SUBROUTINE PSTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, $ B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, $ BERR, WORK, LWORK, IWORK, LIWORK, INFO ) * * -- ScaLAPACK routine (version 1.7) -- * University of Tennessee, Knoxville, Oak Ridge National Laboratory, * and University of California, Berkeley. * May 1, 1997 * * .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER INFO, IA, IB, IX, JA, JB, JX, LIWORK, LWORK, $ N, NRHS * .. * .. Array Arguments .. INTEGER DESCA( * ), DESCB( * ), DESCX( * ), IWORK( * ) REAL A( * ), B( * ), BERR( * ), FERR( * ), $ WORK( * ), X( * ) * .. * * Purpose * ======= * * PSTRRFS provides error bounds and backward error estimates for the * solution to a system of linear equations with a triangular * coefficient matrix. * * The solution matrix X must be computed by PSTRTRS or some other * means before entering this routine. PSTRRFS does not do iterative * refinement because doing so cannot improve the backward error. * * 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 * * In the following comments, sub( A ), sub( X ) and sub( B ) denote * respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and * B(IB:IB+N-1,JB:JB+NRHS-1). * * Arguments * ========= * * UPLO (global input) CHARACTER*1 * = 'U': sub( A ) is upper triangular; * = 'L': sub( A ) is lower triangular. * * TRANS (global input) CHARACTER*1 * Specifies the form of the system of equations. * = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose) * = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose) * = 'C': sub( A )**T * sub( X ) = sub( B ) * (Conjugate transpose = Transpose) * * * DIAG (global input) CHARACTER*1 * = 'N': sub( A ) is non-unit triangular; * = 'U': sub( A ) is unit triangular. * * N (global input) INTEGER * The order of the matrix sub( A ). N >= 0. * * NRHS (global input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrices sub( B ) and sub( X ). NRHS >= 0. * * A (local input) REAL pointer into the local memory * to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This * array contains the local pieces of the original triangular * distributed matrix sub( A ). * If UPLO = 'U', the leading N-by-N upper triangular part of * sub( A ) contains the upper triangular part of the matrix, * and its strictly lower triangular part is not referenced. * If UPLO = 'L', the leading N-by-N lower triangular part of * sub( A ) contains the lower triangular part of the distribu- * ted matrix, and its strictly upper triangular part is not * referenced. * If DIAG = 'U', the diagonal elements of sub( A ) are also * not referenced and are assumed to be 1. * * 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. * * B (local input) REAL pointer into the local memory * to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). * On entry, this array contains the the local pieces of the * right hand sides sub( B ). * * 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. * * X (local input) REAL pointer into the local memory * to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). * On entry, this array contains the the local pieces of the * solution