SUBROUTINE PSGBTRF( N, BWL, BWU, A, JA, DESCA, IPIV, AF, LAF, $ WORK, LWORK, INFO ) * * -- ScaLAPACK routine (version 1.7) -- * University of Tennessee, Knoxville, Oak Ridge National Laboratory, * and University of California, Berkeley. * April 3, 2000 * * .. Scalar Arguments .. INTEGER BWL, BWU, INFO, JA, LAF, LWORK, N * .. * .. Array Arguments .. INTEGER DESCA( * ), IPIV( * ) REAL A( * ), AF( * ), WORK( * ) * .. * * Purpose * ======= * * PSGBTRF computes a LU factorization * of an N-by-N real banded * distributed matrix * with bandwidth BWL, BWU: A(1:N, JA:JA+N-1). * Reordering is used to increase parallelism in the factorization. * This reordering results in factors that are DIFFERENT from those * produced by equivalent sequential codes. These factors cannot * be used directly by users; however, they can be used in * subsequent calls to PSGBTRS to solve linear systems. * * The factorization has the form * * P A(1:N, JA:JA+N-1) Q = L U * * where U is a banded upper triangular matrix and L is banded * lower triangular, and P and Q are permutation matrices. * The matrix Q represents reordering of columns * for parallelism's sake, while P represents * reordering of rows for numerical stability using * classic partial pivoting. * * ===================================================================== * * Arguments * ========= * * * N (global input) INTEGER * The number of rows and columns to be operated on, i.e. the * order of the distributed submatrix A(1:N, JA:JA+N-1). N >= 0. * * BWL (global input) INTEGER * Number of subdiagonals. 0 <= BWL <= N-1 * * BWU (global input) INTEGER * Number of superdiagonals. 0 <= BWU <= N-1 * * A (local input/local output) REAL pointer into * local memory to an array with first dimension * LLD_A >=(2*bwl+2*bwu+1) (stored in DESCA). * On entry, this array contains the local pieces of the * N-by-N unsymmetric banded distributed matrix * A(1:N, JA:JA+N-1) to be factored. * This local portion is stored in the packed banded format * used in LAPACK. Please see the Notes below and the * ScaLAPACK manual for more detail on the format of * distributed matrices. * On exit, this array contains information containing details * of the factorization. * Note that permutations are performed on the matrix, so that * the factors returned are different from those returned * by LAPACK. * * JA (global input) INTEGER * The index in the global array A that points to the start of * the matrix to be operated on (which may be either all of A * or a submatrix of A). * * DESCA (global and local input) INTEGER array of dimension DLEN. * if 1D type (DTYPE_A=501), DLEN >= 7; * if 2D type (DTYPE_A=1), DLEN >= 9 . * The array descriptor for the distributed matrix A. * Contains information of mapping of A to memory. Please * see NOTES below for full description and options. * * IPIV (local output) INTEGER array, dimension >= DESCA( NB ). * Pivot indices for local factorizations. * Users *should not* alter the contents between * factorization and solve. * * AF (local output) REAL array, dimension LAF. * Auxiliary Fillin Space. * Fillin is created during the factorization routine * PSGBTRF and this is stored in AF. If a linear system * is to be solved using PSGBTRS after the factorization * routine, AF *must not be altered* after the factorization. * * LAF (local input) INTEGER * Size of user-input Auxiliary Fillin space AF. Must be >= * (NB+bwu)*(bwl+bwu)+6*(bwl+bwu)*(bwl+2*bwu) * If LAF is not large enough, an error code will be returned * and the minimum acceptable size will be returned in AF( 1 ) * * WORK (local workspace/local output) * REAL temporary workspace. This space may * be overwritten in between calls to routines. WORK must be * the size given