 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ dtpt03()

 subroutine dtpt03 ( character UPLO, character TRANS, character DIAG, integer N, integer NRHS, double precision, dimension( * ) AP, 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 )

DTPT03

Purpose:
``` DTPT03 computes the residual for the solution to a scaled triangular
system of equations A*x = s*b  or  A'*x = s*b  when the triangular
matrix A is stored in packed format.  Here 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] AP ``` AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP((j-1)*j/2 + i) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP((j-1)*(n-j) + j*(j+1)/2 + i-j) = A(i,j) for j<=i<=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 163 of file dtpt03.f.

163 *
164 * -- LAPACK test routine (version 3.7.0) --
165 * -- LAPACK is a software package provided by Univ. of Tennessee, --
166 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
167 * December 2016
168 *
169 * .. Scalar Arguments ..
170  CHARACTER diag, trans, uplo
171  INTEGER ldb, ldx, n, nrhs
172  DOUBLE PRECISION resid, scale, tscal
173 * ..
174 * .. Array Arguments ..
175  DOUBLE PRECISION ap( * ), b( ldb, * ), cnorm( * ), work( * ),
176  \$ x( ldx, * )
177 * ..
178 *
179 * =====================================================================
180 *
181 * .. Parameters ..
182  DOUBLE PRECISION one, zero
183  parameter( one = 1.0d+0, zero = 0.0d+0 )
184 * ..
185 * .. Local Scalars ..
186  INTEGER ix, j, jj
187  DOUBLE PRECISION bignum, eps, err, smlnum, tnorm, xnorm, xscal
188 * ..
189 * .. External Functions ..
190  LOGICAL lsame
191  INTEGER idamax
192  DOUBLE PRECISION dlamch
193  EXTERNAL lsame, idamax, dlamch
194 * ..
195 * .. External Subroutines ..
196  EXTERNAL daxpy, dcopy, dlabad, dscal, dtpmv
197 * ..
198 * .. Intrinsic Functions ..
199  INTRINSIC abs, dble, max
200 * ..
201 * .. Executable Statements ..
202 *
203 * Quick exit if N = 0.
204 *
205  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
206  resid = zero
207  RETURN
208  END IF
209  eps = dlamch( 'Epsilon' )
210  smlnum = dlamch( 'Safe minimum' )
211  bignum = one / smlnum
212  CALL dlabad( smlnum, bignum )
213 *
214 * Compute the norm of the triangular matrix A using the column
215 * norms already computed by DLATPS.
216 *
217  tnorm = zero
218  IF( lsame( diag, 'N' ) ) THEN
219  IF( lsame( uplo, 'U' ) ) THEN
220  jj = 1
221  DO 10 j = 1, n
222  tnorm = max( tnorm, tscal*abs( ap( jj ) )+cnorm( j ) )
223  jj = jj + j + 1
224  10 CONTINUE
225  ELSE
226  jj = 1
227  DO 20 j = 1, n
228  tnorm = max( tnorm, tscal*abs( ap( jj ) )+cnorm( j ) )
229  jj = jj + n - j + 1
230  20 CONTINUE
231  END IF
232  ELSE
233  DO 30 j = 1, n
234  tnorm = max( tnorm, tscal+cnorm( j ) )
235  30 CONTINUE
236  END IF
237 *
238 * Compute the maximum over the number of right hand sides of
239 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
240 *
241  resid = zero
242  DO 40 j = 1, nrhs
243  CALL dcopy( n, x( 1, j ), 1, work, 1 )
244  ix = idamax( n, work, 1 )
245  xnorm = max( one, abs( x( ix, j ) ) )
246  xscal = ( one / xnorm ) / dble( n )
247  CALL dscal( n, xscal, work, 1 )
248  CALL dtpmv( uplo, trans, diag, n, ap, work, 1 )
249  CALL daxpy( n, -scale*xscal, b( 1, j ), 1, work, 1 )
250  ix = idamax( n, work, 1 )
251  err = tscal*abs( work( ix ) )
252  ix = idamax( n, x( 1, j ), 1 )
253  xnorm = abs( x( ix, j ) )
254  IF( err*smlnum.LE.xnorm ) THEN
255  IF( xnorm.GT.zero )
256  \$ err = err / xnorm
257  ELSE
258  IF( err.GT.zero )
259  \$ err = one / eps
260  END IF
261  IF( err*smlnum.LE.tnorm ) THEN
262  IF( tnorm.GT.zero )
263  \$ err = err / tnorm
264  ELSE
265  IF( err.GT.zero )
266  \$ err = one / eps
267  END IF
268  resid = max( resid, err )
269  40 CONTINUE
270 *
271  RETURN
272 *
273 * End of DTPT03
274 *
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:91
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:76
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:81
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:84
subroutine dtpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
DTPMV
Definition: dtpmv.f:144
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:73
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
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