SUBROUTINE PDGERFS( TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF,
$ JAF, DESCAF, IPIV, B, IB, JB, DESCB, X, IX,
$ JX, DESCX, FERR, BERR, WORK, LWORK, IWORK,
$ LIWORK, 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 TRANS
INTEGER IA, IAF, IB, IX, INFO, JA, JAF, JB, JX,
$ LIWORK, LWORK, N, NRHS
* ..
* .. Array Arguments ..
INTEGER DESCA( * ), DESCAF( * ), DESCB( * ),
$ DESCX( * ),IPIV( * ), IWORK( * )
DOUBLE PRECISION A( * ), AF( * ), B( * ), BERR( * ), FERR( * ),
$ WORK( * ), X( * )
* ..
*
* Purpose
* =======
*
* PDGERFS improves the computed solution to a system of linear
* equations and provides error bounds and backward error estimates for
* the solutions.
*
* 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
* =========
*
* 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)
*
*
* 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) DOUBLE PRECISION 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 distributed
* matrix sub( 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.
*
* AF (local input) DOUBLE PRECISION pointer into the local
* memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)).
* This array contains the local pieces of the distributed
* factors of the matrix sub( A ) = P * L * U as computed by
* PDGETRF.
*
* IAF (global input) INTEGER
* The row index in the global array AF indicating the first
* row of sub( AF ).
*
* JAF (global input) INTEGER
* The column index in the global array AF indicating the
* first column of sub( AF ).
*
* DESCAF (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix AF.
*
* IPIV (local input) INTEGER array of dimension LOCr(M_AF)+MB_AF.
* This array contains the pivoting information as computed
* by PDGETRF. IPIV(i) -> The global row local row i
* was swapped with. This array is tied to the distributed
* matrix A.
*
* B (local input) DOUBLE PRECISION pointer into the local
* memory to an array of local dimension
* (LLD_B,LOCc(JB+NRHS-1)). This array contains the local
* pieces of the distributed matrix of 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 and output) DOUBLE PRECISION pointer into the
* local memory to an array of local dimension
* (LLD_X,LOCc(JX+NRHS-1)). On entry, this array contains
* the local pieces of the distributed matrix solution
* sub( X ). On exit, the improved solution vectors.
*
* 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) DOUBLE PRECISION array of local dimension
* LOCc(JB+NRHS-1).
* The estimated forward error bound for each solution vector
* of sub( X ). If XTRUE is the true solution corresponding
* to sub( X ), FERR is an estimated upper bound for the
* magnitude of the largest element in (sub( X ) - XTRUE)
* divided by the magnitude of the largest element 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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.
*
* Internal Parameters
* ===================
*
* ITMAX is the maximum number of steps of iterative refinement.
*
* Notes
* =====
*
* This routine temporarily returns when N <= 1.
*
* The distributed submatrices op( A ) and op( AF ) (respectively
* sub( X ) and sub( B ) ) should be distributed the same way on the
* same processes. These conditions ensure that sub( A ) and sub( AF )
* (resp. sub( X ) and sub( B ) ) are "perfectly" aligned.
*
* Moreover, this routine requires the distributed submatrices sub( A ),
* sub( AF ), 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( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( 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 )
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION TWO, THREE
PARAMETER ( TWO = 2.0D+0, THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, NOTRAN
CHARACTER TRANST
INTEGER COUNT, IACOL, IAFCOL, IAFROW, IAROW, IXBCOL,
$ IXBROW, IXCOL, IXROW, ICOFFA, ICOFFAF, ICOFFB,
$ ICOFFX, ICTXT, ICURCOL, IDUM, II, IIXB, IIW,
$ IOFFXB, IPB, IPR, IPV, IROFFA, IROFFAF, IROFFB,
$ IROFFX, IW, J, JBRHS, JJ, JJFBE, JJXB, JN, JW,
$ K, KASE, LDXB, LIWMIN, LWMIN, MYCOL, MYRHS,
$ MYROW, NP, NP0, NPCOL, NPMOD, NPROW, NZ
DOUBLE PRECISION EPS, EST, LSTRES, S, SAFE1, SAFE2, SAFMIN
* ..
* .. Local Arrays ..
