LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ ztrt05()

 subroutine ztrt05 ( character uplo, character trans, character diag, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, 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 )

ZTRT05

Purpose:
``` ZTRT05 tests the error bounds from iterative refinement for the
computed solution to a system of equations A*X = B, where A is a
triangular 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 matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular``` [in] TRANS ``` TRANS is CHARACTER*1 Specifies the form of the system of equations. = 'N': A * X = B (No transpose) = 'T': A'* X = B (Transpose) = 'C': A'* X = B (Conjugate transpose = Transpose)``` [in] DIAG ``` DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit 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 COMPLEX*16 array, dimension (LDA,N) The triangular matrix A. If UPLO = 'U', the leading n by n upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n by n lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = 'U', the diagonal elements of A are also not referenced and are assumed to be 1.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,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 180 of file ztrt05.f.

182*
183* -- LAPACK test routine --
184* -- LAPACK is a software package provided by Univ. of Tennessee, --
185* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
186*
187* .. Scalar Arguments ..
188 CHARACTER DIAG, TRANS, UPLO
189 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
190* ..
191* .. Array Arguments ..
192 DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * )
193 COMPLEX*16 A( LDA, * ), B( LDB, * ), X( LDX, * ),
194 \$ XACT( LDXACT, * )
195* ..
196*
197* =====================================================================
198*
199* .. Parameters ..
200 DOUBLE PRECISION ZERO, ONE
201 parameter( zero = 0.0d+0, one = 1.0d+0 )
202* ..
203* .. Local Scalars ..
204 LOGICAL NOTRAN, UNIT, UPPER
205 INTEGER I, IFU, IMAX, J, K
206 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
207 COMPLEX*16 ZDUM
208* ..
209* .. External Functions ..
210 LOGICAL LSAME
211 INTEGER IZAMAX
212 DOUBLE PRECISION DLAMCH
213 EXTERNAL lsame, izamax, dlamch
214* ..
215* .. Intrinsic Functions ..
216 INTRINSIC abs, dble, dimag, max, min
217* ..
218* .. Statement Functions ..
219 DOUBLE PRECISION CABS1
220* ..
221* .. Statement Function definitions ..
222 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
223* ..
224* .. Executable Statements ..
225*
226* Quick exit if N = 0 or NRHS = 0.
227*
228 IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
229 reslts( 1 ) = zero
230 reslts( 2 ) = zero
231 RETURN
232 END IF
233*
234 eps = dlamch( 'Epsilon' )
235 unfl = dlamch( 'Safe minimum' )
236 ovfl = one / unfl
237 upper = lsame( uplo, 'U' )
238 notran = lsame( trans, 'N' )
239 unit = lsame( diag, 'U' )
240*
241* Test 1: Compute the maximum of
242* norm(X - XACT) / ( norm(X) * FERR )
243* over all the vectors X and XACT using the infinity-norm.
244*
245 errbnd = zero
246 DO 30 j = 1, nrhs
247 imax = izamax( n, x( 1, j ), 1 )
248 xnorm = max( cabs1( x( imax, j ) ), unfl )
249 diff = zero
250 DO 10 i = 1, n
251 diff = max( diff, cabs1( x( i, j )-xact( i, j ) ) )
252 10 CONTINUE
253*
254 IF( xnorm.GT.one ) THEN
255 GO TO 20
256 ELSE IF( diff.LE.ovfl*xnorm ) THEN
257 GO TO 20
258 ELSE
259 errbnd = one / eps
260 GO TO 30
261 END IF
262*
263 20 CONTINUE
264 IF( diff / xnorm.LE.ferr( j ) ) THEN
265 errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
266 ELSE
267 errbnd = one / eps
268 END IF
269 30 CONTINUE
270 reslts( 1 ) = errbnd
271*
272* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
273* (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
274*
275 ifu = 0
276 IF( unit )
277 \$ ifu = 1
278 DO 90 k = 1, nrhs
279 DO 80 i = 1, n
280 tmp = cabs1( b( i, k ) )
281 IF( upper ) THEN
282 IF( .NOT.notran ) THEN
283 DO 40 j = 1, i - ifu
284 tmp = tmp + cabs1( a( j, i ) )*cabs1( x( j, k ) )
285 40 CONTINUE
286 IF( unit )
287 \$ tmp = tmp + cabs1( x( i, k ) )
288 ELSE
289 IF( unit )
290 \$ tmp = tmp + cabs1( x( i, k ) )
291 DO 50 j = i + ifu, n
292 tmp = tmp + cabs1( a( i, j ) )*cabs1( x( j, k ) )
293 50 CONTINUE
294 END IF
295 ELSE
296 IF( notran ) THEN
297 DO 60 j = 1, i - ifu
298 tmp = tmp + cabs1( a( i, j ) )*cabs1( x( j, k ) )
299 60 CONTINUE
300 IF( unit )
301 \$ tmp = tmp + cabs1( x( i, k ) )
302 ELSE
303 IF( unit )
304 \$ tmp = tmp + cabs1( x( i, k ) )
305 DO 70 j = i + ifu, n
306 tmp = tmp + cabs1( a( j, i ) )*cabs1( x( j, k ) )
307 70 CONTINUE
308 END IF
309 END IF
310 IF( i.EQ.1 ) THEN
311 axbi = tmp
312 ELSE
313 axbi = min( axbi, tmp )
314 END IF
315 80 CONTINUE
316 tmp = berr( k ) / ( ( n+1 )*eps+( n+1 )*unfl /
317 \$ max( axbi, ( n+1 )*unfl ) )
318 IF( k.EQ.1 ) THEN
319 reslts( 2 ) = tmp
320 ELSE
321 reslts( 2 ) = max( reslts( 2 ), tmp )
322 END IF
323 90 CONTINUE
324*
325 RETURN
326*
327* End of ZTRT05
328*
integer function izamax(n, zx, incx)
IZAMAX
Definition izamax.f:71
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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