SUBROUTINE PSPTTRF( N, D, E, JA, DESCA, AF, LAF, WORK, LWORK,
$ INFO )
*
*
*
* -- ScaLAPACK routine (version 1.6) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* November 15, 1997
*
* .. Scalar Arguments ..
INTEGER INFO, JA, LAF, LWORK, N
* ..
* .. Array Arguments ..
INTEGER DESCA( * )
REAL AF( * ), D( * ), E( * ), WORK( * )
* ..
*
*
* Purpose
* =======
*
* PSPTTRF computes a Cholesky factorization
* of an N-by-N real tridiagonal
* symmetric positive definite distributed matrix
* 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 PSPTTRS to solve linear systems.
*
* The factorization has the form
*
* P A(1:N, JA:JA+N-1) P^T = U' D U or
*
* P A(1:N, JA:JA+N-1) P^T = L D L',
*
* where U is a tridiagonal upper triangular matrix and L is tridiagonal
* lower triangular, and P is a permutation matrix.
*
* =====================================================================
*
* 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.
*
* D (local input/local output) REAL pointer to local
* part of global vector storing the main diagonal of the
* matrix.
* On exit, this array contains information containing the
* factors of the matrix.
* Must be of size >= DESCA( NB_ ).
*
* E (local input/local output) REAL pointer to local
* part of global vector storing the upper diagonal of the
* matrix. Globally, DU(n) is not referenced, and DU must be
* aligned with D.
* On exit, this array contains information containing the
* factors of the matrix.
* Must be of size >= DESCA( NB_ ).
*
* 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 or 502), 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.
*
* AF (local output) REAL array, dimension LAF.
* Auxiliary Fillin Space.
* Fillin is created during the factorization routine
* PSPTTRF and this is stored in AF. If a linear system
* is to be solved using PSPTTRS 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+2)
* 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>=
* 8*NPCOL
*
* INFO (local 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
* positive definite, and
* the factorization was not completed.
* If INFO = K>NPROCS, the submatrix stored on processor
* INFO-NPROCS representing interactions with other
* processors was not
* positive definite,
* 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 >= 2
* 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.
* For tridiagonal matrices, it is obvious: N/P >> bw(=1), and so D&C
* algorithms are the appropriate choice.
*
* 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 tridiagonal 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 ((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: tridiagonal 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.
* However, for tridiagonal matrices, since the objects being
* distributed are the individual vectors storing the diagonals, we
* have adopted the convention that both the P-by-1 descriptor and
* the 1-by-P descriptor are allowed and are equivalent for
* tridiagonal matrices. Thus, for tridiagonal matrices,
* DTYPE_A = 501 or 502 can be used interchangeably
* without any other change.
* We require that the distributed vectors storing the diagonals of a
* tridiagonal matrix be aligned with each other. Because of this, a
* single descriptor, DESCA, serves to describe the distribution of
* of all diagonals simultaneously.
*
* 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 OK 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.
* Ignored DESCA( 6 ) Ignored for tridiagonal matrices.
* Reserved DESCA( 7 ) Reserved for future use.
*
*
*
* =====================================================================
*
* Code Developer: Andrew J. Cleary, University of Tennessee.
* Current address: Lawrence Livermore National Labs.
* This version released: May, 1997.
*
* =====================================================================
*
* ..
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0 )
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 COMM_PROC, CSRC, FIRST_PROC, I, ICTXT,
$ ICTXT_NEW, ICTXT_SAVE, IDUM3, INT_TEMP, JA_NEW,
$ LAF_MIN, LEVEL_DIST, LLDA, MYCOL, MYROW,
$ MY_NUM_COLS, NB, NP, NPCOL, NPROW, NP_SAVE,
$ ODD_SIZE, PART_OFFSET, PART_SIZE, RETURN_CODE,
$ STORE_N_A, TEMP, WORK_SIZE_MIN
* ..
* .. Local Arrays ..
