SUBROUTINE PDPBSV( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB,
$ 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 ..
CHARACTER UPLO
INTEGER BW, IB, INFO, JA, LWORK, N, NRHS
* ..
* .. Array Arguments ..
INTEGER DESCA( * ), DESCB( * )
DOUBLE PRECISION A( * ), B( * ), WORK( * )
* ..
*
*
* Purpose
* =======
*
* PDPBSV solves a system of linear equations
*
* A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS)
*
* where A(1:N, JA:JA+N-1) is an N-by-N real
* banded symmetric positive definite distributed
* matrix with bandwidth BW.
*
* Cholesky factorization is used to factor a reordering of
* the matrix into L L'.
*
* See PDPBTRF and PDPBTRS for details.
*
* =====================================================================
*
* Arguments
* =========
*
* UPLO (global input) CHARACTER
* = 'U': Upper triangle of A(1:N, JA:JA+N-1) is stored;
* = 'L': Lower triangle of A(1:N, JA:JA+N-1) is stored.
*
* 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.
*
* BW (global input) INTEGER
* Number of subdiagonals in L or U. 0 <= BW <= N-1
*
* NRHS (global input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the distributed submatrix B(IB:IB+N-1, 1:NRHS).
* NRHS >= 0.
*
* A (local input/local output) DOUBLE PRECISION pointer into
* local memory to an array with first dimension
* LLD_A >=(bw+1) (stored in DESCA).
* On entry, this array contains the local pieces of the
* 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.
*
* B (local input/local output) DOUBLE PRECISION pointer into
* local memory to an array of local lead dimension lld_b>=NB.
* On entry, this array contains the
* the local pieces of the right hand sides
* B(IB:IB+N-1, 1:NRHS).
* On exit, this contains the local piece of the solutions
* distributed matrix X.
*
* IB (global input) INTEGER
* The row index in the global array B that points to the first
* row of the matrix to be operated on (which may be either
* all of B or a submatrix of B).
*
* DESCB (global and local input) INTEGER array of dimension DLEN.
* if 1D type (DTYPE_B=502), DLEN >=7;
* if 2D type (DTYPE_B=1), DLEN >= 9.
* The array descriptor for the distributed matrix B.
* Contains information of mapping of B to memory. Please
* see NOTES below for full description and options.
*
* WORK (local workspace/local output)
* DOUBLE PRECISION 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>=
* (NB+2*bw)*bw
* +max((bw*NRHS), bw*bw)
*
* 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
* 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*BW
* 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 (BW* (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
*
* Note that a consequence of this chart is that it is not possible
* for *both* DTYPE_A and DTYPE_B to be 2D_type(1), as these lead
* to opposite requirements for the orientation of the BLACS grid,
* and as noted before, the *same* BLACS context must be used in
* all descriptors in a single ScaLAPACK subroutine call.
*
* 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.
*
*
*
* =====================================================================
*
* Code Developer: Andrew J. Cleary, University of Tennessee.
* Current address: Lawrence Livermore National Labs.
* This version released: May, 1997.
*
* =====================================================================
*
* ..
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0 )
PARAMETER ( ZERO = 0.0D+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 ICTXT, MYCOL, MYROW, NB, NPCOL, NPROW,
$ WS_FACTOR
* ..
* .. External Subroutines ..
EXTERNAL PDPBTRF, PDPBTRS, PXERBLA
* ..
* .. Executable Statements ..
*
* Note: to avoid duplication, most error checking is not performed
* in this routine and is left to routines
* PDPBTRF and PDPBTRS.
*
* Begin main code
*
INFO = 0
*
* Get block size to calculate workspace requirements
*
IF( DESCA( DTYPE_ ) .EQ. BLOCK_CYCLIC_2D ) THEN
NB = DESCA( NB_ )
ICTXT = DESCA( CTXT_ )
ELSEIF( DESCA( DTYPE_ ) .EQ. 501 ) THEN
NB = DESCA( 4 )
ICTXT = DESCA( 2 )
ELSE
INFO = -( 6*100 + DTYPE_ )
CALL PXERBLA( ICTXT,
$ 'PDPBSV',
$ -INFO )
RETURN
ENDIF
*
CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
*
* Size needed for AF in factorization
*
WS_FACTOR = (NB+2*BW)*BW
*
* Factor the matrix
*
CALL PDPBTRF( UPLO, N, BW, A, JA, DESCA, WORK,
$ MIN( LWORK, WS_FACTOR ), WORK( 1+WS_FACTOR ),
$ LWORK-WS_FACTOR, INFO )
*
* Check info for error conditions
*
IF( INFO.NE.0 ) THEN
IF( INFO .LT. 0 ) THEN
CALL PXERBLA( ICTXT, 'PDPBSV', -INFO )
ENDIF
RETURN
END IF
*
* Solve the system using the factorization
*
CALL PDPBTRS( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, WORK,
$ MIN( LWORK, WS_FACTOR ), WORK( 1+WS_FACTOR),
$ LWORK-WS_FACTOR, INFO )
*
* Check info for error conditions
*
IF( INFO.NE.0 ) THEN
CALL PXERBLA( ICTXT, 'PDPBSV', -INFO )
RETURN
END IF
*
RETURN
*
* End of PDPBSV
*
END