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

 subroutine zpbrfs ( character uplo, integer n, integer kd, integer nrhs, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldafb, * ) afb, integer ldafb, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info )

ZPBRFS

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Purpose:
``` ZPBRFS improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian positive definite
and banded, and provides error bounds and backward error estimates
for the solution.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] KD ``` KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. 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] AFB ``` AFB is COMPLEX*16 array, dimension (LDAFB,N) The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A as computed by ZPBTRF, in the same storage format as A (see AB).``` [in] LDAFB ``` LDAFB is INTEGER The leading dimension of the array AFB. LDAFB >= KD+1.``` [in] B ``` B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [in,out] X ``` X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZPBTRS. On exit, the improved solution matrix X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).``` [out] FERR ``` FERR is DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.``` [out] BERR ``` BERR is DOUBLE PRECISION array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).``` [out] WORK ` WORK is COMPLEX*16 array, dimension (2*N)` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Internal Parameters:
`  ITMAX is the maximum number of steps of iterative refinement.`

Definition at line 187 of file zpbrfs.f.

189*
190* -- LAPACK computational 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 UPLO
196 INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
197* ..
198* .. Array Arguments ..
199 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
200 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
201 \$ WORK( * ), X( LDX, * )
202* ..
203*
204* =====================================================================
205*
206* .. Parameters ..
207 INTEGER ITMAX
208 parameter( itmax = 5 )
209 DOUBLE PRECISION ZERO
210 parameter( zero = 0.0d+0 )
211 COMPLEX*16 ONE
212 parameter( one = ( 1.0d+0, 0.0d+0 ) )
213 DOUBLE PRECISION TWO
214 parameter( two = 2.0d+0 )
215 DOUBLE PRECISION THREE
216 parameter( three = 3.0d+0 )
217* ..
218* .. Local Scalars ..
219 LOGICAL UPPER
220 INTEGER COUNT, I, J, K, KASE, L, NZ
221 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
222 COMPLEX*16 ZDUM
223* ..
224* .. Local Arrays ..
225 INTEGER ISAVE( 3 )
226* ..
227* .. External Subroutines ..
228 EXTERNAL xerbla, zaxpy, zcopy, zhbmv, zlacn2, zpbtrs
229* ..
230* .. Intrinsic Functions ..
231 INTRINSIC abs, dble, dimag, max, min
232* ..
233* .. External Functions ..
234 LOGICAL LSAME
235 DOUBLE PRECISION DLAMCH
236 EXTERNAL lsame, dlamch
237* ..
238* .. Statement Functions ..
239 DOUBLE PRECISION CABS1
240* ..
241* .. Statement Function definitions ..
242 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
243* ..
244* .. Executable Statements ..
245*
246* Test the input parameters.
247*
248 info = 0
249 upper = lsame( uplo, 'U' )
250 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
251 info = -1
252 ELSE IF( n.LT.0 ) THEN
253 info = -2
254 ELSE IF( kd.LT.0 ) THEN
255 info = -3
256 ELSE IF( nrhs.LT.0 ) THEN
257 info = -4
258 ELSE IF( ldab.LT.kd+1 ) THEN
259 info = -6
260 ELSE IF( ldafb.LT.kd+1 ) THEN
261 info = -8
262 ELSE IF( ldb.LT.max( 1, n ) ) THEN
263 info = -10
264 ELSE IF( ldx.LT.max( 1, n ) ) THEN
265 info = -12
266 END IF
267 IF( info.NE.0 ) THEN
268 CALL xerbla( 'ZPBRFS', -info )
269 RETURN
270 END IF
271*
272* Quick return if possible
273*
274 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
275 DO 10 j = 1, nrhs
276 ferr( j ) = zero
277 berr( j ) = zero
278 10 CONTINUE
279 RETURN
280 END IF
281*
282* NZ = maximum number of nonzero elements in each row of A, plus 1
283*
284 nz = min( n+1, 2*kd+2 )
285 eps = dlamch( 'Epsilon' )
286 safmin = dlamch( 'Safe minimum' )
287 safe1 = nz*safmin
288 safe2 = safe1 / eps
289*
290* Do for each right hand side
291*
292 DO 140 j = 1, nrhs
293*
294 count = 1
295 lstres = three
296 20 CONTINUE
297*
298* Loop until stopping criterion is satisfied.
299*
300* Compute residual R = B - A * X
301*
302 CALL zcopy( n, b( 1, j ), 1, work, 1 )
303 CALL zhbmv( uplo, n, kd, -one, ab, ldab, x( 1, j ), 1, one,
304 \$ work, 1 )
305*