INTEGER DESCA_1XP( 7 ), PARAM_CHECK( 7, 3 )
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GET, BLACS_GRIDEXIT, BLACS_GRIDINFO,
$ DESC_CONVERT, GLOBCHK, PXERBLA, RESHAPE, SAXPY,
$ SGEMM, SGERV2D, SGESD2D, SLACPY, SLATCPY,
$ SPBTRF, SPOTRF, SSYRK, STBTRS, STRMM, STRRV2D,
$ STRSD2D, STRSM, STRTRS
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER NUMROC
EXTERNAL LSAME, NUMROC
* ..
* .. Intrinsic Functions ..
INTRINSIC ICHAR, 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
*
TEMP = DESCA( DTYPE_ )
IF( TEMP .EQ. 502 ) THEN
* Temporarily set the descriptor type to 1xP type
DESCA( DTYPE_ ) = 501
ENDIF
*
CALL DESC_CONVERT( DESCA, DESCA_1XP, RETURN_CODE )
*
DESCA( DTYPE_ ) = TEMP
*
IF( RETURN_CODE .NE. 0) THEN
INFO = -( 5*100 + 2 )
ENDIF
*
* 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 = -9
ELSE IF ( LWORK .EQ. -1 ) THEN
IDUM3 = -1
ELSE
IDUM3 = 1
ENDIF
*
IF( N .LT. 0 ) THEN
INFO = -1
ENDIF
*
IF( N+JA-1 .GT. STORE_N_A ) THEN
INFO = -( 5*100 + 6 )
ENDIF
*
* Argument checking that is specific to Divide & Conquer routine
*
IF( NPROW .NE. 1 ) THEN
INFO = -( 5*100+2 )
ENDIF
*
IF( N .GT. NP*NB-MOD( JA-1, NB )) THEN
INFO = -( 1 )
CALL PXERBLA( ICTXT,
$ 'PSPTTRF, D&C alg.: only 1 block per proc',
$ -INFO )
RETURN
ENDIF
*
IF((JA+N-1.GT.NB) .AND. ( NB.LT.2*INT_ONE )) THEN
INFO = -( 5*100+4 )
CALL PXERBLA( ICTXT,
$ 'PSPTTRF, D&C alg.: NB too small',
$ -INFO )
RETURN
ENDIF
*
*
* Check auxiliary storage size
*
LAF_MIN = (12*NPCOL + 3*NB)
*
IF( LAF .LT. LAF_MIN ) THEN
INFO = -7
* put minimum value of laf into AF( 1 )
AF( 1 ) = LAF_MIN
CALL PXERBLA( ICTXT,
$ 'PSPTTRF: auxiliary storage error ',
$ -INFO )
RETURN
ENDIF
*
* Check worksize
*
WORK_SIZE_MIN = 8*NPCOL
*
WORK( 1 ) = WORK_SIZE_MIN
*
IF( LWORK .LT. WORK_SIZE_MIN ) THEN
IF( LWORK .NE. -1 ) THEN
INFO = -9
CALL PXERBLA( ICTXT,
$ 'PSPTTRF: worksize error ',
$ -INFO )
ENDIF
RETURN
ENDIF
*
* Pack params and positions into arrays for global consistency check
*
PARAM_CHECK( 7, 1 ) = DESCA(5)
PARAM_CHECK( 6, 1 ) = DESCA(4)
PARAM_CHECK( 5, 1 ) = DESCA(3)
PARAM_CHECK( 4, 1 ) = DESCA(1)
PARAM_CHECK( 3, 1 ) = JA
PARAM_CHECK( 2, 1 ) = N
PARAM_CHECK( 1, 1 ) = IDUM3
*
PARAM_CHECK( 7, 2 ) = 505
PARAM_CHECK( 6, 2 ) = 504
PARAM_CHECK( 5, 2 ) = 503
PARAM_CHECK( 4, 2 ) = 501
PARAM_CHECK( 3, 2 ) = 4
PARAM_CHECK( 2, 2 ) = 1
PARAM_CHECK( 1, 2 ) = 9
*
* 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, 7, PARAM_CHECK, 7,
$ 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, 'PSPTTRF', -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
ENDIF
*
IF ( MYCOL .LT. CSRC ) THEN
PART_OFFSET = PART_OFFSET - NB
ENDIF
*
* 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
GOTO 1234
ENDIF
*
* ********************************
* 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 )
ENDIF
*
* Size of main (or odd) partition in each processor
*
ODD_SIZE = MY_NUM_COLS
IF ( MYCOL .LT. NP-1 ) THEN
ODD_SIZE = ODD_SIZE - INT_ONE
ENDIF
*
*
* Zero out space for fillin
*
DO 10 I=1, LAF_MIN
AF( I ) = ZERO
10 CONTINUE
*
* Begin main code
*
*
********************************************************************
* PHASE 1: Local computation phase.