in LWORK. * On exit, WORK( 1 ) contains the minimal LWORK. * * LWORK (local input or global input) INTEGER * Size of user-input workspace WORK. * If LWORK is too small, the minimal acceptable size will be * returned in WORK(1) and an error code is returned. LWORK>= * 1 * * INFO (global output) INTEGER * = 0: successful exit * < 0: If the i-th argument is an array and the j-entry had * an illegal value, then INFO = -(i*100+j), if the i-th * argument is a scalar and had an illegal value, then * INFO = -i. * > 0: If INFO = K<=NPROCS, the submatrix stored on processor * INFO and factored locally was not * nonsingular, and * the factorization was not completed. * If INFO = K>NPROCS, the submatrix stored on processor * INFO-NPROCS representing interactions with other * processors was not * nonsingular, * and the factorization was not completed. * * ===================================================================== * * * Restrictions * ============ * * The following are restrictions on the input parameters. Some of these * are temporary and will be removed in future releases, while others * may reflect fundamental technical limitations. * * Non-cyclic restriction: VERY IMPORTANT! * P*NB>= mod(JA-1,NB)+N. * The mapping for matrices must be blocked, reflecting the nature * of the divide and conquer algorithm as a task-parallel algorithm. * This formula in words is: no processor may have more than one * chunk of the matrix. * * Blocksize cannot be too small: * If the matrix spans more than one processor, the following * restriction on NB, the size of each block on each processor, * must hold: * NB >= (BWL+BWU)+1 * The bulk of parallel computation is done on the matrix of size * O(NB) on each processor. If this is too small, divide and conquer * is a poor choice of algorithm. * * Submatrix reference: * JA = IB * Alignment restriction that prevents unnecessary communication. * * * ===================================================================== * * * Notes * ===== * * If the factorization routine and the solve routine are to be called * separately (to solve various sets of righthand sides using the same * coefficient matrix), the auxiliary space AF *must not be altered* * between calls to the factorization routine and the solve routine. * * The best algorithm for solving banded and tridiagonal linear systems * depends on a variety of parameters, especially the bandwidth. * Currently, only algorithms designed for the case N/P >> bw are * implemented. These go by many names, including Divide and Conquer, * Partitioning, domain decomposition-type, etc. * * Algorithm description: Divide and Conquer * * The Divide and Conqer algorithm assumes the matrix is narrowly * banded compared with the number of equations. In this situation, * it is best to distribute the input matrix A one-dimensionally, * with columns atomic and rows divided amongst the processes. * The basic algorithm divides the banded matrix up into * P pieces with one stored on each processor, * and then proceeds in 2 phases for the factorization or 3 for the * solution of a linear system. * 1) Local Phase: * The individual pieces are factored independently and in * parallel. These factors are applied to the matrix creating * fillin, which is stored in a non-inspectable way in auxiliary * space AF. Mathematically, this is equivalent to reordering * the matrix A as P A P^T and then factoring the principal * leading submatrix of size equal to the sum of the sizes of * the matrices factored on each processor. The factors of * these submatrices overwrite the corresponding parts of A * in memory. * 2) Reduced System Phase: * A small (max(bwl,bwu)* (P-1)) system is formed representing * interaction of the