LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dgbt05()

 subroutine dgbt05 ( character trans, integer n, integer kl, integer ku, 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 )

DGBT05

Purpose:
``` DGBT05 tests the error bounds from iterative refinement for the
computed solution to a system of equations op(A)*X = B, where A is a
general band matrix of order n with kl subdiagonals and ku
superdiagonals and op(A) = A or A**T, depending on TRANS.

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(op(A))*abs(X) +abs(b))_i )
and NZ = max. number of nonzeros in any row of A, plus 1```
Parameters
 [in] TRANS ``` TRANS is CHARACTER*1 Specifies the form of the system of equations. = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose = Transpose)``` [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] KL ``` KL is INTEGER The number of subdiagonals within the band of A. KL >= 0.``` [in] KU ``` KU is INTEGER The number of superdiagonals within the band of A. KU >= 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 original band matrix A, stored in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows: AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KL+KU+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 + (*) )```

Definition at line 174 of file dgbt05.f.

176*
177* -- LAPACK test routine --
178* -- LAPACK is a software package provided by Univ. of Tennessee, --
179* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
180*
181* .. Scalar Arguments ..
182 CHARACTER TRANS
183 INTEGER KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS
184* ..
185* .. Array Arguments ..
186 DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ),
187 \$ FERR( * ), RESLTS( * ), X( LDX, * ),
188 \$ XACT( LDXACT, * )
189* ..
190*
191* =====================================================================
192*
193* .. Parameters ..
194 DOUBLE PRECISION ZERO, ONE
195 parameter( zero = 0.0d+0, one = 1.0d+0 )
196* ..
197* .. Local Scalars ..
198 LOGICAL NOTRAN
199 INTEGER I, IMAX, J, K, NZ
200 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
201* ..
202* .. External Functions ..
203 LOGICAL LSAME
204 INTEGER IDAMAX
205 DOUBLE PRECISION DLAMCH
206 EXTERNAL lsame, idamax, dlamch
207* ..
208* .. Intrinsic Functions ..
209 INTRINSIC abs, max, min
210* ..
211* .. Executable Statements ..
212*
213* Quick exit if N = 0 or NRHS = 0.
214*
215 IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
216 reslts( 1 ) = zero
217 reslts( 2 ) = zero
218 RETURN
219 END IF
220*
221 eps = dlamch( 'Epsilon' )
222 unfl = dlamch( 'Safe minimum' )
223 ovfl = one / unfl
224 notran = lsame( trans, 'N' )
225 nz = min( kl+ku+2, n+1 )
226*
227* Test 1: Compute the maximum of
228* norm(X - XACT) / ( norm(X) * FERR )
229* over all the vectors X and XACT using the infinity-norm.
230*
231 errbnd = zero
232 DO 30 j = 1, nrhs
233 imax = idamax( n, x( 1, j ), 1 )
234 xnorm = max( abs( x( imax, j ) ), unfl )
235 diff = zero
236 DO 10 i = 1, n
237 diff = max( diff, abs( x( i, j )-xact( i, j ) ) )
238 10 CONTINUE
239*
240 IF( xnorm.GT.one ) THEN
241 GO TO 20
242 ELSE IF( diff.LE.ovfl*xnorm ) THEN
243 GO TO 20
244 ELSE
245 errbnd = one / eps
246 GO TO 30
247 END IF
248*
249 20 CONTINUE
250 IF( diff / xnorm.LE.ferr( j ) ) THEN
251 errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
252 ELSE
253 errbnd = one / eps
254 END IF
255 30 CONTINUE
256 reslts( 1 ) = errbnd
257*
258* Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
259* (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
260*
261 DO 70 k = 1, nrhs
262 DO 60 i = 1, n
263 tmp = abs( b( i, k ) )
264 IF( notran ) THEN
265 DO 40 j = max( i-kl, 1 ), min( i+ku, n )
266 tmp = tmp + abs( ab( ku+1+i-j, j ) )*abs( x( j, k ) )
267 40 CONTINUE
268 ELSE
269 DO 50 j = max( i-ku, 1 ), min( i+kl, n )
270 tmp = tmp + abs( ab( ku+1+j-i, i ) )*abs( x( j, k ) )
271 50 CONTINUE
272 END IF
273 IF( i.EQ.1 ) THEN
274 axbi = tmp
275 ELSE
276 axbi = min( axbi, tmp )
277 END IF
278 60 CONTINUE
279 tmp = berr( k ) / ( nz*eps+nz*unfl / max( axbi, nz*unfl ) )
280 IF( k.EQ.1 ) THEN
281 reslts( 2 ) = tmp
282 ELSE
283 reslts( 2 ) = max( reslts( 2 ), tmp )
284 END IF
285 70 CONTINUE
286*
287 RETURN
288*
289* End of DGBT05
290*
integer function idamax(n, dx, incx)
IDAMAX
Definition idamax.f:71
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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