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SUBROUTINE ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
$ LDX, FERR, BERR, WORK, RWORK, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ WORK( * ), X( LDX, * )
* ..
*
* Purpose
* =======
*
* ZPORFS improves the computed solution to a system of linear
* equations when the coefficient matrix is Hermitian positive definite,
* and provides error bounds and backward error estimates for the
* solution.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* A (input) COMPLEX*16 array, dimension (LDA,N)
* The Hermitian matrix A. If UPLO = 'U', the leading N-by-N
* upper triangular part of A contains the upper triangular part
* of the matrix A, and the strictly lower triangular part of A
* is not referenced. If UPLO = 'L', the leading N-by-N lower
* triangular part of A contains the lower triangular part of
* the matrix A, and the strictly upper triangular part of A is
* not referenced.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* AF (input) COMPLEX*16 array, dimension (LDAF,N)
* The triangular factor U or L from the Cholesky factorization
* A = U**H*U or A = L*L**H, as computed by ZPOTRF.
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* B (input) COMPLEX*16 array, dimension (LDB,NRHS)
* The right hand side matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
* On entry, the solution matrix X, as computed by ZPOTRS.
* On exit, the improved solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* FERR (output) DOUBLE PRECISION array, dimension (NRHS)
* The estimated forward error bound for each solution vector
* X(j) (the j-th column of the solution matrix X).
* If XTRUE is the true solution corresponding to X(j), FERR(j)
* is an estimated upper bound for the magnitude of the largest
* element in (X(j) - XTRUE) divided by the magnitude of the
* largest element in X(j). The estimate is as reliable as
* the estimate for RCOND, and is almost always a slight
* overestimate of the true error.
*
* BERR (output) DOUBLE PRECISION array, dimension (NRHS)
* The componentwise relative backward error of each solution
* vector X(j) (i.e., the smallest relative change in
* any element of A or B that makes X(j) an exact solution).
*
* WORK (workspace) COMPLEX*16 array, dimension (2*N)
*
* RWORK (workspace) DOUBLE PRECISION array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Internal Parameters
* ===================
*
* ITMAX is the maximum number of steps of iterative refinement.
*
* ====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
COMPLEX*16 ONE
PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER COUNT, I, J, K, KASE, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
COMPLEX*16 ZDUM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZLACN2, ZPOTRS
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DIMAG, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Statement Functions ..
DOUBLE PRECISION CABS1
* ..
* .. Statement Function definitions ..
CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZPORFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 140 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - A * X
*
CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(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.
*
DO 30 I = 1, N
RWORK( I ) = CABS1( B( I, J ) )
30 CONTINUE
*
* Compute abs(A)*abs(X) + abs(B).
*
IF( UPPER ) THEN
DO 50 K = 1, N
S = ZERO
XK = CABS1( X( K, J ) )
DO 40 I = 1, K - 1
RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
40 CONTINUE
RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
XK = CABS1( X( K, J ) )
RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK
DO 60 I = K + 1, N
RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
60 CONTINUE
RWORK( K ) = RWORK( K ) + S
70 CONTINUE
END IF
S = ZERO
DO 80 I = 1, N
IF( RWORK( I ).GT.SAFE2 ) THEN
S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
ELSE
S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
$ ( RWORK( I )+SAFE1 ) )
END IF
80 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(A))*
* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(A) is the inverse of 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(A)*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(A)*abs(X) + abs(B) is less than SAFE2.
*
* Use ZLACN2 to estimate the infinity-norm of the matrix
* inv(A) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
DO 90 I = 1, N
IF( RWORK( I ).GT.SAFE2 ) THEN
RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
ELSE
RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
$ SAFE1
END IF
90 CONTINUE
*
KASE = 0
100 CONTINUE
CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(A').
*
CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
DO 110 I = 1, N
WORK( I ) = RWORK( I )*WORK( I )
110 CONTINUE
ELSE IF( KASE.EQ.2 ) THEN
*
* Multiply by inv(A)*diag(W).
*
DO 120 I = 1, N
WORK( I ) = RWORK( I )*WORK( I )
120 CONTINUE
CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
END IF
GO TO 100
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 130 I = 1, N
LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
130 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
140 CONTINUE
*
RETURN
*
* End of ZPORFS
*
END
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