larger blocks, and is stored (as are its * factors) in the space AF. A parallel Block Cyclic Reduction * algorithm is used. For a linear system, a parallel front solve * followed by an analagous backsolve, both using the structure * of the factored matrix, are performed. * 3) Backsubsitution Phase: * For a linear system, a local backsubstitution is performed on * each processor in parallel. * * * Descriptors * =========== * * Descriptors now have *types* and differ from ScaLAPACK 1.0. * * Note: banded codes can use either the old two dimensional * or new one-dimensional descriptors, though the processor grid in * both cases *must be one-dimensional*. We describe both types below. * * Each global data object is described by an associated description * vector. This vector stores the information required to establish * the mapping between an object element and its corresponding process * and memory location. * * Let A be a generic term for any 2D block cyclicly distributed array. * Such a global array has an associated description vector DESCA. * In the following comments, the character _ should be read as * "of the global array". * * NOTATION STORED IN EXPLANATION * --------------- -------------- -------------------------------------- * DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, * DTYPE_A = 1. * CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating * the BLACS process grid A is distribu- * ted over. The context itself is glo- * bal, but the handle (the integer * value) may vary. * M_A (global) DESCA( M_ ) The number of rows in the global * array A. * N_A (global) DESCA( N_ ) The number of columns in the global * array A. * MB_A (global) DESCA( MB_ ) The blocking factor used to distribute * the rows of the array. * NB_A (global) DESCA( NB_ ) The blocking factor used to distribute * the columns of the array. * RSRC_A (global) DESCA( RSRC_ ) The process row over which the first * row of the array A is distributed. * CSRC_A (global) DESCA( CSRC_ ) The process column over which the * first column of the array A is * distributed. * LLD_A (local) DESCA( LLD_ ) The leading dimension of the local * array. LLD_A >= MAX(1,LOCr(M_A)). * * Let K be the number of rows or columns of a distributed matrix, * and assume that its process grid has dimension p x q. * LOCr( K ) denotes the number of elements of K that a process * would receive if K were distributed over the p processes of its * process column. * Similarly, LOCc( K ) denotes the number of elements of K that a * process would receive if K were distributed over the q processes of * its process row. * The values of LOCr() and LOCc() may be determined via a call to the * ScaLAPACK tool function, NUMROC: * LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), * LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). * An upper bound for these quantities may be computed by: * LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A * LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A * * * One-dimensional descriptors: * * One-dimensional descriptors are a new addition to ScaLAPACK since * version 1.0. They simplify and shorten the descriptor for 1D * arrays. * * Since ScaLAPACK supports two-dimensional arrays as the fundamental * object, we allow 1D arrays to be distributed either over the * first dimension of the array (as if the grid were P-by-1) or the * 2nd dimension (as if the grid were 1-by-P). This choice is * indicated by the descriptor type (501 or 502) * as described below. * * IMPORTANT NOTE: the actual BLACS grid represented by the * CTXT entry in the descriptor may be *either* P-by-1 or 1-by-P * irrespective of which one-dimensional descriptor type * (501 or 502) is input. * This