********************************************************************
*
*
IF ( MYCOL .LT. NP-1 ) THEN
* Transfer last triangle D_i of local matrix to next processor
* which needs it to calculate fillin due to factorization of
* its main (odd) block A_i.
* Overlap the send with the factorization of A_i.
*
CALL STRSD2D( ICTXT, 'U', 'N', 1, 1,
$ E( PART_OFFSET+ODD_SIZE+1 ), LLDA-1, 0,
$ MYCOL+1 )
*
ENDIF
*
*
* Factor main partition A_i = L_i {L_i}^T in each processor
* Or A_i = {U_i}^T {U_i} if E is the upper superdiagonal
*
CALL SPTTRF( ODD_SIZE, D( PART_OFFSET+1 ), E( PART_OFFSET+1 ),
$ INFO )
*
IF( INFO.NE.0 ) THEN
INFO = MYCOL+1
GOTO 1500
ENDIF
*
*
IF ( MYCOL .LT. NP-1 ) THEN
* Apply factorization to odd-even connection block B_i
*
*
* Perform the triangular system solve {L_i}{{B'}_i}^T = {B_i}^T
* by dividing B_i by diagonal element
*
E( PART_OFFSET+ODD_SIZE ) = E( PART_OFFSET+ODD_SIZE )/
$ D( PART_OFFSET+ODD_SIZE )
*
*
*
* Compute contribution to diagonal block(s) of reduced system.
* {C'}_i = {C_i}-{{B'}_i}{{B'}_i}^T
*
D( PART_OFFSET+ODD_SIZE+1 ) = D( PART_OFFSET+ODD_SIZE+1 )-
$ D( PART_OFFSET+ODD_SIZE )*( E( PART_OFFSET+ODD_SIZE )*
$ ( E( PART_OFFSET+ODD_SIZE ) ) )
*
ENDIF
* End of "if ( MYCOL .lt. NP-1 )..." loop
*
*
1500 CONTINUE
* If the processor could not locally factor, it jumps here.
*
IF ( MYCOL .NE. 0 ) THEN
*
* Receive previously transmitted matrix section, which forms
* the right-hand-side for the triangular solve that calculates
* the "spike" fillin.
*
*
CALL STRRV2D( ICTXT, 'U', 'N', 1, 1, AF( 1 ), ODD_SIZE, 0,
$ MYCOL-1 )
*
IF (INFO.EQ.0) THEN
*
* Calculate the "spike" fillin, ${L_i} {{G}_i}^T = {D_i}$ .
*
CALL SPTTRSV( 'N', ODD_SIZE, INT_ONE, D( PART_OFFSET+1 ),
$ E( PART_OFFSET+1 ), AF( 1 ), ODD_SIZE, INFO )
*
* Divide by D
*
DO 20 I=1, ODD_SIZE
AF( I ) = AF( I )/D( PART_OFFSET+I )
20 CONTINUE
*
*
* Calculate the update block for previous proc, E_i = G_i{G_i}^T
*
*
* Since there is no element-by-element vector multiplication in
* the BLAS, this loop must be hardwired in without a BLAS call
*
INT_TEMP = ODD_SIZE*INT_ONE+2+1
AF( INT_TEMP ) = 0
*
DO 30 I=1, ODD_SIZE
AF( INT_TEMP ) = AF( INT_TEMP )-D( PART_OFFSET+I )*
$ ( AF( I )*( AF( I ) ) )
30 CONTINUE
*
*
* Initiate send of E_i to previous processor to overlap
* with next computation.
*
CALL SGESD2D( ICTXT, INT_ONE, INT_ONE, AF( ODD_SIZE+3 ),
$ INT_ONE, 0, MYCOL-1 )
*
*
IF ( MYCOL .LT. NP-1 ) THEN
*
* Calculate off-diagonal block(s) of reduced system.
