LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ zpbt05()

subroutine zpbt05 ( character  uplo,
integer  n,
integer  kd,
integer  nrhs,
complex*16, dimension( ldab, * )  ab,
integer  ldab,
complex*16, dimension( ldb, * )  b,
integer  ldb,
complex*16, dimension( ldx, * )  x,
integer  ldx,
complex*16, dimension( ldxact, * )  xact,
integer  ldxact,
double precision, dimension( * )  ferr,
double precision, dimension( * )  berr,
double precision, dimension( * )  reslts 
)

ZPBT05

Purpose:
 ZPBT05 tests the error bounds from iterative refinement for the
 computed solution to a system of equations A*X = B, where A is a
 Hermitian 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 upper or lower triangular part of the
          Hermitian matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower 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 COMPLEX*16 array, dimension (LDAB,N)
          The upper or lower triangle of the Hermitian 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).
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.
[in]B
          B is COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 169 of file zpbt05.f.

171*
172* -- LAPACK test routine --
173* -- LAPACK is a software package provided by Univ. of Tennessee, --
174* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175*
176* .. Scalar Arguments ..
177 CHARACTER UPLO
178 INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS
179* ..
180* .. Array Arguments ..
181 DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * )
182 COMPLEX*16 AB( LDAB, * ), B( LDB, * ), X( LDX, * ),
183 $ XACT( LDXACT, * )
184* ..
185*
186* =====================================================================
187*
188* .. Parameters ..
189 DOUBLE PRECISION ZERO, ONE
190 parameter( zero = 0.0d+0, one = 1.0d+0 )
191* ..
192* .. Local Scalars ..
193 LOGICAL UPPER
194 INTEGER I, IMAX, J, K, NZ
195 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
196 COMPLEX*16 ZDUM
197* ..
198* .. External Functions ..
199 LOGICAL LSAME
200 INTEGER IZAMAX
201 DOUBLE PRECISION DLAMCH
202 EXTERNAL lsame, izamax, dlamch
203* ..
204* .. Intrinsic Functions ..
205 INTRINSIC abs, dble, dimag, max, min
206* ..
207* .. Statement Functions ..
208 DOUBLE PRECISION CABS1
209* ..
210* .. Statement Function definitions ..
211 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
212* ..
213* .. Executable Statements ..
214*
215* Quick exit if N = 0 or NRHS = 0.
216*
217 IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
218 reslts( 1 ) = zero
219 reslts( 2 ) = zero
220 RETURN
221 END IF
222*
223 eps = dlamch( 'Epsilon' )
224 unfl = dlamch( 'Safe minimum' )
225 ovfl = one / unfl
226 upper = lsame( uplo, 'U' )
227 nz = 2*max( kd, n-1 ) + 1
228*
229* Test 1: Compute the maximum of
230* norm(X - XACT) / ( norm(X) * FERR )
231* over all the vectors X and XACT using the infinity-norm.
232*
233 errbnd = zero
234 DO 30 j = 1, nrhs
235 imax = izamax( n, x( 1, j ), 1 )
236 xnorm = max( cabs1( x( imax, j ) ), unfl )
237 diff = zero
238 DO 10 i = 1, n
239 diff = max( diff, cabs1( x( i, j )-xact( i, j ) ) )
240 10 CONTINUE
241*
242 IF( xnorm.GT.one ) THEN
243 GO TO 20
244 ELSE IF( diff.LE.ovfl*xnorm ) THEN
245 GO TO 20
246 ELSE
247 errbnd = one / eps
248 GO TO 30
249 END IF
250*
251 20 CONTINUE
252 IF( diff / xnorm.LE.ferr( j ) ) THEN
253 errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
254 ELSE
255 errbnd = one / eps
256 END IF
257 30 CONTINUE
258 reslts( 1 ) = errbnd
259*
260* Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
261* (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
262*
263 DO 90 k = 1, nrhs
264 DO 80 i = 1, n
265 tmp = cabs1( b( i, k ) )
266 IF( upper ) THEN
267 DO 40 j = max( i-kd, 1 ), i - 1
268 tmp = tmp + cabs1( ab( kd+1-i+j, i ) )*
269 $ cabs1( x( j, k ) )
270 40 CONTINUE
271 tmp = tmp + abs( dble( ab( kd+1, i ) ) )*
272 $ cabs1( x( i, k ) )
273 DO 50 j = i + 1, min( i+kd, n )
274 tmp = tmp + cabs1( ab( kd+1+i-j, j ) )*
275 $ cabs1( x( j, k ) )
276 50 CONTINUE
277 ELSE
278 DO 60 j = max( i-kd, 1 ), i - 1
279 tmp = tmp + cabs1( ab( 1+i-j, j ) )*cabs1( x( j, k ) )
280 60 CONTINUE
281 tmp = tmp + abs( dble( ab( 1, i ) ) )*cabs1( x( i, k ) )
282 DO 70 j = i + 1, min( i+kd, n )
283 tmp = tmp + cabs1( ab( 1+j-i, i ) )*cabs1( x( j, k ) )
284 70 CONTINUE
285 END IF
286 IF( i.EQ.1 ) THEN
287 axbi = tmp
288 ELSE
289 axbi = min( axbi, tmp )
290 END IF
291 80 CONTINUE
292 tmp = berr( k ) / ( nz*eps+nz*unfl / max( axbi, nz*unfl ) )
293 IF( k.EQ.1 ) THEN
294 reslts( 2 ) = tmp
295 ELSE
296 reslts( 2 ) = max( reslts( 2 ), tmp )
297 END IF
298 90 CONTINUE
299*
300 RETURN
301*
302* End of ZPBT05
303*
izamax
integer function izamax(n, zx, incx)
IZAMAX
Definition izamax.f:71
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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