LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ dtrt03()

 subroutine dtrt03 ( character UPLO, character TRANS, character DIAG, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision SCALE, double precision, dimension( * ) CNORM, double precision TSCAL, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, double precision RESID )

DTRT03

Purpose:
``` DTRT03 computes the residual for the solution to a scaled triangular
system of equations A*x = s*b  or  A'*x = s*b.
Here A is a triangular matrix, A' is the 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 or A' 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'*x = s*b (Transpose) = 'C': A'*x = s*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 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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).``` [out] WORK ` WORK is DOUBLE PRECISION 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 171 of file dtrt03.f.

171 *
172 * -- LAPACK test routine (version 3.7.0) --
173 * -- LAPACK is a software package provided by Univ. of Tennessee, --
174 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175 * December 2016
176 *
177 * .. Scalar Arguments ..
178  CHARACTER diag, trans, uplo
179  INTEGER lda, ldb, ldx, n, nrhs
180  DOUBLE PRECISION resid, scale, tscal
181 * ..
182 * .. Array Arguments ..
183  DOUBLE PRECISION a( lda, * ), b( ldb, * ), cnorm( * ),
184  \$ work( * ), x( ldx, * )
185 * ..
186 *
187 * =====================================================================
188 *
189 * .. Parameters ..
190  DOUBLE PRECISION one, zero
191  parameter( one = 1.0d+0, zero = 0.0d+0 )
192 * ..
193 * .. Local Scalars ..
194  INTEGER ix, j
195  DOUBLE PRECISION bignum, eps, err, smlnum, tnorm, xnorm, xscal
196 * ..
197 * .. External Functions ..
198  LOGICAL lsame
199  INTEGER idamax
200  DOUBLE PRECISION dlamch
201  EXTERNAL lsame, idamax, dlamch
202 * ..
203 * .. External Subroutines ..
204  EXTERNAL daxpy, dcopy, dlabad, dscal, dtrmv
205 * ..
206 * .. Intrinsic Functions ..
207  INTRINSIC abs, dble, max
208 * ..
209 * .. Executable Statements ..
210 *
211 * Quick exit if N = 0
212 *
213  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
214  resid = zero
215  RETURN
216  END IF
217  eps = dlamch( 'Epsilon' )
218  smlnum = dlamch( 'Safe minimum' )
219  bignum = one / smlnum
220  CALL dlabad( smlnum, bignum )
221 *
222 * Compute the norm of the triangular matrix A using the column
223 * norms already computed by DLATRS.
224 *
225  tnorm = zero
226  IF( lsame( diag, 'N' ) ) THEN
227  DO 10 j = 1, n
228  tnorm = max( tnorm, tscal*abs( a( j, j ) )+cnorm( j ) )
229  10 CONTINUE
230  ELSE
231  DO 20 j = 1, n
232  tnorm = max( tnorm, tscal+cnorm( j ) )
233  20 CONTINUE
234  END IF
235 *
236 * Compute the maximum over the number of right hand sides of
237 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
238 *
239  resid = zero
240  DO 30 j = 1, nrhs
241  CALL dcopy( n, x( 1, j ), 1, work, 1 )
242  ix = idamax( n, work, 1 )
243  xnorm = max( one, abs( x( ix, j ) ) )
244  xscal = ( one / xnorm ) / dble( n )
245  CALL dscal( n, xscal, work, 1 )
246  CALL dtrmv( uplo, trans, diag, n, a, lda, work, 1 )
247  CALL daxpy( n, -scale*xscal, b( 1, j ), 1, work, 1 )
248  ix = idamax( n, work, 1 )
249  err = tscal*abs( work( ix ) )
250  ix = idamax( n, x( 1, j ), 1 )
251  xnorm = abs( x( ix, j ) )
252  IF( err*smlnum.LE.xnorm ) THEN
253  IF( xnorm.GT.zero )
254  \$ err = err / xnorm
255  ELSE
256  IF( err.GT.zero )
257  \$ err = one / eps
258  END IF
259  IF( err*smlnum.LE.tnorm ) THEN
260  IF( tnorm.GT.zero )
261  \$ err = err / tnorm
262  ELSE
263  IF( err.GT.zero )
264  \$ err = one / eps
265  END IF
266  resid = max( resid, err )
267  30 CONTINUE
268 *
269  RETURN
270 *
271 * End of DTRT03
272 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:84
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:73
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:91
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:81