INTEGER DESCW( DLEN_ ), IDUM1( 5 ), IDUM2( 5 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ICEIL, INDXG2P, NUMROC
DOUBLE PRECISION PDLAMCH
EXTERNAL ICEIL, INDXG2P, LSAME, NUMROC, PDLAMCH
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, DGAMX2D,
$ DGEBR2D, DGEBS2D, INFOG2L, PCHK2MAT,
$ PDAGEMV, PDAXPY, PDCOPY, PDGEMV,
$ PDGETRS, PDLACON, PXERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, ICHAR, MAX, MIN, MOD
* ..
* .. Executable Statements ..
*
* Get grid parameters
*
ICTXT = DESCA( CTXT_ )
CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
* Test the input parameters.
*
NOTRAN = LSAME( TRANS, 'N' )
*
INFO = 0
IF( NPROW.EQ.-1 ) THEN
INFO = -(700+CTXT_)
ELSE
CALL CHK1MAT( N, 2, N, 2, IA, JA, DESCA, 7, INFO )
CALL CHK1MAT( N, 2, N, 2, IAF, JAF, DESCAF, 11, INFO )
CALL CHK1MAT( N, 2, NRHS, 3, IB, JB, DESCB, 16, INFO )
CALL CHK1MAT( N, 2, NRHS, 3, IX, JX, DESCX, 20, INFO )
IF( INFO.EQ.0 ) THEN
IROFFA = MOD( IA-1, DESCA( MB_ ) )
ICOFFA = MOD( JA-1, DESCA( NB_ ) )
IROFFAF = MOD( IAF-1, DESCAF( MB_ ) )
ICOFFAF = MOD( JAF-1, DESCAF( 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 )
IAFCOL = INDXG2P( JAF, DESCAF( NB_ ), MYCOL,
$ DESCAF( CSRC_ ), NPCOL )
IAFROW = INDXG2P( IAF, DESCAF( MB_ ), MYROW,
$ DESCAF( RSRC_ ), NPROW )
IACOL = INDXG2P( JA, DESCA( NB_ ), MYCOL, DESCA( CSRC_ ),
$ NPCOL )
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
LIWMIN = NPMOD
WORK( 1 ) = DBLE( LWMIN )
IWORK( 1 ) = LIWMIN
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
IF( ( .NOT.NOTRAN ) .AND. ( .NOT.LSAME( TRANS, 'T' ) ) .AND.
$ ( .NOT.LSAME( TRANS, 'C' ) ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( IROFFA.NE.0 ) THEN
INFO = -5
ELSE IF( ICOFFA.NE.0 ) THEN
INFO = -6
ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN
INFO = -( 700 + NB_ )
ELSE IF( DESCA( MB_ ).NE.DESCAF( MB_ ) ) THEN
INFO = -( 1100 + MB_ )
ELSE IF( IROFFAF.NE.0 .OR. IAROW.NE.IAFROW ) THEN
INFO = -9
ELSE IF( DESCA( NB_ ).NE.DESCAF( NB_ ) ) THEN
INFO = -( 1100 + NB_ )
ELSE IF( ICOFFAF.NE.0 .OR. IACOL.NE.IAFCOL ) THEN
INFO = -10
ELSE IF( ICTXT.NE.DESCAF( CTXT_ ) ) THEN
INFO = -( 1100 + CTXT_ )
ELSE IF( IROFFA.NE.IROFFB .OR. IAROW.NE.IXBROW ) THEN
INFO = -14
ELSE IF( DESCA( MB_ ).NE.DESCB( MB_ ) ) THEN
INFO = -( 1600 + MB_ )
ELSE IF( ICTXT.NE.DESCB( CTXT_ ) ) THEN
INFO = -( 1600 + CTXT_ )
ELSE IF( DESCB( MB_ ).NE.DESCX( MB_ ) ) THEN
INFO = -( 2000 + MB_ )
ELSE IF( IROFFX.NE.0 .OR. IXBROW.NE.IXROW ) THEN
INFO = -18
ELSE IF( DESCB( NB_ ).NE.DESCX( NB_ ) ) THEN
INFO = -( 2000 + NB_ )
ELSE IF( ICOFFB.NE.ICOFFX .OR. IXBCOL.NE.IXCOL ) THEN
INFO = -19
ELSE IF( ICTXT.NE.DESCX( CTXT_ ) ) THEN
INFO = -( 2000 + CTXT_ )
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -24
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -26
