SUBROUTINE PSQRT16( TRANS, M, N, NRHS, A, IA, JA, DESCA, X, IX, $ JX, DESCX, B, IB, JB, DESCB, RWORK, RESID ) * * -- ScaLAPACK routine (version 1.7) -- * University of Tennessee, Knoxville, Oak Ridge National Laboratory, * and University of California, Berkeley. * May 1, 1997 * * .. Scalar Arguments .. CHARACTER TRANS INTEGER IA, IB, IX, JA, JB, JX, M, N, NRHS REAL RESID * .. * .. Array Arguments .. INTEGER DESCA( * ), DESCB( * ), DESCX( * ) REAL A( * ), B( * ), RWORK( * ), X( * ) * .. * * Purpose * ======= * * PSQRT16 computes the residual for a solution of a system of linear * equations sub( A )*sub( X ) = B or sub( A' )*sub( X ) = B: * RESID = norm(B - sub( A )*sub( X ) ) / * ( max(m,n) * norm(sub( A ) ) * norm(sub( X ) ) * EPS ), * where EPS is the machine epsilon, sub( A ) denotes * A(IA:IA+N-1,JA,JA+N-1), and sub( X ) denotes * X(IX:IX+N-1, JX:JX+NRHS-1). * * Notes * ===== * * 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 * * Arguments * ========= * * TRANS (global input) CHARACTER*1 * Specifies the form of the system of equations: * = 'N': sub( A )*sub( X ) = sub( B ) * = 'T': sub( A' )*sub( X )= sub( B ), where A' is the * transpose of sub( A ). * = 'C': sub( A' )*sub( X )= B, where A' is the transpose * of sub( A ). * * M (global input) INTEGER * The number of rows to be operated on, i.e. the number of rows * of the distributed submatrix sub( A ). M >= 0. * * N (global input) INTEGER * The number of columns to be operated on, i.e. the number of * columns of the distributed submatrix sub( A ). N >= 0. * * NRHS (global input) INTEGER * The number of right hand sides, i.e., the number of columns * of the distributed submatrix sub( B ). NRHS >= 0. * * A (local input) REAL pointer into the local * memory to an array of dimension (LLD_A,LOCc(JA+N-1)). * The original M x N matrix A. * * IA (global input) INTEGER * The row index in the global array A indicating the first * row of sub( A ). * * JA (global input) INTEGER * The column index in the global array A indicating the * first column of sub( A ). * * DESCA (global and local input) INTEGER array of dimension DLEN_. * The array descriptor for the distributed matrix A. * * X (local input) REAL pointer into the local * memory to an array of dimension (LLD_X,LOCc(JX+NRHS-1)). This * array contains the local pieces of the computed solution * distributed vectors for the system of linear equations. * * IX (global input) INTEGER * The row index in the global array X indicating the first * row of sub( X ). * * JX (global input) INTEGER * The column index in the global array X indicating the * first column of sub( X ). * * DESCX (global and local input) INTEGER array of dimension DLEN_. * The array descriptor for the distributed matrix X. * * B (local input/local output) REAL pointer into * the local memory to an array of dimension * (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the * local pieces of the distributes right hand side vectors for * the system of linear equations. On exit, sub( B ) is over- * written with the difference sub( B ) - sub( A )*sub( X ) or * sub( B ) - sub( A )'*sub( X ). * * IB (global input) INTEGER * The row index in the global array B indicating the first * row of sub( B ). * * JB (global input) INTEGER * The column index in the global array B indicating the * first column of sub( B ). * * DESCB (global and local input) INTEGER array of dimension DLEN_. * The array descriptor for the distributed matrix B. * * RWORK (local workspace) REAL array, dimension (LRWORK) * LWORK >= Nq0 if TRANS = 'N', and LRWORK >= Mp0 otherwise. * * where * * IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), * IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), * IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), * Mp0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), * Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), * * INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, * MYCOL, NPROW and NPCOL can be determined by calling the * subroutine BLACS_GRIDINFO. * * RESID (global output) REAL * The maximum over the number of right hand sides of * norm( sub( B )- sub( A )*sub( X ) ) / * ( max(m,n) * norm( sub( A ) ) * norm( sub( X ) ) * EPS ). * * ===================================================================== * * .. Parameters .. 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 ) REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. INTEGER ICTXT, IDUMM, J, MYCOL, MYROW, N1, N2, NPCOL, $ NPROW REAL ANORM, BNORM, EPS, XNORM * .. * .. Local Arrays .. REAL TEMP( 2 ) * .. * .. External Functions .. LOGICAL LSAME REAL PSLAMCH, PSLANGE EXTERNAL LSAME, PSLAMCH, PSLANGE * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDINFO, PSASUM, PSGEMM, SGAMX2D * .. * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Get grid parameters * ICTXT = DESCA( CTXT_ ) CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) * * Quick exit if M = 0 or N = 0 or NRHS = 0 * IF( M.LE.0 .OR. N.LE.0 .OR. NRHS.EQ.0 ) THEN RESID = ZERO RETURN END IF * IF( LSAME( TRANS, 'T' ) .OR. LSAME( TRANS, 'C' ) ) THEN ANORM = PSLANGE( 'I', M, N, A, IA, JA, DESCA, RWORK ) N1 = N N2 = M ELSE ANORM = PSLANGE( '1', M, N, A, IA, JA, DESCA, RWORK ) N1 = M N2 = N END IF * EPS = PSLAMCH( ICTXT, 'Epsilon' ) * * Compute B - sub( A )*sub( X ) (or B - sub( A' )*sub( X ) ) and * store in B. * CALL PSGEMM( TRANS, 'No transpose', N1, NRHS, N2, -ONE, A, IA, $ JA, DESCA, X, IX, JX, DESCX, ONE, B, IB, JB, DESCB ) * * Compute the maximum over the number of right hand sides of * norm( sub( B ) - sub( A )*sub( X ) ) / * ( max(m,n) * norm( sub( A ) ) * norm( sub( X ) ) * EPS ). * RESID = ZERO DO 10 J = 1, NRHS * CALL PSASUM( N1, BNORM, B, IB, JB+J-1, DESCB, 1 ) CALL PSASUM( N2, XNORM, X, IX, JX+J-1, DESCX, 1 ) * * Only the process columns owning the vector operands will have * the correct result, the other will have zero. * TEMP( 1 ) = BNORM TEMP( 2 ) = XNORM IDUMM = 0 CALL SGAMX2D( ICTXT, 'All', ' ', 2, 1, TEMP, 2, IDUMM, IDUMM, $ -1, -1, IDUMM ) BNORM = TEMP( 1 ) XNORM = TEMP( 2 ) * * Every processes have ANORM, BNORM and XNORM now. * IF( ANORM.EQ.ZERO .AND. BNORM.EQ.ZERO ) THEN RESID = ZERO ELSE IF( ANORM.LE.ZERO .OR. XNORM.LE.ZERO ) THEN RESID = ONE / EPS ELSE RESID = MAX( RESID, ( ( BNORM / ANORM ) / XNORM ) / $ ( MAX( M, N )*EPS ) ) END IF * 10 CONTINUE * RETURN * * End of PSQRT16 * END