vectors sub( X ). * * 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. * * FERR (local output) REAL array of local dimension * LOCc(JB+NRHS-1). The estimated forward error bounds for * each solution vector of sub( X ). If XTRUE is the true * solution, FERR bounds the magnitude of the largest entry * in (sub( X ) - XTRUE) divided by the magnitude of the * largest entry in sub( X ). The estimate is as reliable as * the estimate for RCOND, and is almost always a slight * overestimate of the true error. * This array is tied to the distributed matrix X. * * BERR (local output) REAL array of local dimension * LOCc(JB+NRHS-1). The componentwise relative backward * error of each solution vector (i.e., the smallest re- * lative change in any entry of sub( A ) or sub( B ) * that makes sub( X ) an exact solution). * This array is tied to the distributed matrix X. * * WORK (local workspace/local output) REAL array, * dimension (LWORK) * On exit, WORK(1) returns the minimal and optimal LWORK. * * LWORK (local or global input) INTEGER * The dimension of the array WORK. * LWORK is local input and must be at least * LWORK >= 3*LOCr( N + MOD( IA-1, MB_A ) ). * * If LWORK = -1, then LWORK is global input and a workspace * query is assumed; the routine only calculates the minimum * and optimal size for all work arrays. Each of these * values is returned in the first entry of the corresponding * work array, and no error message is issued by PXERBLA. * * IWORK (local workspace/local output) INTEGER array, * dimension (LIWORK) * On exit, IWORK(1) returns the minimal and optimal LIWORK. * * LIWORK (local or global input) INTEGER * The dimension of the array IWORK. * LIWORK is local input and must be at least * LIWORK >= LOCr( N + MOD( IB-1, MB_B ) ). * * If LIWORK = -1, then LIWORK is global input and a workspace * query is assumed; the routine only calculates the minimum * and optimal size for all work arrays. Each of these * values is returned in the first entry of the corresponding * work array, and no error message is issued by PXERBLA. * * * 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. * * Notes * ===== * * This routine temporarily returns when N <= 1. * * The distributed submatrices sub( X ) and sub( B ) should be * distributed the same way on the same processes. These conditions * ensure that sub( X ) and sub( B ) are "perfectly" aligned. * * Moreover, this routine requires the distributed submatrices sub( A ), * sub( X ), and sub( B ) to be aligned on a block boundary, * i.e., if f(x,y) = MOD( x-1, y ): * f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, * f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and * f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. * * ===================================================================== * * .. 