LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ ctrt03()

subroutine ctrt03 ( character  UPLO,
character  TRANS,
character  DIAG,
integer  N,
integer  NRHS,
complex, dimension( lda, * )  A,
integer  LDA,
real  SCALE,
real, dimension( * )  CNORM,
real  TSCAL,
complex, dimension( ldx, * )  X,
integer  LDX,
complex, dimension( ldb, * )  B,
integer  LDB,
complex, dimension( * )  WORK,
real  RESID 
)

CTRT03

Purpose:
 CTRT03 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 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 REAL
          The scaling factor s used in solving the triangular system.
[in]CNORM
          CNORM is REAL array, dimension (N)
          The 1-norms of the columns of A, not counting the diagonal.
[in]TSCAL
          TSCAL is REAL
          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 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 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 array, dimension (N)
[out]RESID
          RESID is REAL
          The maximum over the number of right hand sides of
          norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
December 2016

Definition at line 173 of file ctrt03.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  REAL resid, scale, tscal
183 * ..
184 * .. Array Arguments ..
185  REAL cnorm( * )
186  COMPLEX a( lda, * ), b( ldb, * ), work( * ),
187  $ x( ldx, * )
188 * ..
189 *
190 * =====================================================================
191 *
192 * .. Parameters ..
193  REAL one, zero
194  parameter( one = 1.0e+0, zero = 0.0e+0 )
195 * ..
196 * .. Local Scalars ..
197  INTEGER ix, j
198  REAL eps, err, smlnum, tnorm, xnorm, xscal
199 * ..
200 * .. External Functions ..
201  LOGICAL lsame
202  INTEGER icamax
203  REAL slamch
204  EXTERNAL lsame, icamax, slamch
205 * ..
206 * .. External Subroutines ..
207  EXTERNAL caxpy, ccopy, csscal, ctrmv
208 * ..
209 * .. Intrinsic Functions ..
210  INTRINSIC abs, cmplx, max, real
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 = slamch( 'Epsilon' )
221  smlnum = slamch( 'Safe minimum' )
222 *
223 * Compute the norm of the triangular matrix A using the column
224 * norms already computed by CLATRS.
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 ccopy( n, x( 1, j ), 1, work, 1 )
243  ix = icamax( n, work, 1 )
244  xnorm = max( one, abs( x( ix, j ) ) )
245  xscal = ( one / xnorm ) / REAL( n )
246  CALL csscal( n, xscal, work, 1 )
247  CALL ctrmv( uplo, trans, diag, n, a, lda, work, 1 )
248  CALL caxpy( n, cmplx( -scale*xscal ), b( 1, j ), 1, work, 1 )
249  ix = icamax( n, work, 1 )
250  err = tscal*abs( work( ix ) )
251  ix = icamax( 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 CTRT03
273 *
integer function icamax(N, CX, INCX)
ICAMAX
Definition: icamax.f:73
subroutine ctrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
CTRMV
Definition: ctrmv.f:149
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:83
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:90
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:80
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