END IF
END IF
*
IF( NOTRAN ) THEN
IDUM1( 1 ) = ICHAR( 'N' )
ELSE IF( LSAME( TRANS, 'T' ) ) THEN
IDUM1( 1 ) = ICHAR( 'T' )
ELSE
IDUM1( 1 ) = ICHAR( 'C' )
END IF
IDUM2( 1 ) = 1
IDUM1( 2 ) = N
IDUM2( 2 ) = 2
IDUM1( 3 ) = NRHS
IDUM2( 3 ) = 3
IF( LWORK.EQ.-1 ) THEN
IDUM1( 4 ) = -1
ELSE
IDUM1( 4 ) = 1
END IF
IDUM2( 4 ) = 24
IF( LIWORK.EQ.-1 ) THEN
IDUM1( 5 ) = -1
ELSE
IDUM1( 5 ) = 1
END IF
IDUM2( 5 ) = 26
CALL PCHK2MAT( N, 2, N, 2, IA, JA, DESCA, 7, N, 2, N, 2, IAF,
$ JAF, DESCAF, 11, 5, IDUM1, IDUM2, INFO )
CALL PCHK2MAT( N, 2, NRHS, 3, IB, JB, DESCB, 16, N, 2, NRHS, 3,
$ IX, JX, DESCX, 20, 5, IDUM1, IDUM2, INFO )
END IF
IF( INFO.NE.0 ) THEN
CALL PXERBLA( ICTXT, 'PDGERFS', -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 = 1 + maximum number of nonzero entries in each row of sub( A )
*
NZ = N + 1
EPS = PDLAMCH( ICTXT, 'Epsilon' )
SAFMIN = PDLAMCH( 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 100 K = 0, JBRHS-1
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = sub(B) - op(sub(A)) * sub(X),
* where op(sub(A)) = sub(A), or sub(A)' (A**T or A**H),
* depending on TRANS.
*
CALL PDCOPY( N, B, IB, JB+K, DESCB, 1, WORK( IPR ), IW, JW,
$ DESCW, 1 )
CALL PDGEMV( TRANS, N, N, -ONE, A, IA, JA, DESCA, X, IX,
$ JX+K, DESCX, 1, ONE, 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 30 II = IIXB, IIXB + NP - 1
WORK( IIW+II-IIXB ) = ABS( B( II+IOFFXB ) )
30 CONTINUE
END IF
END IF
*
CALL PDAGEMV( TRANS, N, 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 40 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
40 CONTINUE
END IF
END IF
*
CALL DGAMX2D( ICTXT, 'All', ' ', 1, 1, S, 1, IDUM, IDUM, 1,
$ -1, MYCOL )
IF( MYCOL.EQ.IXBCOL )
$ BERR( JJFBE ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J+K) is larger than machine epsilon,
* and
* 2) BERR(J+K) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( S.GT.EPS .AND. TWO*S.LE.LSTRES .AND. COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL PDGETRS( TRANS, N, 1, AF, IAF, JAF, DESCAF, IPIV,
$ WORK( IPR ), IW, JW, DESCW, INFO )
CALL PDAXPY( N, ONE, WORK( IPR ), IW, JW, DESCW, 1, X, IX,
$ JX+K, DESCX, 1 )
LSTRES = S
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(sub(X) - XTRUE) / norm(sub(X)) .le. FERR =
* norm( abs(inv(op(sub(A))))*
* ( abs(R) + NZ*EPS*(
* abs(op(sub(A)))*abs(sub(X))+abs(sub(B)))))/norm(sub(X))
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(op(sub(A))) is the inverse of op(sub(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 sub(A),
* plus 1
* EPS is machine epsilon
*
* The i-th component of
* abs(R)+NZ*EPS*(abs(op(sub(A)))*abs(sub(X))+abs(sub(B)))
* is incremented by SAFE1 if the i-th component of
* abs(op(sub(A)))*abs(sub(X)) + abs(sub(B)) is less than
* SAFE2.
