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 ).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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 *
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
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:76
subroutine dtpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
DTPMV
Definition: dtpmv.f:144
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