 LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine dgbt05 ( character TRANS, integer N, integer KL, integer KU, integer NRHS, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( ldxact, * ) XACT, integer LDXACT, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) RESLTS )

DGBT05

Purpose:
``` DGBT05 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 + (*) )```
Date
November 2011

Definition at line 178 of file dgbt05.f.

178 *
179 * -- LAPACK test routine (version 3.4.0) --
180 * -- LAPACK is a software package provided by Univ. of Tennessee, --
181 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
182 * November 2011
183 *
184 * .. Scalar Arguments ..
185  CHARACTER trans
186  INTEGER kl, ku, ldab, ldb, ldx, ldxact, n, nrhs
187 * ..
188 * .. Array Arguments ..
189  DOUBLE PRECISION ab( ldab, * ), b( ldb, * ), berr( * ),
190  \$ ferr( * ), reslts( * ), x( ldx, * ),
191  \$ xact( ldxact, * )
192 * ..
193 *
194 * =====================================================================
195 *
196 * .. Parameters ..
197  DOUBLE PRECISION zero, one
198  parameter ( zero = 0.0d+0, one = 1.0d+0 )
199 * ..
200 * .. Local Scalars ..
201  LOGICAL notran
202  INTEGER i, imax, j, k, nz
203  DOUBLE PRECISION axbi, diff, eps, errbnd, ovfl, tmp, unfl, xnorm
204 * ..
205 * .. External Functions ..
206  LOGICAL lsame
207  INTEGER idamax
208  DOUBLE PRECISION dlamch
209  EXTERNAL lsame, idamax, dlamch
210 * ..
211 * .. Intrinsic Functions ..
212  INTRINSIC abs, max, min
213 * ..
214 * .. Executable Statements ..
215 *
216 * Quick exit if N = 0 or NRHS = 0.
217 *
218  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
219  reslts( 1 ) = zero
220  reslts( 2 ) = zero
221  RETURN
222  END IF
223 *
224  eps = dlamch( 'Epsilon' )
225  unfl = dlamch( 'Safe minimum' )
226  ovfl = one / unfl
227  notran = lsame( trans, 'N' )
228  nz = min( kl+ku+2, n+1 )
229 *
230 * Test 1: Compute the maximum of
231 * norm(X - XACT) / ( norm(X) * FERR )
232 * over all the vectors X and XACT using the infinity-norm.
233 *
234  errbnd = zero
235  DO 30 j = 1, nrhs
236  imax = idamax( n, x( 1, j ), 1 )
237  xnorm = max( abs( x( imax, j ) ), unfl )
238  diff = zero
239  DO 10 i = 1, n
240  diff = max( diff, abs( x( i, j )-xact( i, j ) ) )
241  10 CONTINUE
242 *
243  IF( xnorm.GT.one ) THEN
244  GO TO 20
245  ELSE IF( diff.LE.ovfl*xnorm ) THEN
246  GO TO 20
247  ELSE
248  errbnd = one / eps
249  GO TO 30
250  END IF
251 *
252  20 CONTINUE
253  IF( diff / xnorm.LE.ferr( j ) ) THEN
254  errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
255  ELSE
256  errbnd = one / eps
257  END IF
258  30 CONTINUE
259  reslts( 1 ) = errbnd
260 *
261 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
262 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
263 *
264  DO 70 k = 1, nrhs
265  DO 60 i = 1, n
266  tmp = abs( b( i, k ) )
267  IF( notran ) THEN
268  DO 40 j = max( i-kl, 1 ), min( i+ku, n )
269  tmp = tmp + abs( ab( ku+1+i-j, j ) )*abs( x( j, k ) )
270  40 CONTINUE
271  ELSE
272  DO 50 j = max( i-ku, 1 ), min( i+kl, n )
273  tmp = tmp + abs( ab( ku+1+j-i, i ) )*abs( x( j, k ) )
274  50 CONTINUE
275  END IF
276  IF( i.EQ.1 ) THEN
277  axbi = tmp
278  ELSE
279  axbi = min( axbi, tmp )
280  END IF
281  60 CONTINUE
282  tmp = berr( k ) / ( nz*eps+nz*unfl / max( axbi, nz*unfl ) )
283  IF( k.EQ.1 ) THEN
284  reslts( 2 ) = tmp
285  ELSE
286  reslts( 2 ) = max( reslts( 2 ), tmp )
287  END IF
288  70 CONTINUE
289 *
290  RETURN
291 *
292 * End of DGBT05
293 *
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:53
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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