LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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 ).```

Definition at line 159 of file dtpt03.f.

161*
162* -- LAPACK test routine --
163* -- LAPACK is a software package provided by Univ. of Tennessee, --
164* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
165*
166* .. Scalar Arguments ..
167 CHARACTER DIAG, TRANS, UPLO
168 INTEGER LDB, LDX, N, NRHS
169 DOUBLE PRECISION RESID, SCALE, TSCAL
170* ..
171* .. Array Arguments ..
172 DOUBLE PRECISION AP( * ), B( LDB, * ), CNORM( * ), WORK( * ),
173 \$ X( LDX, * )
174* ..
175*
176* =====================================================================
177*
178* .. Parameters ..
179 DOUBLE PRECISION ONE, ZERO
180 parameter( one = 1.0d+0, zero = 0.0d+0 )
181* ..
182* .. Local Scalars ..
183 INTEGER IX, J, JJ
184 DOUBLE PRECISION BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
185* ..
186* .. External Functions ..
187 LOGICAL LSAME
188 INTEGER IDAMAX
189 DOUBLE PRECISION DLAMCH
190 EXTERNAL lsame, idamax, dlamch
191* ..
192* .. External Subroutines ..
193 EXTERNAL daxpy, dcopy, dlabad, dscal, dtpmv
194* ..
195* .. Intrinsic Functions ..
196 INTRINSIC abs, dble, max
197* ..
198* .. Executable Statements ..
199*
200* Quick exit if N = 0.
201*
202 IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
203 resid = zero
204 RETURN
205 END IF
206 eps = dlamch( 'Epsilon' )
207 smlnum = dlamch( 'Safe minimum' )
208 bignum = one / smlnum
209 CALL dlabad( smlnum, bignum )
210*
211* Compute the norm of the triangular matrix A using the column
212* norms already computed by DLATPS.
213*
214 tnorm = zero
215 IF( lsame( diag, 'N' ) ) THEN
216 IF( lsame( uplo, 'U' ) ) THEN
217 jj = 1
218 DO 10 j = 1, n
219 tnorm = max( tnorm, tscal*abs( ap( jj ) )+cnorm( j ) )
220 jj = jj + j + 1
221 10 CONTINUE
222 ELSE
223 jj = 1
224 DO 20 j = 1, n
225 tnorm = max( tnorm, tscal*abs( ap( jj ) )+cnorm( j ) )
226 jj = jj + n - j + 1
227 20 CONTINUE
228 END IF
229 ELSE
230 DO 30 j = 1, n
231 tnorm = max( tnorm, tscal+cnorm( j ) )
232 30 CONTINUE
233 END IF
234*
235* Compute the maximum over the number of right hand sides of
236* norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
237*
238 resid = zero
239 DO 40 j = 1, nrhs
240 CALL dcopy( n, x( 1, j ), 1, work, 1 )
241 ix = idamax( n, work, 1 )
242 xnorm = max( one, abs( x( ix, j ) ) )
243 xscal = ( one / xnorm ) / dble( n )
244 CALL dscal( n, xscal, work, 1 )
245 CALL dtpmv( uplo, trans, diag, n, ap, work, 1 )
246 CALL daxpy( n, -scale*xscal, b( 1, j ), 1, work, 1 )
247 ix = idamax( n, work, 1 )
248 err = tscal*abs( work( ix ) )
249 ix = idamax( n, x( 1, j ), 1 )
250 xnorm = abs( x( ix, j ) )
251 IF( err*smlnum.LE.xnorm ) THEN
252 IF( xnorm.GT.zero )
253 \$ err = err / xnorm
254 ELSE
255 IF( err.GT.zero )
256 \$ err = one / eps
257 END IF
258 IF( err*smlnum.LE.tnorm ) THEN
259 IF( tnorm.GT.zero )
260 \$ err = err / tnorm
261 ELSE
262 IF( err.GT.zero )
263 \$ err = one / eps
264 END IF
265 resid = max( resid, err )
266 40 CONTINUE
267*
268 RETURN
269*
270* End of DTPT03
271*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dtpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
DTPMV
Definition: dtpmv.f:142
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