sgbt05()

subroutine sgbt05	(	character	TRANS,
		integer	N,
		integer	KL,
		integer	KU,
		integer	NRHS,
		real, dimension( ldab, * )	AB,
		integer	LDAB,
		real, dimension( ldb, * )	B,
		integer	LDB,
		real, dimension( ldx, * )	X,
		integer	LDX,
		real, dimension( ldxact, * )	XACT,
		integer	LDXACT,
		real, dimension( * )	FERR,
		real, dimension( * )	BERR,
		real, dimension( * )	RESLTS
	)

SGBT05

Purpose:

 SGBT05 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 band matrix of order n with kl subdiagonals and ku
 superdiagonals 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 / ( NZ*EPS + (*) ), where
             (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
             and NZ = max. number of nonzeros in any row of A, plus 1

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, B, and XACT, and the order of the matrix A. N >= 0.
[in]	KL	KL is INTEGER The number of subdiagonals within the band of A. KL >= 0.
[in]	KU	KU is INTEGER The number of superdiagonals within the band of A. KU >= 0.
[in]	NRHS	NRHS is INTEGER The number of columns of the matrices X, B, and XACT. NRHS >= 0.
[in]	AB	AB is REAL array, dimension (LDAB,N) The original band matrix A, stored in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows: AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
[in]	LDAB	LDAB is INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1.
[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]	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 / ( NZ*EPS + (*) )

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

Definition at line 178 of file sgbt05.f.

*
*  -- LAPACK test routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS
*     ..
*     .. Array Arguments ..
      REAL               AB( LDAB, * ), 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, NZ
      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' )
      nz = min( kl+ku+2, n+1 )
*
*     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
      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
      reslts( 1 ) = errbnd
*
*     Test 2:  Compute the maximum of BERR / ( NZ*EPS + (*) ), where
*     (*) = NZ*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 = max( i-kl, 1 ), min( i+ku, n )
                  tmp = tmp + abs( ab( ku+1+i-j, j ) )*abs( x( j, k ) )
   40          CONTINUE
            ELSE
               DO 50 j = max( i-ku, 1 ), min( i+kl, n )
                  tmp = tmp + abs( ab( ku+1+j-i, 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 ) / ( nz*eps+nz*unfl / max( axbi, nz*unfl ) )
         IF( k.EQ.1 ) THEN
            reslts( 2 ) = tmp
         ELSE
            reslts( 2 ) = max( reslts( 2 ), tmp )
         END IF
   70 CONTINUE
*
      RETURN
*
*     End of SGBT05
*

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