* Note: for ease of use in solution of reduced system, store
* L's off-diagonal block in transpose form.
* {F_i}^T = {H_i}{{B'}_i}^T
*
AF( ODD_SIZE+1 ) =
$ - D( PART_OFFSET+ODD_SIZE )
$ * ( E( PART_OFFSET+ODD_SIZE )
$ * AF( ODD_SIZE ) )
*
*
ENDIF
*
ENDIF
* End of "if ( MYCOL .ne. 0 )..."
*
ENDIF
* End of "if (info.eq.0) then"
*
*
* Check to make sure no processors have found errors
*
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 )
ENDIF
*
IF ( INFO.NE.0 ) THEN
GOTO 1000
ENDIF
* No errors found, continue
*
*
********************************************************************
* PHASE 2: Formation and factorization of Reduced System.
********************************************************************
*
* Gather up local sections of reduced system
*
*
* The last processor does not participate in the factorization of
* the reduced system, having sent its E_i already.
IF( MYCOL .EQ. NPCOL-1 ) THEN
GOTO 14
ENDIF
*
* Initiate send of off-diag block(s) to overlap with next part.
* Off-diagonal block needed on neighboring processor to start
* algorithm.
*
IF( (MOD( MYCOL+1, 2 ) .EQ. 0) .AND. ( MYCOL .GT. 0 ) ) THEN
*
CALL SGESD2D( ICTXT, INT_ONE, INT_ONE,
$ AF( ODD_SIZE+1 ),
$ INT_ONE, 0, MYCOL-1 )
*
ENDIF
*
* Copy last diagonal block into AF storage for subsequent
* operations.
*
AF( ODD_SIZE+2 ) =
$ REAL( D( PART_OFFSET+ODD_SIZE+1 ) )
*
* Receive cont. to diagonal block that is stored on this proc.
*
IF( MYCOL.LT. NPCOL-1 ) THEN
*
CALL SGERV2D( ICTXT, INT_ONE, INT_ONE,
$ AF( ODD_SIZE+2+1 ),
$ INT_ONE, 0,
$ MYCOL+1 )
*
* Add contribution to diagonal block
*
AF( ODD_SIZE+2 ) = AF( ODD_SIZE+2 )+AF( ODD_SIZE+3 )
*
ENDIF
*
*
* *************************************
* Modification Loop
*
* The distance for sending and receiving for each level starts
* at 1 for the first level.
LEVEL_DIST = 1
*
* Do until this proc is needed to modify other procs' equations
*
12 CONTINUE
IF( MOD( (MYCOL+1)/LEVEL_DIST, 2) .NE. 0 ) GOTO 11
*
* Receive and add contribution to diagonal block from the left
*
IF( MYCOL-LEVEL_DIST .GE. 0 ) THEN
CALL SGERV2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ),
$ INT_ONE, 0, MYCOL-LEVEL_DIST )
*
AF( ODD_SIZE+2 ) = AF( ODD_SIZE+2 )+WORK( 1 )
*
ENDIF
*
* Receive and add contribution to diagonal block from the right
*
IF( MYCOL+LEVEL_DIST .LT. NPCOL-1 ) THEN
CALL SGERV2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ),
$ INT_ONE, 0, MYCOL+LEVEL_DIST )
*
AF( ODD_SIZE+2 ) = AF( ODD_SIZE+2 )+WORK( 1 )
*
ENDIF
*
LEVEL_DIST = LEVEL_DIST*2
*
GOTO 12
11 CONTINUE
* [End of GOTO Loop]
*
*
* *********************************
* Calculate and use this proc's blocks to modify other procs'...
IF( AF( ODD_SIZE+2 ) .EQ. ZERO ) THEN
INFO = NPCOL + MYCOL
ENDIF
*
* ****************************************************************
* Receive offdiagonal block from processor to right.
* If this is the first group of processors, the receive comes
* from a different processor than otherwise.
*
IF( LEVEL_DIST .EQ. 1 )THEN
COMM_PROC = MYCOL + 1
*
* Move block into place that it will be expected to be for
* calcs.
