LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
Loading...
Searching...
No Matches

◆ dtbt05()

subroutine dtbt05 ( character uplo,
character trans,
character diag,
integer n,
integer kd,
integer nrhs,
double precision, dimension( ldab, * ) ab,
integer ldab,
double precision, dimension( ldb, * ) b,
integer ldb,
double precision, dimension( ldx, * ) x,
integer ldx,
double precision, dimension( ldxact, * ) xact,
integer ldxact,
double precision, dimension( * ) ferr,
double precision, dimension( * ) berr,
double precision, dimension( * ) reslts )

DTBT05

Purpose:
!>
!> DTBT05 tests the error bounds from iterative refinement for the
!> computed solution to a system of equations A*X = B, where A is a
!> triangular band matrix.
!>
!> RESLTS(1) = test of the error bound
!>           = norm(X - XACT) / ( norm(X) * FERR )
!>
!> A large value is returned if this ratio is not less than one.
!>
!> RESLTS(2) = residual from the iterative refinement routine
!>           = the maximum of BERR / ( NZ*EPS + (*) ), where
!>             (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
!>             and NZ = max. number of nonzeros in any row of A, plus 1
!>
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 form of the system of equations.
!>          = 'N':  A * X = B  (No transpose)
!>          = 'T':  A'* X = B  (Transpose)
!>          = 'C':  A'* X = 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 number of rows of the matrices X, B, and XACT, and the
!>          order of the matrix A.  N >= 0.
!>
[in]KD
!>          KD is INTEGER
!>          The number of super-diagonals of the matrix A if UPLO = 'U',
!>          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.
!>
[in]NRHS
!>          NRHS is INTEGER
!>          The number of columns of the matrices X, B, and XACT.
!>          NRHS >= 0.
!>
[in]AB
!>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
!>          The upper or lower triangular band matrix A, stored in the
!>          first kd+1 rows of the array. The j-th column of A is stored
!>          in the j-th column of the array AB as follows:
!>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
!>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
!>          If DIAG = 'U', the diagonal elements of A are not referenced
!>          and are assumed to be 1.
!>
[in]LDAB
!>          LDAB is INTEGER
!>          The leading dimension of the array AB.  LDAB >= KD+1.
!>
[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).
!>
[in]X
!>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
!>          The computed solution vectors.  Each vector is stored as a
!>          column of the matrix X.
!>
[in]LDX
!>          LDX is INTEGER
!>          The leading dimension of the array X.  LDX >= max(1,N).
!>
[in]XACT
!>          XACT is DOUBLE PRECISION array, dimension (LDX,NRHS)
!>          The exact solution vectors.  Each vector is stored as a
!>          column of the matrix XACT.
!>
[in]LDXACT
!>          LDXACT is INTEGER
!>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
!>
[in]FERR
!>          FERR is DOUBLE PRECISION array, dimension (NRHS)
!>          The estimated forward error bounds for each solution vector
!>          X.  If XTRUE is the true solution, FERR bounds the magnitude
!>          of the largest entry in (X - XTRUE) divided by the magnitude
!>          of the largest entry in X.
!>
[in]BERR
!>          BERR is DOUBLE PRECISION array, dimension (NRHS)
!>          The componentwise relative backward error of each solution
!>          vector (i.e., the smallest relative change in any entry of A
!>          or B that makes X an exact solution).
!>
[out]RESLTS
!>          RESLTS is DOUBLE PRECISION array, dimension (2)
!>          The maximum over the NRHS solution vectors of the ratios:
!>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
!>          RESLTS(2) = BERR / ( NZ*EPS + (*) )
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 187 of file dtbt05.f.