routine will interpret the grid properly either way. * ScaLAPACK routines *do not support intercontext operations* so that * the grid passed to a single ScaLAPACK routine *must be the same* * for all array descriptors passed to that routine. * * NOTE: In all cases where 1D descriptors are used, 2D descriptors * may also be used, since a one-dimensional array is a special case * of a two-dimensional array with one dimension of size unity. * The two-dimensional array used in this case *must* be of the * proper orientation: * If the appropriate one-dimensional descriptor is DTYPEA=501 * (1 by P type), then the two dimensional descriptor must * have a CTXT value that refers to a 1 by P BLACS grid; * If the appropriate one-dimensional descriptor is DTYPEA=502 * (P by 1 type), then the two dimensional descriptor must * have a CTXT value that refers to a P by 1 BLACS grid. * * * Summary of allowed descriptors, types, and BLACS grids: * DTYPE 501 502 1 1 * BLACS grid 1xP or Px1 1xP or Px1 1xP Px1 * ----------------------------------------------------- * A OK NO OK NO * B NO OK NO OK * * Let A be a generic term for any 1D block cyclicly distributed array. * Such a global array has an associated description vector DESCA. * In the following comments, the character _ should be read as * "of the global array". * * NOTATION STORED IN EXPLANATION * --------------- ---------- ------------------------------------------ * DTYPE_A(global) DESCA( 1 ) The descriptor type. For 1D grids, * TYPE_A = 501: 1-by-P grid. * TYPE_A = 502: P-by-1 grid. * CTXT_A (global) DESCA( 2 ) The BLACS context handle, indicating * the BLACS process grid A is distribu- * ted over. The context itself is glo- * bal, but the handle (the integer * value) may vary. * N_A (global) DESCA( 3 ) The size of the array dimension being * distributed. * NB_A (global) DESCA( 4 ) The blocking factor used to distribute * the distributed dimension of the array. * SRC_A (global) DESCA( 5 ) The process row or column over which the * first row or column of the array * is distributed. * LLD_A (local) DESCA( 6 ) The leading dimension of the local array * storing the local blocks of the distri- * buted array A. Minimum value of LLD_A * depends on TYPE_A. * TYPE_A = 501: LLD_A >= * size of undistributed dimension, 1. * TYPE_A = 502: LLD_A >=NB_A, 1. * Reserved DESCA( 7 ) Reserved for future use. * * ===================================================================== * * Implemented for ScaLAPACK by: * Andrew J. Cleary, Livermore National Lab and University of Tenn., * and Markus Hegland, Australian National University. Feb., 1997. * Based on code written by : Peter Arbenz, ETH Zurich, 1996. * Last modified by: Peter Arbenz, Institute of Scientific Computing, * ETH, Zurich. * * ===================================================================== * * .. Parameters .. REAL ONE PARAMETER ( ONE = 1.0E+0 ) REAL ZERO PARAMETER ( ZERO = 0.0E+0 ) INTEGER INT_ONE PARAMETER ( INT_ONE = 1 ) INTEGER DESCMULT, BIGNUM PARAMETER ( DESCMULT = 100, BIGNUM = DESCMULT*DESCMULT ) INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_, $ LLD_, MB_, M_, NB_, N_, RSRC_ PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, $ CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, $ RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) * .. * .. Local Scalars .. INTEGER APTR, BBPTR, BIPTR, BM, BM1, BM2, BMN, BN, BW, $ CSRC, DBPTR, FIRST_PROC, I, I1, I2, ICTXT, $ ICTXT_NEW, ICTXT_SAVE, IDUM3, J, JA_NEW, JPTR, $ L, LAF_MIN, LBWL, LBWU, LDB, LDBB, LLDA, LM, $ LMJ, LN, LNJ, LPTR, MYCOL, MYROW, MY_NUM_COLS, $ NB, NEICOL, NP, NPACT, NPCOL, NPROW, NPSTR, $ NP_SAVE, NRHS, ODD_N, ODD_SIZE, ODPTR, OFST, $ PART_OFFSET, PART_SIZE, RETURN_CODE, STORE_N_A, $ WORK_SIZE_MIN * .. * .. Local Arrays .. INTEGER DESCA_1XP( 7 ), PARAM_CHECK( 9, 3 ) * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDEXIT, BLACS_GRIDINFO, DESC_CONVERT, $ SGBTRF, SGEMM, SGER, SGERV2D, SGESD2D, SGETRF, $ SLACPY, SLASWP, SLATCPY, SSWAP, STRRV2D, $ STRSD2D, STRSM, GLOBCHK, IGAMX2D, IGEBR2D, $ IGEBS2D, PXERBLA, RESHAPE * .. * .. External Functions .. INTEGER NUMROC EXTERNAL NUMROC * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, MOD * .. * .. Executable Statements .. * * * Test the input parameters * INFO = 0 * * Convert descriptor into standard form for easy access to * parameters, check that grid is of right shape. * DESCA_1XP( 1 ) = 501 * CALL DESC_CONVERT( DESCA, DESCA_1XP, RETURN_CODE ) * IF( RETURN_CODE.NE.0 ) THEN INFO = -( 6*100+2 ) END IF * * Get values out of descriptor for use in code. * ICTXT = DESCA_1XP( 2 ) CSRC = DESCA_1XP( 5 ) NB = DESCA_1XP( 4 ) LLDA = DESCA_1XP( 6 ) STORE_N_A = DESCA_1XP( 3 ) * * Get grid parameters * * CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) NP = NPROW*NPCOL * * * IF( LWORK.LT.-1 ) THEN INFO = -11 ELSE IF( LWORK.EQ.-1 ) THEN IDUM3 = -1 ELSE IDUM3 = 1 END IF * IF( N.LT.0 ) THEN INFO = -1 END IF * IF( N+JA-1.GT.STORE_N_A ) THEN INFO = -( 6*100+6 ) END IF * IF( ( BWL.GT.N-1 ) .OR. ( BWL.LT.0 ) ) THEN INFO = -2 END IF * IF( ( BWU.GT.N-1 ) .OR. ( BWU.LT.0 ) ) THEN INFO = -3 END IF * IF( LLDA.LT.( 2*BWL+2*BWU+1 ) ) THEN INFO = -( 6*100+6 ) END IF * IF( NB.LE.0 ) THEN INFO = -( 6*100+4 ) END IF * BW = BWU + BWL * * Argument checking that is specific to Divide & Conquer routine * IF( NPROW.NE.1 ) THEN INFO = -( 6*100+2 ) END IF * IF( N.GT.NP*NB-MOD( JA-1, NB ) ) THEN INFO = -( 1 ) CALL PXERBLA( ICTXT, 'PSGBTRF, D&C alg.: only 1 block per proc' $ , -INFO ) RETURN END IF * IF( ( JA+N-1.GT.NB ) .AND. ( NB.LT.( BWL+BWU+1 ) ) ) THEN INFO = -( 6*100+4 ) CALL PXERBLA( ICTXT, 'PSGBTRF, D&C alg.: NB too small', -INFO ) RETURN END IF * * * Check auxiliary storage size * LAF_MIN = ( NB+BWU )*( BWL+BWU ) + 6*( BWL+BWU )*( BWL+2*BWU ) * IF( LAF.LT.LAF_MIN ) THEN INFO = -9 * put minimum value of laf into AF( 1 ) AF( 1 ) = LAF_MIN CALL PXERBLA( ICTXT, 'PSGBTRF: auxiliary storage error ', $ -INFO ) RETURN END IF * * Check worksize * WORK_SIZE_MIN = 1 * WORK( 1 ) = WORK_SIZE_MIN * IF( LWORK.LT.WORK_SIZE_MIN ) THEN IF( LWORK.NE.-1 ) THEN INFO = -11 * put minimum value of work into work( 1 ) WORK( 1 ) = WORK_SIZE_MIN CALL PXERBLA( ICTXT, 'PSGBTRF: worksize error ', -INFO ) END IF RETURN END IF * * Pack params and positions into arrays for global consistency check * PARAM_CHECK( 9, 1 ) = DESCA( 5 ) PARAM_CHECK( 8, 1 ) = DESCA( 4 ) PARAM_CHECK( 7, 1 ) = DESCA( 3 ) PARAM_CHECK( 6, 1 ) = DESCA( 1 ) PARAM_CHECK( 5, 1 ) = JA PARAM_CHECK( 4, 1 ) = BWU PARAM_CHECK( 3, 1 ) = BWL PARAM_CHECK( 2, 1 ) = N PARAM_CHECK( 1, 1 ) = IDUM3 * PARAM_CHECK( 9, 2 ) = 605 PARAM_CHECK( 8, 2 ) = 604 PARAM_CHECK( 7, 2 ) = 603 PARAM_CHECK( 6, 2 ) = 601 PARAM_CHECK( 5, 2 ) = 5 PARAM_CHECK( 4, 2 ) = 3 PARAM_CHECK( 3, 2 ) = 2 PARAM_CHECK( 2, 2 ) = 1 PARAM_CHECK( 1, 2 ) = 11 * * Want to find errors with MIN( ), so if no error, set it to a big * number. If there already is an error, multiply by the the * descriptor multiplier. * IF( INFO.GE.0 ) THEN INFO = BIGNUM ELSE IF( INFO.LT.-DESCMULT ) THEN INFO = -INFO ELSE INFO = -INFO*DESCMULT END IF * * Check consistency across processors * CALL GLOBCHK( ICTXT, 9, PARAM_CHECK, 9, PARAM_CHECK( 1, 3 ), $ INFO ) * * Prepare output: set info = 0 if no error, and divide by DESCMULT * if error is not in a descriptor entry. * IF( INFO.EQ.BIGNUM ) THEN INFO = 0 ELSE IF( MOD( INFO, DESCMULT ).EQ.0 ) THEN INFO = -INFO / DESCMULT ELSE INFO = -INFO END IF * IF( INFO.LT.0 ) THEN CALL PXERBLA( ICTXT, 'PSGBTRF', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * * Adjust addressing into matrix space to properly get into * the beginning part of the relevant data * PART_OFFSET = NB*( ( JA-1 ) / ( NPCOL*NB ) ) * IF( ( MYCOL-CSRC ).LT.