sget07()

subroutine sget07	(	character	trans,
		integer	n,
		integer	nrhs,
		real, dimension( lda, * )	a,
		integer	lda,
		real, dimension( ldb, * )	b,
		integer	ldb,
		real, dimension( ldx, * )	x,
		integer	ldx,
		real, dimension( ldxact, * )	xact,
		integer	ldxact,
		real, dimension( * )	ferr,
		logical	chkferr,
		real, dimension( * )	berr,
		real, dimension( * )	reslts
	)

SGET07

Purpose:

 SGET07 tests the error bounds from iterative refinement for the
 computed solution to a system of equations op(A)*X = B, where A is a
 general n by n matrix and op(A) = A or A**T, depending on TRANS.

 RESLTS(1) = test of the error bound
           = norm(X - XACT) / ( norm(X) * FERR )

 A large value is returned if this ratio is not less than one.

 RESLTS(2) = residual from the iterative refinement routine
           = the maximum of BERR / ( (n+1)*EPS + (*) ), where
             (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )

Parameters

[in]	TRANS	TRANS is CHARACTER*1 Specifies the form of the system of equations. = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose = Transpose)
[in]	N	N is INTEGER The number of rows of the matrices X and XACT. N >= 0.
[in]	NRHS	NRHS is INTEGER The number of columns of the matrices X and XACT. NRHS >= 0.
[in]	A	A is REAL array, dimension (LDA,N) The original n by n matrix A.
[in]	LDA	LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
[in]	B	B is REAL array, dimension (LDB,NRHS) The right hand side vectors for the system of linear equations.
[in]	LDB	LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
[in]	X	X is REAL array, dimension (LDX,NRHS) The computed solution vectors. Each vector is stored as a column of the matrix X.
[in]	LDX	LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).
[in]	XACT	XACT is REAL array, dimension (LDX,NRHS) The exact solution vectors. Each vector is stored as a column of the matrix XACT.
[in]	LDXACT	LDXACT is INTEGER The leading dimension of the array XACT. LDXACT >= max(1,N).
[in]	FERR	FERR is REAL array, dimension (NRHS) The estimated forward error bounds for each solution vector X. If XTRUE is the true solution, FERR bounds the magnitude of the largest entry in (X - XTRUE) divided by the magnitude of the largest entry in X.
[in]	CHKFERR	CHKFERR is LOGICAL Set to .TRUE. to check FERR, .FALSE. not to check FERR. When the test system is ill-conditioned, the "true" solution in XACT may be incorrect.
[in]	BERR	BERR is REAL array, dimension (NRHS) The componentwise relative backward error of each solution vector (i.e., the smallest relative change in any entry of A or B that makes X an exact solution).
[out]	RESLTS	RESLTS is REAL array, dimension (2) The maximum over the NRHS solution vectors of the ratios: RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) RESLTS(2) = BERR / ( (n+1)*EPS + (*) )

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 163 of file sget07.f.

*
*  -- LAPACK test routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      LOGICAL            CHKFERR
      INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
     $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e+0, one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN
      INTEGER            I, IMAX, J, K
      REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ISAMAX
      REAL               SLAMCH
      EXTERNAL           lsame, isamax, slamch
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min
*     ..
*     .. Executable Statements ..
*
*     Quick exit if N = 0 or NRHS = 0.
*
      IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
         reslts( 1 ) = zero
         reslts( 2 ) = zero
         RETURN
      END IF
*
      eps = slamch( 'Epsilon' )
      unfl = slamch( 'Safe minimum' )
      ovfl = one / unfl
      notran = lsame( trans, 'N' )
*
*     Test 1:  Compute the maximum of
*        norm(X - XACT) / ( norm(X) * FERR )
*     over all the vectors X and XACT using the infinity-norm.
*
      errbnd = zero
      IF( chkferr ) THEN
         DO 30 j = 1, nrhs
            imax = isamax( n, x( 1, j ), 1 )
            xnorm = max( abs( x( imax, j ) ), unfl )
            diff = zero
            DO 10 i = 1, n
               diff = max( diff, abs( x( i, j )-xact( i, j ) ) )
 10         CONTINUE
*
            IF( xnorm.GT.one ) THEN
               GO TO 20
            ELSE IF( diff.LE.ovfl*xnorm ) THEN
               GO TO 20
            ELSE
               errbnd = one / eps
               GO TO 30
            END IF
*
 20         CONTINUE
            IF( diff / xnorm.LE.ferr( j ) ) THEN
               errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
            ELSE
               errbnd = one / eps
            END IF
 30      CONTINUE
      END IF
      reslts( 1 ) = errbnd
*
*     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
*     (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
*
      DO 70 k = 1, nrhs
         DO 60 i = 1, n
            tmp = abs( b( i, k ) )
            IF( notran ) THEN
               DO 40 j = 1, n
                  tmp = tmp + abs( a( i, j ) )*abs( x( j, k ) )
   40          CONTINUE
            ELSE
               DO 50 j = 1, n
                  tmp = tmp + abs( a( j, i ) )*abs( x( j, k ) )
   50          CONTINUE
            END IF
            IF( i.EQ.1 ) THEN
               axbi = tmp
            ELSE
               axbi = min( axbi, tmp )
            END IF
   60    CONTINUE
         tmp = berr( k ) / ( ( n+1 )*eps+( n+1 )*unfl /
     $         max( axbi, ( n+1 )*unfl ) )
         IF( k.EQ.1 ) THEN
            reslts( 2 ) = tmp
         ELSE
            reslts( 2 ) = max( reslts( 2 ), tmp )
         END IF
   70 CONTINUE
*
      RETURN
*
*     End of SGET07
*

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