LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ ztrt03()

 subroutine ztrt03 ( character UPLO, character TRANS, character DIAG, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, double precision SCALE, double precision, dimension( * ) CNORM, double precision TSCAL, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) WORK, double precision RESID )

ZTRT03

Purpose:
``` ZTRT03 computes the residual for the solution to a scaled triangular
system of equations A*x = s*b,  A**T *x = s*b,  or  A**H *x = s*b.
Here A is a triangular matrix, A**T denotes the transpose of A, A**H
denotes the conjugate transpose of A, s is a scalar, and x and b are
N by NRHS matrices.  The test ratio is the maximum over the number of
right hand sides of
norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
where op(A) denotes A, A**T, or A**H, and EPS is the machine epsilon.```
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 operation applied to A. = 'N': A *x = s*b (No transpose) = 'T': A**T *x = s*b (Transpose) = 'C': A**H *x = s*b (Conjugate 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 order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices X and B. 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] SCALE ``` SCALE is DOUBLE PRECISION The scaling factor s used in solving the triangular system.``` [in] CNORM ``` CNORM is DOUBLE PRECISION array, dimension (N) The 1-norms of the columns of A, not counting the diagonal.``` [in] TSCAL ``` TSCAL is DOUBLE PRECISION The scaling factor used in computing the 1-norms in CNORM. CNORM actually contains the column norms of TSCAL*A.``` [in] X ``` X is COMPLEX*16 array, dimension (LDX,NRHS) The computed solution vectors for the system of linear equations.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= 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).``` [out] WORK ` WORK is COMPLEX*16 array, dimension (N)` [out] RESID ``` RESID is DOUBLE PRECISION The maximum over the number of right hand sides of norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).```
Date
December 2016

Definition at line 173 of file ztrt03.f.

173 *
174 * -- LAPACK test routine (version 3.7.0) --
175 * -- LAPACK is a software package provided by Univ. of Tennessee, --
176 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177 * December 2016
178 *
179 * .. Scalar Arguments ..
180  CHARACTER diag, trans, uplo
181  INTEGER lda, ldb, ldx, n, nrhs
182  DOUBLE PRECISION resid, scale, tscal
183 * ..
184 * .. Array Arguments ..
185  DOUBLE PRECISION cnorm( * )
186  COMPLEX*16 a( lda, * ), b( ldb, * ), work( * ),
187  \$ x( ldx, * )
188 * ..
189 *
190 * =====================================================================
191 *
192 * .. Parameters ..
193  DOUBLE PRECISION one, zero
194  parameter( one = 1.0d+0, zero = 0.0d+0 )
195 * ..
196 * .. Local Scalars ..
197  INTEGER ix, j
198  DOUBLE PRECISION eps, err, smlnum, tnorm, xnorm, xscal
199 * ..
200 * .. External Functions ..
201  LOGICAL lsame
202  INTEGER izamax
203  DOUBLE PRECISION dlamch
204  EXTERNAL lsame, izamax, dlamch
205 * ..
206 * .. External Subroutines ..
207  EXTERNAL zaxpy, zcopy, zdscal, ztrmv
208 * ..
209 * .. Intrinsic Functions ..
210  INTRINSIC abs, dble, dcmplx, max
211 * ..
212 * .. Executable Statements ..
213 *
214 * Quick exit if N = 0
215 *
216  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
217  resid = zero
218  RETURN
219  END IF
220  eps = dlamch( 'Epsilon' )
221  smlnum = dlamch( 'Safe minimum' )
222 *
223 * Compute the norm of the triangular matrix A using the column
224 * norms already computed by ZLATRS.
225 *
226  tnorm = zero
227  IF( lsame( diag, 'N' ) ) THEN
228  DO 10 j = 1, n
229  tnorm = max( tnorm, tscal*abs( a( j, j ) )+cnorm( j ) )
230  10 CONTINUE
231  ELSE
232  DO 20 j = 1, n
233  tnorm = max( tnorm, tscal+cnorm( j ) )
234  20 CONTINUE
235  END IF
236 *
237 * Compute the maximum over the number of right hand sides of
238 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
239 *
240  resid = zero
241  DO 30 j = 1, nrhs
242  CALL zcopy( n, x( 1, j ), 1, work, 1 )
243  ix = izamax( n, work, 1 )
244  xnorm = max( one, abs( x( ix, j ) ) )
245  xscal = ( one / xnorm ) / dble( n )
246  CALL zdscal( n, xscal, work, 1 )
247  CALL ztrmv( uplo, trans, diag, n, a, lda, work, 1 )
248  CALL zaxpy( n, dcmplx( -scale*xscal ), b( 1, j ), 1, work, 1 )
249  ix = izamax( n, work, 1 )
250  err = tscal*abs( work( ix ) )
251  ix = izamax( n, x( 1, j ), 1 )
252  xnorm = abs( x( ix, j ) )
253  IF( err*smlnum.LE.xnorm ) THEN
254  IF( xnorm.GT.zero )
255  \$ err = err / xnorm
256  ELSE
257  IF( err.GT.zero )
258  \$ err = one / eps
259  END IF
260  IF( err*smlnum.LE.tnorm ) THEN
261  IF( tnorm.GT.zero )
262  \$ err = err / tnorm
263  ELSE
264  IF( err.GT.zero )
265  \$ err = one / eps
266  END IF
267  resid = max( resid, err )
268  30 CONTINUE
269 *
270  RETURN
271 *
272 * End of ZTRT03
273 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:83
integer function izamax(N, ZX, INCX)
IZAMAX
Definition: izamax.f:73
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine ztrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
ZTRMV
Definition: ztrmv.f:149
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:80
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:90
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