LAPACK  3.10.1
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 ).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 169 of file ztrt03.f.

171 *
172 * -- LAPACK test routine --
173 * -- LAPACK is a software package provided by Univ. of Tennessee, --
174 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175 *
176 * .. Scalar Arguments ..
177  CHARACTER DIAG, TRANS, UPLO
178  INTEGER LDA, LDB, LDX, N, NRHS
179  DOUBLE PRECISION RESID, SCALE, TSCAL
180 * ..
181 * .. Array Arguments ..
182  DOUBLE PRECISION CNORM( * )
183  COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ),
184  $ 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 EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
196 * ..
197 * .. External Functions ..
198  LOGICAL LSAME
199  INTEGER IZAMAX
200  DOUBLE PRECISION DLAMCH
201  EXTERNAL lsame, izamax, dlamch
202 * ..
203 * .. External Subroutines ..
204  EXTERNAL zaxpy, zcopy, zdscal, ztrmv
205 * ..
206 * .. Intrinsic Functions ..
207  INTRINSIC abs, dble, dcmplx, 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 *
220 * Compute the norm of the triangular matrix A using the column
221 * norms already computed by ZLATRS.
222 *
223  tnorm = zero
224  IF( lsame( diag, 'N' ) ) THEN
225  DO 10 j = 1, n
226  tnorm = max( tnorm, tscal*abs( a( j, j ) )+cnorm( j ) )
227  10 CONTINUE
228  ELSE
229  DO 20 j = 1, n
230  tnorm = max( tnorm, tscal+cnorm( j ) )
231  20 CONTINUE
232  END IF
233 *
234 * Compute the maximum over the number of right hand sides of
235 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
236 *
237  resid = zero
238  DO 30 j = 1, nrhs
239  CALL zcopy( n, x( 1, j ), 1, work, 1 )
240  ix = izamax( n, work, 1 )
241  xnorm = max( one, abs( x( ix, j ) ) )
242  xscal = ( one / xnorm ) / dble( n )
243  CALL zdscal( n, xscal, work, 1 )
244  CALL ztrmv( uplo, trans, diag, n, a, lda, work, 1 )
245  CALL zaxpy( n, dcmplx( -scale*xscal ), b( 1, j ), 1, work, 1 )
246  ix = izamax( n, work, 1 )
247  err = tscal*abs( work( ix ) )
248  ix = izamax( n, x( 1, j ), 1 )
249  xnorm = abs( x( ix, j ) )
250  IF( err*smlnum.LE.xnorm ) THEN
251  IF( xnorm.GT.zero )
252  $ err = err / xnorm
253  ELSE
254  IF( err.GT.zero )
255  $ err = one / eps
256  END IF
257  IF( err*smlnum.LE.tnorm ) THEN
258  IF( tnorm.GT.zero )
259  $ err = err / tnorm
260  ELSE
261  IF( err.GT.zero )
262  $ err = one / eps
263  END IF
264  resid = max( resid, err )
265  30 CONTINUE
266 *
267  RETURN
268 *
269 * End of ZTRT03
270 *
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
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine ztrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
ZTRMV
Definition: ztrmv.f:147
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