*
* Use PDLACON to estimate the infinity-norm of the matrix
* inv(op(sub(A))) * diag(W), where
* W = abs(R)+NZ*EPS*(abs(op(sub(A)))*abs(sub(X))+abs(sub(B))).
*
IF( MYCOL.EQ.IXBCOL ) THEN
IF( NP.GT.0 ) THEN
DO 50 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
50 CONTINUE
END IF
END IF
*
KASE = 0
60 CONTINUE
IF( MYCOL.EQ.IXBCOL ) THEN
CALL DGEBS2D( ICTXT, 'Rowwise', ' ', NP, 1, WORK( IPR ),
$ DESCW( LLD_ ) )
ELSE
CALL DGEBR2D( ICTXT, 'Rowwise', ' ', NP, 1, WORK( IPR ),
$ DESCW( LLD_ ), MYROW, IXBCOL )
END IF
DESCW( CSRC_ ) = MYCOL
CALL PDLACON( 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(sub(A))').
*
CALL PDGETRS( TRANST, N, 1, AF, IAF, JAF, DESCAF,
$ IPIV, WORK( IPR ), IW, JW, DESCW, INFO )
*
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
ELSE
*
* Multiply by inv(op(sub(A)))*diag(W).
*
IF( MYCOL.EQ.IXBCOL ) THEN
IF( NP.GT.0 ) THEN
DO 80 II = IIW-1, IIW+NP-2
WORK( IPR+II ) = WORK( IPB+II )*WORK( IPR+II )
80 CONTINUE
END IF
END IF
*
CALL PDGETRS( TRANS, N, 1, AF, IAF, JAF, DESCAF, IPIV,
$ WORK( IPR ), IW, JW, DESCW, INFO )
END IF
GO TO 60
END IF
*
* Normalize error.
*
LSTRES = ZERO
IF( MYCOL.EQ.IXBCOL ) THEN
IF( NP.GT.0 ) THEN
DO 90 II = IIXB, IIXB+NP-1
LSTRES = MAX( LSTRES, ABS( X( IOFFXB+II ) ) )
90 CONTINUE
END IF
CALL DGAMX2D( 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
*
100 CONTINUE
*
ICURCOL = MOD( IXBCOL+1, NPCOL )
*
* Do for each right hand side
*
DO 200 J = JN+1, JB+NRHS-1, DESCB( NB_ )
JBRHS = MIN( JB+NRHS-J, DESCB( NB_ ) )
DESCW( CSRC_ ) = ICURCOL
*
DO 190 K = 0, JBRHS-1
*
COUNT = 1
LSTRES = THREE
110 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = sub(B) - op(sub(A)) * sub(X),
* where op(sub(A)) = sub(A), or sub(A)' (A**T or A**H),
* depending on TRANS.
*
CALL PDCOPY( N, B, IB, J+K, DESCB, 1, WORK( IPR ), IW, JW,
$ DESCW, 1 )
CALL PDGEMV( TRANS, N, N, -ONE, A, IA, JA, DESCA, X,
$ IX, J+K, DESCX, 1, ONE, WORK( IPR ), IW, JW,
$ DESCW, 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) (abs(R(i))/(abs(op(sub(A)))*abs(sub(X)) +
* abs(sub(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.ICURCOL ) THEN
IF( NP.GT.0 ) THEN
DO 120 II = IIXB, IIXB+NP-1
WORK( IIW+II-IIXB ) = ABS( B( II+IOFFXB ) )
120 CONTINUE
END IF
END IF
*
CALL PDAGEMV( TRANS, N, N, ONE, A, IA, JA, DESCA, X, IX,
$ J+K, DESCX, 1, ONE, WORK( IPB ), IW, JW,
$ DESCW, 1 )
*
S = ZERO
IF( MYCOL.EQ.ICURCOL ) THEN
IF( NP.GT.0 )THEN
DO 130 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
130 CONTINUE
END IF
END IF
*
CALL DGAMX2D( ICTXT, 'All', ' ', 1, 1, S, 1, IDUM, IDUM, 1,
$ -1, MYCOL )
IF( MYCOL.EQ.ICURCOL )
$ BERR( JJFBE ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J+K) is larger than machine epsilon,
* and