( JA-PART_OFFSET-1 ) / NB ) THEN PART_OFFSET = PART_OFFSET + NB END IF * IF( MYCOL.LT.CSRC ) THEN PART_OFFSET = PART_OFFSET - NB END IF * * Form a new BLACS grid (the "standard form" grid) with only procs * holding part of the matrix, of size 1xNP where NP is adjusted, * starting at csrc=0, with JA modified to reflect dropped procs. * * First processor to hold part of the matrix: * FIRST_PROC = MOD( ( JA-1 ) / NB+CSRC, NPCOL ) * * Calculate new JA one while dropping off unused processors. * JA_NEW = MOD( JA-1, NB ) + 1 * * Save and compute new value of NP * NP_SAVE = NP NP = ( JA_NEW+N-2 ) / NB + 1 * * Call utility routine that forms "standard-form" grid * CALL RESHAPE( ICTXT, INT_ONE, ICTXT_NEW, INT_ONE, FIRST_PROC, $ INT_ONE, NP ) * * Use new context from standard grid as context. * ICTXT_SAVE = ICTXT ICTXT = ICTXT_NEW DESCA_1XP( 2 ) = ICTXT_NEW * * Get information about new grid. * CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) * * Drop out processors that do not have part of the matrix. * IF( MYROW.LT.0 ) THEN GO TO 210 END IF * * ******************************** * Values reused throughout routine * * User-input value of partition size * PART_SIZE = NB * * Number of columns in each processor * MY_NUM_COLS = NUMROC( N, PART_SIZE, MYCOL, 0, NPCOL ) * * Offset in columns to beginning of main partition in each proc * IF( MYCOL.EQ.0 ) THEN PART_OFFSET = PART_OFFSET + MOD( JA_NEW-1, PART_SIZE ) MY_NUM_COLS = MY_NUM_COLS - MOD( JA_NEW-1, PART_SIZE ) END IF * * Offset in elements * OFST = PART_OFFSET*LLDA * * Size of main (or odd) partition in each processor * ODD_SIZE = NUMROC( N, PART_SIZE, MYCOL, 0, NPCOL ) * * * Zero out space for fillin * DO 10 I = 1, LAF_MIN AF( I ) = ZERO 10 CONTINUE * DO 30 J = 1, ODD_SIZE DO 20 I = 1, BW A( I+( J-1 )*LLDA ) = ZERO 20 CONTINUE 30 CONTINUE * * Begin main code * ******************************************************************** * PHASE 1: Local computation phase. ******************************************************************** * * * Transfer triangle B_i of local matrix to next processor * for fillin. Overlap the send with the factorization of A_i. * IF( MYCOL.LE.NPCOL-2 ) THEN * * The last processor does not need to send anything. * BIPTR = location of triangle B_i in memory BIPTR = ( NB-BW )*LLDA + 2*BW + 1 * CALL STRSD2D( ICTXT, 'U', 'N', $ MIN( BW, BWU+NUMROC( N, NB, MYCOL+1, 0, $ NPCOL ) ), BW, A( BIPTR ), LLDA-1, 0, MYCOL+1 ) * END IF * * Factor main partition P_i A_i = L_i U_i on each processor * * LBWL, LBWU: lower and upper bandwidth of local solver * Note that for MYCOL > 0 one has lower triangular blocks! * LM is the number of rows which is usually NB except for * MYCOL = 0 where it is BWU less and MYCOL=NPCOL-1 where it * is NR+BWU where NR is the number of columns on the last processor * Finally APTR is the pointer to the first element of A. As LAPACK * has a slightly different matrix format than Scalapack the pointer * has to be adjusted on processor MYCOL=0. * IF( MYCOL.NE.0 ) THEN LBWL = BW LBWU = 0 APTR = 1 ELSE LBWL = BWL LBWU = BWU APTR = 1 + BWU END IF * IF( MYCOL.NE.NPCOL-1 ) THEN LM = NB - LBWU LN = NB - BW ELSE IF( MYCOL.NE.0 ) THEN LM = ODD_SIZE + BWU LN = MAX( ODD_SIZE-BW, 0 ) ELSE LM = N LN = MAX( N-BW, 0 ) END IF * IF( LN.GT.0 ) THEN * CALL SGBTRF( LM, LN, LBWL, LBWU, A( APTR ), LLDA, IPIV, INFO ) * IF( INFO.NE.0 ) THEN INFO = INFO + NB*MYCOL GO TO 80 END IF * NRHS = BW LDB = LLDA - 1 * * Update the last BW columns of A_i (code modified