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 .. LOGICAL LQUERY, NOTRAN, NOUNIT, UPPER CHARACTER TRANST INTEGER IAROW, IXBCOL, IXBROW, IXCOL, IXROW, ICOFFA, $ ICOFFB, ICOFFX, ICTXT, ICURCOL, IDUM, II, IIXB, $ IIW, IOFFXB, IPB, IPR, IPV, IROFFA, IROFFB, $ IROFFX, IW, J, JBRHS, JJ, JJFBE, JJXB, JN, JW, $ K, KASE, LDXB, LIWMIN, LWMIN, MYCOL, MYRHS, $ MYROW, NP, NP0, NPCOL, NPMOD, NPROW, NZ REAL EPS, EST, LSTRES, S, SAFE1, SAFE2, SAFMIN * .. * .. Local Arrays .. INTEGER DESCW( DLEN_ ), IDUM1( 5 ), IDUM2( 5 ) * .. * .. External Functions .. LOGICAL LSAME INTEGER ICEIL, INDXG2P, NUMROC REAL PSLAMCH EXTERNAL ICEIL, INDXG2P, LSAME, NUMROC, PSLAMCH * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, INFOG2L, $ PCHK1MAT, PCHK2MAT, PSATRMV, PSAXPY, $ PSCOPY, PSLACON, PSTRSV, PSTRMV, $ PXERBLA, SGAMX2D, SGEBR2D, SGEBS2D * .. * .. Intrinsic Functions .. INTRINSIC ABS, ICHAR, MAX, MIN, MOD, REAL * .. * .. Executable Statements .. * * Get grid parameters * ICTXT = DESCA( CTXT_ ) CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) * * Test the input parameters. * INFO = 0 IF( NPROW.EQ.-1 ) THEN INFO = -( 900+CTXT_ ) ELSE CALL CHK1MAT( N, 4, N, 4, IA, JA, DESCA, 9, INFO ) CALL CHK1MAT( N, 4, NRHS, 5, IB, JB, DESCB, 13, INFO ) CALL CHK1MAT( N, 4, NRHS, 5, IX, JX, DESCX, 17, INFO ) IF( INFO.EQ.0 ) THEN UPPER = LSAME( UPLO, 'U' ) NOTRAN = LSAME( TRANS, 'N' ) NOUNIT = LSAME( DIAG, 'N' ) IROFFA = MOD( IA-1, DESCA( MB_ ) ) ICOFFA = MOD( JA-1, DESCA( NB_ ) ) IROFFB = MOD( IB-1, DESCB( MB_ ) ) ICOFFB = MOD( JB-1, DESCB( NB_ ) ) IROFFX = MOD( IX-1, DESCX( MB_ ) ) ICOFFX = MOD( JX-1, DESCX( NB_ ) ) IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ), $ NPROW ) CALL INFOG2L( IB, JB, DESCB, NPROW, NPCOL, MYROW, MYCOL, $ IIXB, JJXB, IXBROW, IXBCOL ) IXROW = INDXG2P( IX, DESCX( MB_ ), MYROW, DESCX( RSRC_ ), $ NPROW ) IXCOL = INDXG2P( JX, DESCX( NB_ ), MYCOL, DESCX( CSRC_ ), $ NPCOL ) NPMOD = NUMROC( N+IROFFA, DESCA( MB_ ), MYROW, IAROW, $ NPROW ) LWMIN = 3*NPMOD WORK( 1 ) = REAL( LWMIN ) LIWMIN = NPMOD IWORK( 1 ) = LIWMIN LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) * IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. $ .NOT.LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( IROFFA.NE.0 ) THEN INFO = -7 ELSE IF( ICOFFA.NE.0 ) THEN INFO = -8 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN INFO = -( 900+NB_ ) ELSE IF( IROFFA.NE.IROFFB .OR. IAROW.NE.IXBROW ) THEN INFO = -11 ELSE IF( DESCA( MB_ ).NE.DESCB( MB_ ) ) THEN INFO = -( 1300+MB_ ) ELSE IF( ICTXT.NE.DESCB( CTXT_ ) ) THEN INFO = -( 1300+CTXT_ ) ELSE IF( IROFFX.NE.0 .OR. IXBROW.NE.IXROW ) THEN INFO = -15 ELSE IF( ICOFFB.NE.ICOFFX .OR. IXBCOL.NE.IXCOL ) THEN INFO = -16 ELSE IF( DESCB( MB_ ).NE.DESCX( MB_ ) ) THEN INFO = -( 1700+MB_ ) ELSE IF( DESCB( NB_ ).NE.DESCX( NB_ ) ) THEN INFO = -( 1700+NB_ ) ELSE IF( ICTXT.NE.DESCX( CTXT_ ) ) THEN INFO = -( 1700+CTXT_ ) ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -21 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -23 END IF END IF * IF( UPPER ) THEN IDUM1( 1 ) = ICHAR( 'U' ) ELSE IDUM1( 1 ) = ICHAR( 'L' ) END IF IDUM2( 1 ) = 1 IF( NOTRAN ) THEN IDUM1( 2 ) = ICHAR( 'N' ) ELSE IF( LSAME( TRANS, 'T' ) ) THEN IDUM1( 2 ) = ICHAR( 'T' ) ELSE IDUM1( 2 ) = ICHAR( 'C' ) END IF IDUM2( 2 ) = 2 IF( NOUNIT ) THEN IDUM1( 3 ) = ICHAR( 'N' ) ELSE IDUM1( 3 ) = ICHAR( 'U' ) END IF IDUM2( 3 ) = 3 IF( LWORK.EQ.-1 ) THEN IDUM1( 4 ) = -1 ELSE IDUM1( 4 ) = 1 END IF IDUM2( 4 ) = 21 IF( LIWORK.EQ.