 895* End of "if ( info.eq.0 ) then" 896* 897* Calculate contribution from this block to next diagonal block 898* 899 WORK( 1 ) = -ONE*AF( ODD_SIZE+1 )*AF( ODD_SIZE+2 )* 900 $ ( AF( ODD_SIZE+1 ) ) 901* 902* Send contribution to diagonal block's owning processor. 903* 904 CALL SGESD2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE, 0, 905 $ MYCOL+LEVEL_DIST ) 906* 907 END IF 908* End of "if( mycol/level_dist .le. (npcol-1)/level_dist-2 )..." 909* 910* 911* **************************************************************** 912* Receive off_diagonal block from left and use to finish with this 913* processor. 914* 915 IF( ( MYCOL / LEVEL_DIST.GT.0 ) .AND. 916 $ ( MYCOL / LEVEL_DIST.LE.( NPCOL-1 ) / LEVEL_DIST-1 ) ) THEN 917* 918 IF( LEVEL_DIST.GT.1 ) THEN 919* 920* Receive offdiagonal block(s) from proc level_dist/2 to the 921* left 922* 923 CALL SGERV2D( ICTXT, INT_ONE, INT_ONE, AF( ODD_SIZE+2+1 ), 924 $ INT_ONE, 0, MYCOL-LEVEL_DIST / 2 ) 925* 926 END IF 927* 928* 929 IF( INFO.EQ.0 ) THEN 930* 931* Use diagonal block(s) to modify this offdiagonal block 932* 933 AF( ODD_SIZE+3 ) = ( AF( ODD_SIZE+3 ) ) / AF( ODD_SIZE+2 ) 934* 935 END IF 936* End of "if( info.eq.0 ) then" 937* 938* Use offdiag block(s) to calculate modification to diag block 939* of processor to the left 940* 941 WORK( 1 ) = -ONE*AF( ODD_SIZE+3 )*AF( ODD_SIZE+2 )* 942 $ ( AF( ODD_SIZE+3 ) ) 943* 944* Send contribution to diagonal block's owning processor. 945* 946 CALL SGESD2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE, 0, 947 $ MYCOL-LEVEL_DIST ) 948* 949* ******************************************************* 950* 951 IF( MYCOL / LEVEL_DIST.LE.( NPCOL-1 ) / LEVEL_DIST-2 ) THEN 952* 953* Decide which processor offdiagonal block(s) goes to 954* 955 IF( ( MOD( MYCOL / ( 2*LEVEL_DIST ), 2 ) ).EQ.0 ) THEN 956 COMM_PROC = MYCOL + LEVEL_DIST 957 ELSE 958 COMM_PROC = MYCOL - LEVEL_DIST 959 END IF 960* 961* Use offdiagonal blocks to calculate offdiag 962* block to send to neighboring processor. Depending 963* on circumstances, may need to transpose the matrix. 964* 965 WORK( 1 ) = -ONE*AF( ODD_SIZE+3 )*AF( ODD_SIZE+2 )* 966 $ AF( ODD_SIZE+1 ) 967* 968* Send contribution to offdiagonal block's owning processor. 969* 970 CALL SGESD2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE, 971 $ 0, COMM_PROC ) 972* 973 END IF 974* 975 END IF 976* End of "if( mycol/level_dist.le. (npcol-1)/level_dist -1 )..." 977* 978 70 CONTINUE 979* 980* 981 80 CONTINUE 982* 983* 984* Free BLACS space used to hold standard-form grid. 985* 986 IF( ICTXT_SAVE.NE.ICTXT_NEW ) THEN 987 CALL BLACS_GRIDEXIT( ICTXT_NEW ) 988 END IF 989* 990 90 CONTINUE 991* 992* Restore saved input parameters 993* 994 ICTXT = ICTXT_SAVE 995 NP = NP_SAVE 996* 997* Output minimum worksize 998* 999 WORK( 1 ) = WORK_SIZE_MIN 1000* 1001* Make INFO consistent across processors 1002* 1003 CALL IGAMX2D( ICTXT, 'A', ' ', 1, 1, INFO, 1, INFO, INFO, -1, 0, 1004 $ 0 ) 1005* 1006 IF( MYCOL.EQ.0 ) THEN 1007 CALL IGEBS2D( ICTXT, 'A', ' ', 1, 1, INFO, 1 ) 1008 ELSE 1009 CALL IGEBR2D( ICTXT, 'A', ' ', 1, 1, INFO, 1, 0, 0 ) 1010 END IF 1011* 1012* 1013 RETURN 1014* 1015* End of PSPTTRF 1016* 1017 END 1018