from DGBTRS) * * Only the eliminations of unknowns > LN-BW have an effect on * the last BW columns. Loop over them... * DO 40 J = MAX( LN-BW+1, 1 ), LN * LMJ = MIN( LBWL, LM-J ) LNJ = MIN( BW, J+BW-LN+APTR-1 ) * L = IPIV( J ) * JPTR = J - ( LN+1 ) + 2*BW + 1 - LBWL + LN*LLDA * IF( L.NE.J ) THEN * * Element (L,LN+1) is swapped with element (J,LN+1) etc * Furthermore, the elements in the same row are LDB=LLDA-1 apart * The complicated formulas are to cope with the banded * data format: * LPTR = L - ( LN+1 ) + 2*BW + 1 - LBWL + LN*LLDA * CALL SSWAP( LNJ, A( LPTR ), LDB, A( JPTR ), LDB ) * END IF * * LPTR is the pointer to the beginning of the * coefficients of L * LPTR = BW + 1 + APTR + ( J-1 )*LLDA * CALL SGER( LMJ, LNJ, -ONE, A( LPTR ), 1, A( JPTR ), LDB, $ A( JPTR+1 ), LDB ) 40 CONTINUE * END IF * * Compute spike fill-in, L_i F_i = P_i B_{i-1} * * Receive triangle B_{i-1} from previous processor * IF( MYCOL.GT.0 ) THEN CALL STRRV2D( ICTXT, 'U', 'N', MIN( BW, LM ), BW, AF( 1 ), BW, $ 0, MYCOL-1 ) * * Transpose transmitted upper triangular (trapezoidal) matrix * DO 60 I2 = 1, MIN( BW, LM ) DO 50 I1 = I2 + 1, BW AF( I1+( I2-1 )*BW ) = AF( I2+( I1-1 )*BW ) AF( I2+( I1-1 )*BW ) = ZERO 50 CONTINUE 60 CONTINUE * * Permutation and forward elimination (triang. solve) * DO 70 J = 1, LN * LMJ = MIN( LBWL, LM-J ) L = IPIV( J ) * IF( L.NE.J ) THEN CALL SSWAP( BW, AF( ( L-1 )*BW+1 ), 1, $ AF( ( J-1 )*BW+1 ), 1 ) END IF * LPTR = BW + 1 + APTR + ( J-1 )*LLDA * CALL SGER( NRHS, LMJ, -ONE, AF( ( J-1 )*BW+1 ), 1, $ A( LPTR ), 1, AF( J*BW+1 ), BW ) * 70 CONTINUE * END IF * 80 CONTINUE * ******************************************************************** * PHASE 2: Formation and factorization of Reduced System. ******************************************************************** * * Define the initial dimensions of the diagonal blocks * The offdiagonal blocks (for MYCOL > 0) are of size BM by BW * IF( MYCOL.NE.NPCOL-1 ) THEN BM = BW - LBWU BN = BW ELSE BM = MIN( BW, ODD_SIZE ) + BWU BN = MIN( BW, ODD_SIZE ) END IF * * Pointer to first element of block bidiagonal matrix in AF * Leading dimension of block bidiagonal system * BBPTR = ( NB+BWU )*BW + 1 LDBB = 2*BW + BWU * * Copy from A and AF into block bidiagonal matrix (tail of AF) * * DBPTR = Pointer to diagonal blocks in A DBPTR = BW + 1 + LBWU + LN*LLDA * CALL SLACPY( 'G', BM, BN, A( DBPTR ), LLDA-1, AF( BBPTR+BW*LDBB ), $ LDBB ) * * Zero out any junk entries that were copied * DO 100 J = 1, BM DO 90 I = J + LBWL, BM - 1 AF( BBPTR+BW*LDBB+( J-1 )*LDBB+I ) = ZERO 90 CONTINUE 100 CONTINUE * IF( MYCOL.NE.0 ) THEN * * ODPTR = Pointer to offdiagonal blocks in A * ODPTR = ( LM-BM )*BW + 1 CALL SLATCPY( 'G', BW, BM, AF( ODPTR ), BW, $ AF( BBPTR+2*BW*LDBB ), LDBB ) END IF * IF( NPCOL.EQ.1 ) THEN * * In this case the loop over the levels will not be * performed. CALL SGETRF( N-LN, N-LN, AF( BBPTR+BW*LDBB ), LDBB, $ IPIV( LN+1 ), INFO ) * END IF * * Loop over levels ... only occurs if npcol > 1 * * The two integers NPACT (nu. of active processors) and NPSTR * (stride between active processors) are used to control the * loop. * NPACT = NPCOL NPSTR = 1 * * Begin loop over levels * 110 CONTINUE IF( NPACT.LE.1 ) $ GO TO 190 * * Test if processor is active * IF( MOD( MYCOL, NPSTR ).EQ.0 ) THEN * * Send/Receive blocks * * IF( MOD( MYCOL, 2*NPSTR ).EQ.0 ) THEN * * This node will potentially do more work later * NEICOL = MYCOL + NPSTR * IF( NEICOL / NPSTR.LT.NPACT-1 ) THEN BMN = BW ELSE IF( NEICOL / NPSTR.EQ.NPACT-1 ) THEN ODD_N = NUMROC( N, NB, NPCOL-1, 0, NPCOL ) BMN = MIN( BW, ODD_N ) + BWU ELSE * * Last processor skips to next level GO TO 180 END IF * * BM1 = M for 1st block on proc pair, BM2 2nd block * BM1 = BM BM2 = BMN * IF( NEICOL / NPSTR.LE.NPACT-1 ) THEN * CALL SGESD2D( ICTXT, BM, 2*BW, AF( BBPTR+BW*LDBB ), LDBB, $ 0, NEICOL ) * CALL SGERV2D( ICTXT, BMN, 2*BW, AF( BBPTR+BM ), LDBB, 0, $ NEICOL ) * IF( NPACT.EQ.2 ) THEN * * Copy diagonal block to align whole system * CALL SLACPY( 'G', BMN, BW, AF( BBPTR+BM ), LDBB, $ AF( BBPTR+2*BW*LDBB+BM ), LDBB ) END IF * END IF * ELSE * * This node stops work after this stage -- an extra copy * is required to make the odd and even frontal matrices * look identical * NEICOL = MYCOL - NPSTR * IF( NEICOL.EQ.0 ) THEN BMN = BW - BWU ELSE BMN = BW END IF * BM1 = BMN BM2 = BM * CALL SGESD2D( ICTXT, BM, 2*BW, AF( BBPTR+BW*LDBB ), LDBB, 0, $ NEICOL ) * CALL SLACPY( 'G', BM, 2*BW, AF( BBPTR+BW*LDBB ), LDBB, $ AF( BBPTR+BMN ), LDBB ) * DO 130 J = BBPTR + 2*BW*LDBB, BBPTR + 3*BW*LDBB - 1, LDBB DO 120 I = 0, LDBB - 1 AF( I+J ) = ZERO 120 CONTINUE 130 CONTINUE * CALL SGERV2D( ICTXT, BMN, 2*BW, AF( BBPTR+BW*LDBB ), LDBB, $ 0, NEICOL ) * IF( NPACT.EQ.2 ) THEN * * Copy diagonal block to align whole system * CALL SLACPY( 'G', BM, BW, AF( BBPTR+BMN ), LDBB, $ AF( BBPTR+2*BW*LDBB+BMN ), LDBB ) END IF * END IF * * LU factorization with partial pivoting * IF( NPACT.NE.2 ) THEN * CALL SGETRF( BM+BMN, BW, AF( BBPTR+BW*LDBB ), LDBB, $ IPIV( LN+1 ), INFO ) * * Backsolve left side * DO 150 J = BBPTR, BBPTR + BW*LDBB - 1, LDBB DO 140 I = 0, BM1 - 1 AF( I+J ) = ZERO 140 CONTINUE 150 CONTINUE * CALL SLASWP( BW, AF( BBPTR ), LDBB, 1, BW, IPIV( LN+1 ), 1 ) * CALL STRSM( 'L', 'L', 'N', 'U', BW, BW, ONE, $ AF( BBPTR+BW*LDBB ), LDBB, AF( BBPTR ), LDBB ) * * Use partial factors to update remainder * CALL SGEMM( 'N', 'N', BM+BMN-BW, BW, BW, -ONE, $ AF( BBPTR+BW*LDBB+BW ), LDBB, AF( BBPTR ), LDBB, $ ONE, AF( BBPTR+BW ), LDBB ) * * Backsolve right side * NRHS = BW * CALL SLASWP( NRHS, AF( BBPTR+2*BW*LDBB ), LDBB, 1, BW, $ IPIV( LN+1 ), 1 ) * CALL STRSM( 'L', 'L', 'N', 'U', BW, NRHS, ONE, $ AF( BBPTR+BW*LDBB ), LDBB, $ AF( BBPTR+2*BW*LDBB ), LDBB ) * * Use partial factors to update remainder * CALL SGEMM( 'N', 'N', BM+BMN-BW, NRHS, BW, -ONE, $ AF( BBPTR+BW*LDBB+BW ), LDBB, $ AF( BBPTR+2*BW*LDBB ), LDBB, ONE, $ AF( BBPTR+2*BW*LDBB+BW ), LDBB ) * * * Test if processor is active in next round * IF( MOD( MYCOL, 2*NPSTR ).EQ.0 ) THEN * * Reset BM * BM = BM1 + BM2 - BW * * Local copying in the block bidiagonal area * * CALL SLACPY( 'G', BM, BW, AF( BBPTR+BW ), LDBB, $ AF( BBPTR+BW*LDBB ), LDBB ) CALL SLACPY( 'G', BM, BW, AF( BBPTR+2*BW*LDBB+BW ), LDBB, $ AF( BBPTR+2*BW*LDBB ), LDBB ) * * Zero out space that held original copy * DO 170 J = 0, BW - 1 DO 160 I = 0, BM - 1 AF( BBPTR+2*BW*LDBB+BW+J*LDBB+I ) = ZERO 160 CONTINUE 170 CONTINUE * END IF * ELSE * * Factor the final 2 by 2 block matrix * CALL SGETRF( BM+BMN, BM+BMN, AF( BBPTR+BW*LDBB ), LDBB, $ IPIV( LN+1 ), INFO ) END IF * END IF * * Last processor in an odd-sized NPACT skips to here * 180 CONTINUE * NPACT = ( NPACT+1 ) / 2 NPSTR = NPSTR*2 GO TO 110 * 190 CONTINUE * End loop over levels * 200 CONTINUE * If error was found in Phase 1, processors jump here. * * Free BLACS space used to hold standard-form grid. * ICTXT = ICTXT_SAVE IF( ICTXT.NE.ICTXT_NEW ) THEN CALL BLACS_GRIDEXIT( ICTXT_NEW ) END IF * 210 CONTINUE * If this processor did not hold part of the grid it * jumps here. * * Restore saved input parameters * ICTXT = ICTXT_SAVE NP = NP_SAVE * * Output worksize * WORK( 1 ) = WORK_SIZE_MIN * * Make INFO consistent across processors * CALL IGAMX2D( ICTXT, 'A', ' ', 1, 1, INFO, 1, INFO, INFO, -1, 0, $ 0 ) * IF( MYCOL.EQ.0 ) THEN CALL IGEBS2D( ICTXT, 'A', ' ', 1, 1, INFO, 1 ) ELSE CALL IGEBR2D( ICTXT, 'A', ' ', 1, 1, INFO, 1, 0, 0 ) END IF * * RETURN * * End of PSGBTRF * END

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