-1 ) THEN IDUM1( 5 ) = -1 ELSE IDUM1( 5 ) = 1 END IF IDUM2( 5 ) = 23 CALL PCHK1MAT( N, 4, N, 4, IA, JA, DESCA, 9, 0, IDUM1, IDUM2, $ INFO ) CALL PCHK2MAT( N, 4, NRHS, 5, IB, JB, DESCB, 13, N, 4, NRHS, 5, $ IX, JX, DESCX, 17, 5, IDUM1, IDUM2, INFO ) END IF IF( INFO.NE.0 ) THEN CALL PXERBLA( ICTXT, 'PSTRRFS', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * JJFBE = JJXB MYRHS = NUMROC( JB+NRHS-1, DESCB( NB_ ), MYCOL, DESCB( CSRC_ ), $ NPCOL ) * * Quick return if possible * IF( N.LE.1 .OR. NRHS.EQ.0 ) THEN DO 10 JJ = JJFBE, MYRHS FERR( JJ ) = ZERO BERR( JJ ) = ZERO 10 CONTINUE RETURN END IF * IF( NOTRAN ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF * NP0 = NUMROC( N+IROFFB, DESCB( MB_ ), MYROW, IXBROW, NPROW ) CALL DESCSET( DESCW, N+IROFFB, 1, DESCA( MB_ ), 1, IXBROW, IXBCOL, $ ICTXT, MAX( 1, NP0 ) ) IPB = 1 IPR = IPB + NP0 IPV = IPR + NP0 IF( MYROW.EQ.IXBROW ) THEN IIW = 1 + IROFFB NP = NP0 - IROFFB ELSE IIW = 1 NP = NP0 END IF IW = 1 + IROFFB JW = 1 LDXB = DESCB( LLD_ ) IOFFXB = ( JJXB-1 )*LDXB * * NZ = maximum number of nonzero entries in each row of A, plus 1 * NZ = N + 1 EPS = PSLAMCH( ICTXT, 'Epsilon' ) SAFMIN = PSLAMCH( ICTXT, 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS JN = MIN( ICEIL( JB, DESCB( NB_ ) )*DESCB( NB_ ), JB+NRHS-1 ) * * Handle first block separately * JBRHS = JN - JB + 1 DO 90 K = 0, JBRHS - 1 * * Compute residual R = B - op(A) * X, * where op(A) = A or A', depending on TRANS. * CALL PSCOPY( N, X, IX, JX+K, DESCX, 1, WORK( IPR ), IW, JW, $ DESCW, 1 ) CALL PSTRMV( UPLO, TRANS, DIAG, N, A, IA, JA, DESCA, $ WORK( IPR ), IW, JW, DESCW, 1 ) CALL PSAXPY( N, -ONE, B, IB, JB+K, DESCB, 1, WORK( IPR ), IW, $ JW, DESCW, 1 ) * * Compute componentwise relative backward error from formula * * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) * * where abs(Z) is the componentwise absolute value of the matrix * or vector Z. If the i-th component of the denominator is less * than SAFE2, then SAFE1 is added to the i-th components of the * numerator and denominator before dividing. * IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 20 II = IIXB, IIXB + NP - 1 WORK( IIW+II-IIXB ) = ABS( B( II+IOFFXB ) ) 20 CONTINUE END IF END IF * CALL PSATRMV( UPLO, TRANS, DIAG, N, ONE, A, IA, JA, DESCA, X, $ IX, JX+K, DESCX, 1, ONE, WORK( IPB ), IW, JW, $ DESCW, 1 ) * S = ZERO IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 30 II = IIW - 1, IIW + NP - 2 IF( WORK( IPB+II ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( IPR+II ) ) / $ WORK( IPB+II ) ) ELSE S = MAX( S, ( ABS( WORK( IPR+II ) )+SAFE1 ) / $ ( WORK( IPB+II )+SAFE1 ) ) END IF 30 CONTINUE END IF END IF * CALL SGAMX2D( ICTXT, 'All', ' ', 1, 1, S, 1, IDUM, IDUM, 1, $ -1, MYCOL ) IF( MYCOL.EQ.IXBCOL ) $ BERR( JJFBE ) = S * * Bound error from formula * * norm(X - XTRUE) / norm(X) .le. FERR = * norm( abs(inv(op(A)))* * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) * * where * norm(Z) is the magnitude of the largest component of Z * inv(op(A)) is the inverse of op(A) * abs(Z) is the componentwise absolute value of the matrix or * vector Z * NZ is the maximum number of nonzeros in any row of