 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zppt05()

 subroutine zppt05 ( character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( ldxact, * ) XACT, integer LDXACT, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) RESLTS )

ZPPT05

Purpose:
``` ZPPT05 tests the error bounds from iterative refinement for the
computed solution to a system of equations A*X = B, where A is a
Hermitian matrix in packed storage format.

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 Hermitian 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] AP ``` AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.``` [in] B ``` B is COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 155 of file zppt05.f.

157 *
158 * -- LAPACK test routine --
159 * -- LAPACK is a software package provided by Univ. of Tennessee, --
160 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
161 *
162 * .. Scalar Arguments ..
163  CHARACTER UPLO
164  INTEGER LDB, LDX, LDXACT, N, NRHS
165 * ..
166 * .. Array Arguments ..
167  DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * )
168  COMPLEX*16 AP( * ), B( LDB, * ), X( LDX, * ),
169  \$ XACT( LDXACT, * )
170 * ..
171 *
172 * =====================================================================
173 *
174 * .. Parameters ..
175  DOUBLE PRECISION ZERO, ONE
176  parameter( zero = 0.0d+0, one = 1.0d+0 )
177 * ..
178 * .. Local Scalars ..
179  LOGICAL UPPER
180  INTEGER I, IMAX, J, JC, K
181  DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
182  COMPLEX*16 ZDUM
183 * ..
184 * .. External Functions ..
185  LOGICAL LSAME
186  INTEGER IZAMAX
187  DOUBLE PRECISION DLAMCH
188  EXTERNAL lsame, izamax, dlamch
189 * ..
190 * .. Intrinsic Functions ..
191  INTRINSIC abs, dble, dimag, max, min
192 * ..
193 * .. Statement Functions ..
194  DOUBLE PRECISION CABS1
195 * ..
196 * .. Statement Function definitions ..
197  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
198 * ..
199 * .. Executable Statements ..
200 *
201 * Quick exit if N = 0 or NRHS = 0.
202 *
203  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
204  reslts( 1 ) = zero
205  reslts( 2 ) = zero
206  RETURN
207  END IF
208 *
209  eps = dlamch( 'Epsilon' )
210  unfl = dlamch( 'Safe minimum' )
211  ovfl = one / unfl
212  upper = lsame( uplo, 'U' )
213 *
214 * Test 1: Compute the maximum of
215 * norm(X - XACT) / ( norm(X) * FERR )
216 * over all the vectors X and XACT using the infinity-norm.
217 *
218  errbnd = zero
219  DO 30 j = 1, nrhs
220  imax = izamax( n, x( 1, j ), 1 )
221  xnorm = max( cabs1( x( imax, j ) ), unfl )
222  diff = zero
223  DO 10 i = 1, n
224  diff = max( diff, cabs1( x( i, j )-xact( i, j ) ) )
225  10 CONTINUE
226 *
227  IF( xnorm.GT.one ) THEN
228  GO TO 20
229  ELSE IF( diff.LE.ovfl*xnorm ) THEN
230  GO TO 20
231  ELSE
232  errbnd = one / eps
233  GO TO 30
234  END IF
235 *
236  20 CONTINUE
237  IF( diff / xnorm.LE.ferr( j ) ) THEN
238  errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
239  ELSE
240  errbnd = one / eps
241  END IF
242  30 CONTINUE
243  reslts( 1 ) = errbnd
244 *
245 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
246 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
247 *
248  DO 90 k = 1, nrhs
249  DO 80 i = 1, n
250  tmp = cabs1( b( i, k ) )
251  IF( upper ) THEN
252  jc = ( ( i-1 )*i ) / 2
253  DO 40 j = 1, i - 1
254  tmp = tmp + cabs1( ap( jc+j ) )*cabs1( x( j, k ) )
255  40 CONTINUE
256  tmp = tmp + abs( dble( ap( jc+i ) ) )*cabs1( x( i, k ) )
257  jc = jc + i + i
258  DO 50 j = i + 1, n
259  tmp = tmp + cabs1( ap( jc ) )*cabs1( x( j, k ) )
260  jc = jc + j
261  50 CONTINUE
262  ELSE
263  jc = i
264  DO 60 j = 1, i - 1
265  tmp = tmp + cabs1( ap( jc ) )*cabs1( x( j, k ) )
266  jc = jc + n - j
267  60 CONTINUE
268  tmp = tmp + abs( dble( ap( jc ) ) )*cabs1( x( i, k ) )
269  DO 70 j = i + 1, n
270  tmp = tmp + cabs1( ap( jc+j-i ) )*cabs1( x( j, k ) )
271  70 CONTINUE
272  END IF
273  IF( i.EQ.1 ) THEN
274  axbi = tmp
275  ELSE
276  axbi = min( axbi, tmp )
277  END IF
278  80 CONTINUE
279  tmp = berr( k ) / ( ( n+1 )*eps+( n+1 )*unfl /
280  \$ max( axbi, ( n+1 )*unfl ) )
281  IF( k.EQ.1 ) THEN
282  reslts( 2 ) = tmp
283  ELSE
284  reslts( 2 ) = max( reslts( 2 ), tmp )
285  END IF
286  90 CONTINUE
287 *
288  RETURN
289 *
290 * End of ZPPT05
291 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
integer function izamax(N, ZX, INCX)
IZAMAX
Definition: izamax.f:71
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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