*
AF( ODD_SIZE+3 ) = AF( ODD_SIZE+1 )
*
ELSE
COMM_PROC = MYCOL + LEVEL_DIST/2
ENDIF
*
IF( MYCOL/LEVEL_DIST .LE. (NPCOL-1)/LEVEL_DIST-2 )THEN
*
CALL SGERV2D( ICTXT, INT_ONE, INT_ONE,
$ AF( ODD_SIZE+1 ),
$ INT_ONE, 0, COMM_PROC )
*
IF( INFO .EQ. 0 ) THEN
*
*
* Modify upper off_diagonal block with diagonal block
*
*
AF( ODD_SIZE+1 ) = AF( ODD_SIZE+1 )/AF( ODD_SIZE+2 )
*
ENDIF
* End of "if ( info.eq.0 ) then"
*
* Calculate contribution from this block to next diagonal block
*
WORK( 1 ) = -ONE*AF( ODD_SIZE+1 )*AF( ODD_SIZE+2 )
$ *( AF( ODD_SIZE+1 ) )
*
* Send contribution to diagonal block's owning processor.
*
CALL SGESD2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE,
$ 0, MYCOL+LEVEL_DIST )
*
ENDIF
* End of "if( mycol/level_dist .le. (npcol-1)/level_dist-2 )..."
*
*
* ****************************************************************
* Receive off_diagonal block from left and use to finish with this
* processor.
*
IF( (MYCOL/LEVEL_DIST .GT. 0 ).AND.
$ ( MYCOL/LEVEL_DIST .LE. (NPCOL-1)/LEVEL_DIST-1 ) ) THEN
*
IF( LEVEL_DIST .GT. 1)THEN
*
* Receive offdiagonal block(s) from proc level_dist/2 to the left
*
CALL SGERV2D( ICTXT, INT_ONE, INT_ONE,
$ AF( ODD_SIZE+2+1 ),
$ INT_ONE, 0, MYCOL-LEVEL_DIST/2 )
*
ENDIF
*
*
IF( INFO.EQ.0 ) THEN
*
* Use diagonal block(s) to modify this offdiagonal block
*
AF( ODD_SIZE+3 ) = ( AF( ODD_SIZE+3 ) )
$ /AF( ODD_SIZE+2 )
*
ENDIF
* End of "if( info.eq.0 ) then"
*
* Use offdiag block(s) to calculate modification to diag block
* of processor to the left
*
WORK( 1 ) = -ONE*AF( ODD_SIZE+3 )*AF( ODD_SIZE+2 )
$ *( AF( ODD_SIZE+3 ) )
*
* Send contribution to diagonal block's owning processor.
*
CALL SGESD2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE,
$ 0, MYCOL-LEVEL_DIST )
*
* *******************************************************
*
IF( MYCOL/LEVEL_DIST .LE. (NPCOL-1)/LEVEL_DIST-2 ) THEN
*
* Decide which processor offdiagonal block(s) goes to
*
IF( ( MOD( MYCOL/( 2*LEVEL_DIST ),2 )) .EQ.0 ) THEN
COMM_PROC = MYCOL + LEVEL_DIST
ELSE
COMM_PROC = MYCOL - LEVEL_DIST
ENDIF
*
* Use offdiagonal blocks to calculate offdiag
* block to send to neighboring processor. Depending
* on circumstances, may need to transpose the matrix.
*
WORK( 1 ) = -ONE*AF( ODD_SIZE+3 )
$ * AF( ODD_SIZE+2 )
$ * AF( ODD_SIZE+1 )
*
* Send contribution to offdiagonal block's owning processor.
*
CALL SGESD2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE,
$ 0, COMM_PROC )
*
ENDIF
*
ENDIF
* End of "if( mycol/level_dist.le. (npcol-1)/level_dist -1 )..."
*
14 CONTINUE
*
*
1000 CONTINUE
*
*
* Free BLACS space used to hold standard-form grid.
*
IF( ICTXT_SAVE .NE. ICTXT_NEW ) THEN
CALL BLACS_GRIDEXIT( ICTXT_NEW )
ENDIF
*
1234 CONTINUE
*
* Restore saved input parameters
*
ICTXT = ICTXT_SAVE
NP = NP_SAVE
*
* Output minimum 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 )
ENDIF
*
*
RETURN
*
* End of PSPTTRF
*
END