A, plus 1 * EPS is machine epsilon * * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) * is incremented by SAFE1 if the i-th component of * abs(op(A))*abs(X) + abs(B) is less than SAFE2. * * Use PSLACON to estimate the infinity-norm of the matrix * inv(op(A)) * diag(W), * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) * IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 40 II = IIW - 1, IIW + NP - 2 IF( WORK( IPB+II ).GT.SAFE2 ) THEN WORK( IPB+II ) = ABS( WORK( IPR+II ) ) + $ NZ*EPS*WORK( IPB+II ) ELSE WORK( IPB+II ) = ABS( WORK( IPR+II ) ) + $ NZ*EPS*WORK( IPB+II ) + SAFE1 END IF 40 CONTINUE END IF END IF * KASE = 0 50 CONTINUE IF( MYCOL.EQ.IXBCOL ) THEN CALL SGEBS2D( ICTXT, 'Rowwise', ' ', NP, 1, WORK( IPR ), $ DESCW( LLD_ ) ) ELSE CALL SGEBR2D( ICTXT, 'Rowwise', ' ', NP, 1, WORK( IPR ), $ DESCW( LLD_ ), MYROW, IXBCOL ) END IF DESCW( CSRC_ ) = MYCOL CALL PSLACON( N, WORK( IPV ), IW, JW, DESCW, WORK( IPR ), $ IW, JW, DESCW, IWORK, EST, KASE ) DESCW( CSRC_ ) = IXBCOL * IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * * Multiply by diag(W)*inv(op(A)'). * CALL PSTRSV( UPLO, TRANST, DIAG, N, A, IA, JA, DESCA, $ WORK( IPR ), IW, JW, DESCW, 1 ) IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 60 II = IIW - 1, IIW + NP - 2 WORK( IPR+II ) = WORK( IPB+II )*WORK( IPR+II ) 60 CONTINUE END IF END IF ELSE * * Multiply by inv(op(A))*diag(W). * IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 70 II = IIW - 1, IIW + NP - 2 WORK( IPR+II ) = WORK( IPB+II )*WORK( IPR+II ) 70 CONTINUE END IF END IF CALL PSTRSV( UPLO, TRANS, DIAG, N, A, IA, JA, DESCA, $ WORK( IPR ), IW, JW, DESCW, 1 ) END IF GO TO 50 END IF * * Normalize error. * LSTRES = ZERO IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 80 II = IIXB, IIXB + NP - 1 LSTRES = MAX( LSTRES, ABS( X( IOFFXB+II ) ) ) 80 CONTINUE END IF CALL SGAMX2D( ICTXT, 'Column', ' ', 1, 1, LSTRES, 1, IDUM, $ IDUM, 1, -1, MYCOL ) IF( LSTRES.NE.ZERO ) $ FERR( JJFBE ) = EST / LSTRES * JJXB = JJXB + 1 JJFBE = JJFBE + 1 IOFFXB = IOFFXB + LDXB * END IF * 90 CONTINUE * ICURCOL = MOD( IXBCOL+1, NPCOL ) * * Do for each right hand side * DO 180 J = JN + 1, JB + NRHS - 1, DESCB( NB_ ) JBRHS = MIN( JB+NRHS-J, DESCB( NB_ ) ) DESCW( CSRC_ ) = ICURCOL * DO 170 K = 0, JBRHS - 1 * * Compute residual R = B - op(A) * X, * where op(A) = A or A', depending on TRANS. * CALL PSCOPY( N, X, IX, J+K, DESCX, 1, WORK( IPR ), IW, JW, $ DESCW, 1 ) CALL PSTRMV( UPLO, TRANS, DIAG, N, A, IA, JA, DESCA, $ WORK( IPR ), IW, JW, DESCW, 1 ) CALL PSAXPY( N, -ONE, B, IB, J+K, DESCB, 1, WORK( IPR ), $ IW, JW, DESCW, 1 ) * * Compute componentwise relative backward error from formula * * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) * * where abs(Z) is the componentwise absolute value of the * matrix or vector Z. If the i-th component of the * denominator is less than SAFE2, then SAFE1 is added to the * i-th components of the numerator and denominator before * dividing. * IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 100 II = IIXB, IIXB + NP - 1 WORK( IIW+II-IIXB ) = ABS( B( II+IOFFXB ) ) 100 CONTINUE END IF END IF * CALL PSATRMV( UPLO, TRANS, DIAG, N, ONE, A, IA, JA, DESCA, $ X, IX, J+K, DESCX, 1, ONE, WORK( IPB ), IW, $ JW, DESCW, 1 ) * S = ZERO IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 110 II = IIW - 1, IIW + NP - 2 IF( WORK( IPB+II ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( IPR+II ) ) / $ WORK( IPB+II ) ) ELSE S = MAX( S, ( ABS( WORK( IPR+II ) )+SAFE1 ) / $ ( WORK( IPB+II )+SAFE1 ) ) END IF 110 CONTINUE END IF END IF * CALL SGAMX2D( ICTXT, 'All', ' ', 1, 1, S, 1, IDUM, IDUM, 1, $ -1, MYCOL ) IF( MYCOL.EQ.IXBCOL ) $ BERR( JJFBE ) = S * * Bound error from formula * * norm(X - XTRUE) / norm(X) .le. FERR = * norm( abs(inv(op(A)))* * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))/norm(X) * * where * norm(Z) is the magnitude of the largest component of Z * inv(op(A)) is the inverse of op(A) * abs(Z) is the componentwise absolute value of the matrix * or vector Z * NZ is the maximum number of nonzeros in any row of A, * plus 1 * EPS is machine epsilon * * The i-th component of * abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) * is incremented by SAFE1 if the i-th component of * abs(op(A))*abs(X) + abs(B) is less than SAFE2. * * Use PSLACON to estimate the infinity-norm of the matrix * inv(op(A)) * diag(W), * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) * IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 120 II = IIW - 1, IIW + NP - 2 IF( WORK( IPB+II ).GT.SAFE2 ) THEN WORK( IPB+II ) = ABS( WORK( IPR+II ) ) + $ NZ*EPS*WORK( IPB+II ) ELSE WORK( IPB+II ) = ABS( WORK( IPR+II ) ) + $ NZ*EPS*WORK( IPB+II ) + SAFE1 END IF 120 CONTINUE END IF END IF * KASE = 0 130 CONTINUE IF( MYCOL.EQ.IXBCOL ) THEN CALL SGEBS2D( ICTXT, 'Rowwise', ' ', NP, 1, WORK( IPR ), $ DESCW( LLD_ ) ) ELSE CALL SGEBR2D( ICTXT, 'Rowwise', ' ', NP, 1, WORK( IPR ), $ DESCW( LLD_ ), MYROW, IXBCOL ) END IF DESCW( CSRC_ ) = MYCOL CALL PSLACON( N, WORK( IPV ), IW, JW, DESCW, WORK( IPR ), $ IW, JW, DESCW, IWORK, EST, KASE ) DESCW( CSRC_ ) = IXBCOL * IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * * Multiply by diag(W)*inv(op(A)'). * CALL PSTRSV( UPLO, TRANST, DIAG, N, A, IA, JA, DESCA, $ WORK( IPR ), IW, JW, DESCW, 1 ) IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 140 II = IIW - 1, IIW + NP - 2 WORK( IPR+II ) = WORK( IPB+II )* $ WORK( IPR+II ) 140 CONTINUE END IF END IF ELSE * * Multiply by inv(op(A))*diag(W). * IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 150 II = IIW - 1, IIW + NP - 2 WORK( IPR+II ) = WORK( IPB+II )* $ WORK( IPR+II ) 150 CONTINUE END IF END IF CALL PSTRSV( UPLO, TRANS, DIAG, N, A, IA, JA, DESCA, $ WORK( IPR ), IW, JW, DESCW, 1 ) END IF GO TO 130 END IF * * Normalize error. * LSTRES = ZERO IF( MYCOL.EQ.IXBCOL ) THEN IF( NP.GT.0 ) THEN DO 160 II = IIXB, IIXB + NP - 1 LSTRES = MAX( LSTRES, ABS( X( IOFFXB+II ) ) ) 160 CONTINUE END IF CALL SGAMX2D( ICTXT, 'Column', ' ', 1, 1, LSTRES, 1, $ IDUM, IDUM, 1, -1, MYCOL ) IF( LSTRES.NE.ZERO ) $ FERR( JJFBE ) = EST / LSTRES * JJXB = JJXB + 1 JJFBE = JJFBE + 1 IOFFXB = IOFFXB + LDXB * END IF * 170 CONTINUE * ICURCOL = MOD( ICURCOL+1, NPCOL ) * 180 CONTINUE * WORK( 1 ) = REAL( LWMIN ) IWORK( 1 ) = LIWMIN * RETURN * * End of PSTRRFS * END