306* Compute componentwise relative backward error from formula
307*
308* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
309*
310* where abs(Z) is the componentwise absolute value of the matrix
311* or vector Z. If the i-th component of the denominator is less
312* than SAFE2, then SAFE1 is added to the i-th components of the
313* numerator and denominator before dividing.
314*
315 DO 30 i = 1, n
316 rwork( i ) = cabs1( b( i, j ) )
317 30 CONTINUE
318*
319* Compute abs(A)*abs(X) + abs(B).
320*
321 IF( upper ) THEN
322 DO 50 k = 1, n
323 s = zero
324 xk = cabs1( x( k, j ) )
325 l = kd + 1 - k
326 DO 40 i = max( 1, k-kd ), k - 1
327 rwork( i ) = rwork( i ) + cabs1( ab( l+i, k ) )*xk
328 s = s + cabs1( ab( l+i, k ) )*cabs1( x( i, j ) )
329 40 CONTINUE
330 rwork( k ) = rwork( k ) + abs( dble( ab( kd+1, k ) ) )*
331 \$ xk + s
332 50 CONTINUE
333 ELSE
334 DO 70 k = 1, n
335 s = zero
336 xk = cabs1( x( k, j ) )
337 rwork( k ) = rwork( k ) + abs( dble( ab( 1, k ) ) )*xk
338 l = 1 - k
339 DO 60 i = k + 1, min( n, k+kd )
340 rwork( i ) = rwork( i ) + cabs1( ab( l+i, k ) )*xk
341 s = s + cabs1( ab( l+i, k ) )*cabs1( x( i, j ) )
342 60 CONTINUE
343 rwork( k ) = rwork( k ) + s
344 70 CONTINUE
345 END IF
346 s = zero
347 DO 80 i = 1, n
348 IF( rwork( i ).GT.safe2 ) THEN
349 s = max( s, cabs1( work( i ) ) / rwork( i ) )
350 ELSE
351 s = max( s, ( cabs1( work( i ) )+safe1 ) /
352 \$ ( rwork( i )+safe1 ) )
353 END IF
354 80 CONTINUE
355 berr( j ) = s
356*
357* Test stopping criterion. Continue iterating if
358* 1) The residual BERR(J) is larger than machine epsilon, and
359* 2) BERR(J) decreased by at least a factor of 2 during the
360* last iteration, and
361* 3) At most ITMAX iterations tried.
362*
363 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
364 \$ count.LE.itmax ) THEN
365*
366* Update solution and try again.
367*
368 CALL zpbtrs( uplo, n, kd, 1, afb, ldafb, work, n, info )
369 CALL zaxpy( n, one, work, 1, x( 1, j ), 1 )
370 lstres = berr( j )
371 count = count + 1
372 GO TO 20
373 END IF
374*
375* Bound error from formula
376*
377* norm(X - XTRUE) / norm(X) .le. FERR =
378* norm( abs(inv(A))*
379* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
380*
381* where
382* norm(Z) is the magnitude of the largest component of Z
383* inv(A) is the inverse of A
384* abs(Z) is the componentwise absolute value of the matrix or
385* vector Z
386* NZ is the maximum number of nonzeros in any row of A, plus 1
387* EPS is machine epsilon
388*
389* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
390* is incremented by SAFE1 if the i-th component of
391* abs(A)*abs(X) + abs(B) is less than SAFE2.
392*
393* Use ZLACN2 to estimate the infinity-norm of the matrix
394* inv(A) * diag(W),
395* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
396*
397 DO 90 i = 1, n
398 IF( rwork( i ).GT.safe2 ) THEN
399 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
400 ELSE
401 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
402 \$ safe1
403 END IF
404 90 CONTINUE
405*
406 kase = 0
407 100 CONTINUE
408 CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
409 IF( kase.NE.0 ) THEN
410 IF( kase.EQ.1 ) THEN
411*
412* Multiply by diag(W)*inv(A**H).
413*
414 CALL zpbtrs( uplo, n, kd, 1, afb, ldafb, work, n, info )
415 DO 110 i = 1, n
416 work( i ) = rwork( i )*work( i )
417 110 CONTINUE
418 ELSE IF( kase.EQ.2 ) THEN
419*
420* Multiply by inv(A)*diag(W).
421*
422 DO 120 i = 1, n
423 work( i ) = rwork( i )*work( i )
424 120 CONTINUE
425 CALL zpbtrs( uplo, n, kd, 1, afb, ldafb, work, n, info )
426 END IF
427 GO TO 100
428 END IF
429*
430* Normalize error.
431*
432 lstres = zero
433 DO 130 i = 1, n
434 lstres = max( lstres, cabs1( x( i, j ) ) )
435 130 CONTINUE
436 IF( lstres.NE.zero )
437 \$ ferr( j ) = ferr( j ) / lstres
438*
439 140 CONTINUE
440*
441 RETURN
442*
443* End of ZPBRFS
444*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zaxpy(n, za, zx, incx, zy, incy)
ZAXPY
Definition zaxpy.f:88
subroutine zcopy(n, zx, incx, zy, incy)
ZCOPY
Definition zcopy.f:81
subroutine zhbmv(uplo, n, k, alpha, a, lda, x, incx, beta, y, incy)
ZHBMV
Definition zhbmv.f:187
subroutine zlacn2(n, v, x, est, kase, isave)
ZLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition zlacn2.f:133
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine zpbtrs(uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
ZPBTRS
Definition zpbtrs.f:121
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