189*
190* -- LAPACK test routine --
191* -- LAPACK is a software package provided by Univ. of Tennessee, --
192* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
193*
194* .. Scalar Arguments ..
195 CHARACTER DIAG, TRANS, UPLO
196 INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS
197* ..
198* .. Array Arguments ..
199 DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ),
200 $ FERR( * ), RESLTS( * ), X( LDX, * ),
201 $ XACT( LDXACT, * )
202* ..
203*
204* =====================================================================
205*
206* .. Parameters ..
207 DOUBLE PRECISION ZERO, ONE
208 parameter( zero = 0.0d+0, one = 1.0d+0 )
209* ..
210* .. Local Scalars ..
211 LOGICAL NOTRAN, UNIT, UPPER
212 INTEGER I, IFU, IMAX, J, K, NZ
213 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
214* ..
215* .. External Functions ..
216 LOGICAL LSAME
217 INTEGER IDAMAX
218 DOUBLE PRECISION DLAMCH
219 EXTERNAL lsame, idamax, dlamch
220* ..
221* .. Intrinsic Functions ..
222 INTRINSIC abs, max, min
223* ..
224* .. Executable Statements ..
225*
226* Quick exit if N = 0 or NRHS = 0.
227*
228 IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
229 reslts( 1 ) = zero
230 reslts( 2 ) = zero
231 RETURN
232 END IF
233*
234 eps = dlamch( 'Epsilon' )
235 unfl = dlamch( 'Safe minimum' )
236 ovfl = one / unfl
237 upper = lsame( uplo, 'U' )
238 notran = lsame( trans, 'N' )
239 unit = lsame( diag, 'U' )
240 nz = min( kd, n-1 ) + 1
241*
242* Test 1: Compute the maximum of
243* norm(X - XACT) / ( norm(X) * FERR )
244* over all the vectors X and XACT using the infinity-norm.
245*
246 errbnd = zero
247 DO 30 j = 1, nrhs
248 imax = idamax( n, x( 1, j ), 1 )
249 xnorm = max( abs( x( imax, j ) ), unfl )
250 diff = zero
251 DO 10 i = 1, n
252 diff = max( diff, abs( x( i, j )-xact( i, j ) ) )
253 10 CONTINUE
254*
255 IF( xnorm.GT.one ) THEN
256 GO TO 20
257 ELSE IF( diff.LE.ovfl*xnorm ) THEN
258 GO TO 20
259 ELSE
260 errbnd = one / eps
261 GO TO 30
262 END IF
263*
264 20 CONTINUE
265 IF( diff / xnorm.LE.ferr( j ) ) THEN
266 errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
267 ELSE
268 errbnd = one / eps
269 END IF
270 30 CONTINUE
271 reslts( 1 ) = errbnd
272*
273* Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
274* (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
275*
276 ifu = 0
277 IF( unit )
278 $ ifu = 1
279 DO 90 k = 1, nrhs
280 DO 80 i = 1, n
281 tmp = abs( b( i, k ) )
282 IF( upper ) THEN
283 IF( .NOT.notran ) THEN
284 DO 40 j = max( i-kd, 1 ), i - ifu
285 tmp = tmp + abs( ab( kd+1-i+j, i ) )*
286 $ abs( x( j, k ) )
287 40 CONTINUE
288 IF( unit )
289 $ tmp = tmp + abs( x( i, k ) )
290 ELSE
291 IF( unit )
292 $ tmp = tmp + abs( x( i, k ) )
293 DO 50 j = i + ifu, min( i+kd, n )
294 tmp = tmp + abs( ab( kd+1+i-j, j ) )*
295 $ abs( x( j, k ) )
296 50 CONTINUE
297 END IF
298 ELSE
299 IF( notran ) THEN
300 DO 60 j = max( i-kd, 1 ), i - ifu
301 tmp = tmp + abs( ab( 1+i-j, j ) )*abs( x( j, k ) )
302 60 CONTINUE
303 IF( unit )
304 $ tmp = tmp + abs( x( i, k ) )
305 ELSE
306 IF( unit )
307 $ tmp = tmp + abs( x( i, k ) )
308 DO 70 j = i + ifu, min( i+kd, n )
309 tmp = tmp + abs( ab( 1+j-i, i ) )*abs( x( j, k ) )
310 70 CONTINUE
311 END IF
312 END IF
313 IF( i.EQ.1 ) THEN
314 axbi = tmp
315 ELSE
316 axbi = min( axbi, tmp )
317 END IF
318 80 CONTINUE
319 tmp = berr( k ) / ( nz*eps+nz*unfl / max( axbi, nz*unfl ) )
320 IF( k.EQ.1 ) THEN
321 reslts( 2 ) = tmp
322 ELSE
323 reslts( 2 ) = max( reslts( 2 ), tmp )
324 END IF
325 90 CONTINUE
326*
327 RETURN
328*
329* End of DTBT05
330*
idamax
integer function idamax(n, dx, incx)
IDAMAX
Definition idamax.f:71
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
Here is the caller graph for this function: