 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dpot05()

 subroutine dpot05 ( character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, 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 )

DPOT05

Purpose:
``` DPOT05 tests the error bounds from iterative refinement for the
computed solution to a system of equations A*X = B, where A is a
symmetric n by n matrix.

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(A)*abs(X) +abs(b))_i )```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular``` [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] NRHS ``` NRHS is INTEGER The number of columns of the matrices X, B, and XACT. NRHS >= 0.``` [in] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) The symmetric 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.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [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 / ( (n+1)*EPS + (*) )```

Definition at line 162 of file dpot05.f.

164*
165* -- LAPACK test routine --
166* -- LAPACK is a software package provided by Univ. of Tennessee, --
167* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
168*
169* .. Scalar Arguments ..
170 CHARACTER UPLO
171 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
172* ..
173* .. Array Arguments ..
174 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
175 \$ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
176* ..
177*
178* =====================================================================
179*
180* .. Parameters ..
181 DOUBLE PRECISION ZERO, ONE
182 parameter( zero = 0.0d+0, one = 1.0d+0 )
183* ..
184* .. Local Scalars ..
185 LOGICAL UPPER
186 INTEGER I, IMAX, J, K
187 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
188* ..
189* .. External Functions ..
190 LOGICAL LSAME
191 INTEGER IDAMAX
192 DOUBLE PRECISION DLAMCH
193 EXTERNAL lsame, idamax, dlamch
194* ..
195* .. Intrinsic Functions ..
196 INTRINSIC abs, max, min
197* ..
198* .. Executable Statements ..
199*
200* Quick exit if N = 0 or NRHS = 0.
201*
202 IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
203 reslts( 1 ) = zero
204 reslts( 2 ) = zero
205 RETURN
206 END IF
207*
208 eps = dlamch( 'Epsilon' )
209 unfl = dlamch( 'Safe minimum' )
210 ovfl = one / unfl
211 upper = lsame( uplo, 'U' )
212*
213* Test 1: Compute the maximum of
214* norm(X - XACT) / ( norm(X) * FERR )
215* over all the vectors X and XACT using the infinity-norm.
216*
217 errbnd = zero
218 DO 30 j = 1, nrhs
219 imax = idamax( n, x( 1, j ), 1 )
220 xnorm = max( abs( x( imax, j ) ), unfl )
221 diff = zero
222 DO 10 i = 1, n
223 diff = max( diff, abs( x( i, j )-xact( i, j ) ) )
224 10 CONTINUE
225*
226 IF( xnorm.GT.one ) THEN
227 GO TO 20
228 ELSE IF( diff.LE.ovfl*xnorm ) THEN
229 GO TO 20
230 ELSE
231 errbnd = one / eps
232 GO TO 30
233 END IF
234*
235 20 CONTINUE
236 IF( diff / xnorm.LE.ferr( j ) ) THEN
237 errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
238 ELSE
239 errbnd = one / eps
240 END IF
241 30 CONTINUE
242 reslts( 1 ) = errbnd
243*
244* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
245* (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
246*
247 DO 90 k = 1, nrhs
248 DO 80 i = 1, n
249 tmp = abs( b( i, k ) )
250 IF( upper ) THEN
251 DO 40 j = 1, i
252 tmp = tmp + abs( a( j, i ) )*abs( x( j, k ) )
253 40 CONTINUE
254 DO 50 j = i + 1, n
255 tmp = tmp + abs( a( i, j ) )*abs( x( j, k ) )
256 50 CONTINUE
257 ELSE
258 DO 60 j = 1, i - 1
259 tmp = tmp + abs( a( i, j ) )*abs( x( j, k ) )
260 60 CONTINUE
261 DO 70 j = i, n
262 tmp = tmp + abs( a( j, i ) )*abs( x( j, k ) )
263 70 CONTINUE
264 END IF
265 IF( i.EQ.1 ) THEN
266 axbi = tmp
267 ELSE
268 axbi = min( axbi, tmp )
269 END IF
270 80 CONTINUE
271 tmp = berr( k ) / ( ( n+1 )*eps+( n+1 )*unfl /
272 \$ max( axbi, ( n+1 )*unfl ) )
273 IF( k.EQ.1 ) THEN
274 reslts( 2 ) = tmp
275 ELSE
276 reslts( 2 ) = max( reslts( 2 ), tmp )
277 END IF
278 90 CONTINUE
279*
280 RETURN
281*
282* End of DPOT05
283*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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