LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ 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 *
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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