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