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

 subroutine zporfs ( character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, 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 )

ZPORFS

Purpose:
``` ZPORFS improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian positive definite,
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] 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] A ``` A is COMPLEX*16 array, dimension (LDA,N) The Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] AF ``` AF is COMPLEX*16 array, dimension (LDAF,N) The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by ZPOTRF.``` [in] LDAF ``` LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).``` [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 ZPOTRS. 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 181 of file zporfs.f.

183*
184* -- LAPACK computational routine --
185* -- LAPACK is a software package provided by Univ. of Tennessee, --
186* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
187*
188* .. Scalar Arguments ..
189 CHARACTER UPLO
190 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
191* ..
192* .. Array Arguments ..
193 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
194 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
195 \$ WORK( * ), X( LDX, * )
196* ..
197*
198* ====================================================================
199*
200* .. Parameters ..
201 INTEGER ITMAX
202 parameter( itmax = 5 )
203 DOUBLE PRECISION ZERO
204 parameter( zero = 0.0d+0 )
205 COMPLEX*16 ONE
206 parameter( one = ( 1.0d+0, 0.0d+0 ) )
207 DOUBLE PRECISION TWO
208 parameter( two = 2.0d+0 )
209 DOUBLE PRECISION THREE
210 parameter( three = 3.0d+0 )
211* ..
212* .. Local Scalars ..
213 LOGICAL UPPER
214 INTEGER COUNT, I, J, K, KASE, NZ
215 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
216 COMPLEX*16 ZDUM
217* ..
218* .. Local Arrays ..
219 INTEGER ISAVE( 3 )
220* ..
221* .. External Subroutines ..
222 EXTERNAL xerbla, zaxpy, zcopy, zhemv, zlacn2, zpotrs
223* ..
224* .. Intrinsic Functions ..
225 INTRINSIC abs, dble, dimag, max
226* ..
227* .. External Functions ..
228 LOGICAL LSAME
229 DOUBLE PRECISION DLAMCH
230 EXTERNAL lsame, dlamch
231* ..
232* .. Statement Functions ..
233 DOUBLE PRECISION CABS1
234* ..
235* .. Statement Function definitions ..
236 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
237* ..
238* .. Executable Statements ..
239*
240* Test the input parameters.
241*
242 info = 0
243 upper = lsame( uplo, 'U' )
244 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
245 info = -1
246 ELSE IF( n.LT.0 ) THEN
247 info = -2
248 ELSE IF( nrhs.LT.0 ) THEN
249 info = -3
250 ELSE IF( lda.LT.max( 1, n ) ) THEN
251 info = -5
252 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
253 info = -7
254 ELSE IF( ldb.LT.max( 1, n ) ) THEN
255 info = -9
256 ELSE IF( ldx.LT.max( 1, n ) ) THEN
257 info = -11
258 END IF
259 IF( info.NE.0 ) THEN
260 CALL xerbla( 'ZPORFS', -info )
261 RETURN
262 END IF
263*
264* Quick return if possible
265*
266 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
267 DO 10 j = 1, nrhs
268 ferr( j ) = zero
269 berr( j ) = zero
270 10 CONTINUE
271 RETURN
272 END IF
273*
274* NZ = maximum number of nonzero elements in each row of A, plus 1
275*
276 nz = n + 1
277 eps = dlamch( 'Epsilon' )
278 safmin = dlamch( 'Safe minimum' )
279 safe1 = nz*safmin
280 safe2 = safe1 / eps
281*
282* Do for each right hand side
283*
284 DO 140 j = 1, nrhs
285*
286 count = 1
287 lstres = three
288 20 CONTINUE
289*
290* Loop until stopping criterion is satisfied.
291*