* 2) BERR(J+K) decreased by at least a factor of 2 during
* the last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( S.GT.EPS .AND. TWO*S.LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL PDGETRS( TRANS, N, 1, AF, IAF, JAF, DESCAF, IPIV,
$ WORK( IPR ), IW, JW, DESCW, INFO )
CALL PDAXPY( N, ONE, WORK( IPR ), IW, JW, DESCW, 1, X,
$ IX, J+K, DESCX, 1 )
LSTRES = S
COUNT = COUNT + 1
GO TO 110
END IF
*
* Bound error from formula
*
* norm(sub(X) - XTRUE) / norm(sub(X)) .le. FERR =
* norm( abs(inv(op(sub(A))))*
* ( abs(R) + NZ*EPS*(
* abs(op(sub(A)))*abs(sub(X))+abs(sub(B)))))/norm(sub(X))
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(op(sub(A))) is the inverse of op(sub(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 sub(A),
* plus 1
* EPS is machine epsilon
*
* The i-th component of
* abs(R)+NZ*EPS*(abs(op(sub(A)))*abs(sub(X))+abs(sub(B)))
* is incremented by SAFE1 if the i-th component of
* abs(op(sub(A)))*abs(sub(X)) + abs(sub(B)) is less than
* SAFE2.
*
* Use PDLACON to estimate the infinity-norm of the matrix
* inv(op(sub(A))) * diag(W), where
* W = abs(R)+NZ*EPS*(abs(op(sub(A)))*abs(sub(X))+abs(sub(B))).
*
IF( MYCOL.EQ.ICURCOL ) THEN
IF( NP.GT.0 ) THEN
DO 140 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
140 CONTINUE
END IF
END IF
*
KASE = 0
150 CONTINUE
IF( MYCOL.EQ.ICURCOL ) THEN
CALL DGEBS2D( ICTXT, 'Rowwise', ' ', NP, 1, WORK( IPR ),
$ DESCW( LLD_ ) )
ELSE
CALL DGEBR2D( ICTXT, 'Rowwise', ' ', NP, 1, WORK( IPR ),
$ DESCW( LLD_ ), MYROW, ICURCOL )
END IF
DESCW( CSRC_ ) = MYCOL
CALL PDLACON( N, WORK( IPV ), IW, JW, DESCW, WORK( IPR ),
$ IW, JW, DESCW, IWORK, EST, KASE )
DESCW( CSRC_ ) = ICURCOL
*
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(op(sub(A))').
*
CALL PDGETRS( TRANST, N, 1, AF, IAF, JAF, DESCAF,
$ IPIV, WORK( IPR ), IW, JW, DESCW, INFO )
*
IF( MYCOL.EQ.ICURCOL ) THEN
IF( NP.GT.0 ) THEN
DO 160 II = IIW-1, IIW+NP-2
WORK( IPR+II ) = WORK( IPB+II )*
$ WORK( IPR+II )
160 CONTINUE
END IF
END IF
ELSE
*
* Multiply by inv(op(sub(A)))*diag(W).
*
IF( MYCOL.EQ.ICURCOL ) THEN
IF( NP.GT.0 ) THEN
DO 170 II = IIW-1, IIW+NP-2
WORK( IPR+II ) = WORK( IPB+II )*
$ WORK( IPR+II )
170 CONTINUE
END IF
END IF
*
CALL PDGETRS( TRANS, N, 1, AF, IAF, JAF, DESCAF,
$ IPIV, WORK( IPR ), IW, JW, DESCW,
$ INFO )
END IF
GO TO 150
END IF
*
* Normalize error.
*
LSTRES = ZERO
IF( MYCOL.EQ.ICURCOL ) THEN
IF( NP.GT.0 ) THEN
DO 180 II = IIXB, IIXB+NP-1
LSTRES = MAX( LSTRES, ABS( X( IOFFXB+II ) ) )
180 CONTINUE
END IF
CALL DGAMX2D( 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
*
190 CONTINUE
*
ICURCOL = MOD( ICURCOL+1, NPCOL )
*
200 CONTINUE
*
WORK( 1 ) = DBLE( LWMIN )
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of PDGERFS
*
END