292* Compute residual R = B - A * X
293*
294 CALL zcopy( n, b( 1, j ), 1, work, 1 )
295 CALL zhemv( uplo, n, -one, a, lda, x( 1, j ), 1, one, work, 1 )
296*
297* Compute componentwise relative backward error from formula
298*
299* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
300*
301* where abs(Z) is the componentwise absolute value of the matrix
302* or vector Z. If the i-th component of the denominator is less
303* than SAFE2, then SAFE1 is added to the i-th components of the
304* numerator and denominator before dividing.
305*
306 DO 30 i = 1, n
307 rwork( i ) = cabs1( b( i, j ) )
308 30 CONTINUE
309*
310* Compute abs(A)*abs(X) + abs(B).
311*
312 IF( upper ) THEN
313 DO 50 k = 1, n
314 s = zero
315 xk = cabs1( x( k, j ) )
316 DO 40 i = 1, k - 1
317 rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
318 s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
319 40 CONTINUE
320 rwork( k ) = rwork( k ) + abs( dble( a( k, k ) ) )*xk + s
321 50 CONTINUE
322 ELSE
323 DO 70 k = 1, n
324 s = zero
325 xk = cabs1( x( k, j ) )
326 rwork( k ) = rwork( k ) + abs( dble( a( k, k ) ) )*xk
327 DO 60 i = k + 1, n
328 rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
329 s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
330 60 CONTINUE
331 rwork( k ) = rwork( k ) + s
332 70 CONTINUE
333 END IF
334 s = zero
335 DO 80 i = 1, n
336 IF( rwork( i ).GT.safe2 ) THEN
337 s = max( s, cabs1( work( i ) ) / rwork( i ) )
338 ELSE
339 s = max( s, ( cabs1( work( i ) )+safe1 ) /
340 \$ ( rwork( i )+safe1 ) )
341 END IF
342 80 CONTINUE
343 berr( j ) = s
344*
345* Test stopping criterion. Continue iterating if
346* 1) The residual BERR(J) is larger than machine epsilon, and
347* 2) BERR(J) decreased by at least a factor of 2 during the
348* last iteration, and
349* 3) At most ITMAX iterations tried.
350*
351 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
352 \$ count.LE.itmax ) THEN
353*
354* Update solution and try again.
355*
356 CALL zpotrs( uplo, n, 1, af, ldaf, work, n, info )
357 CALL zaxpy( n, one, work, 1, x( 1, j ), 1 )
358 lstres = berr( j )
359 count = count + 1
360 GO TO 20
361 END IF
362*
363* Bound error from formula
364*
365* norm(X - XTRUE) / norm(X) .le. FERR =
366* norm( abs(inv(A))*
367* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
368*
369* where
370* norm(Z) is the magnitude of the largest component of Z
371* inv(A) is the inverse of A
372* abs(Z) is the componentwise absolute value of the matrix or
373* vector Z
374* NZ is the maximum number of nonzeros in any row of A, plus 1
375* EPS is machine epsilon
376*
377* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
378* is incremented by SAFE1 if the i-th component of
379* abs(A)*abs(X) + abs(B) is less than SAFE2.
380*
381* Use ZLACN2 to estimate the infinity-norm of the matrix
382* inv(A) * diag(W),
383* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
384*
385 DO 90 i = 1, n
386 IF( rwork( i ).GT.safe2 ) THEN
387 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
388 ELSE
389 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
390 \$ safe1
391 END IF
392 90 CONTINUE
393*
394 kase = 0
395 100 CONTINUE
396 CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
397 IF( kase.NE.0 ) THEN
398 IF( kase.EQ.1 ) THEN
399*
400* Multiply by diag(W)*inv(A**H).
401*
402 CALL zpotrs( uplo, n, 1, af, ldaf, work, n, info )
403 DO 110 i = 1, n
404 work( i ) = rwork( i )*work( i )
405 110 CONTINUE
406 ELSE IF( kase.EQ.2 ) THEN
407*
408* Multiply by inv(A)*diag(W).
409*
410 DO 120 i = 1, n
411 work( i ) = rwork( i )*work( i )
412 120 CONTINUE
413 CALL zpotrs( uplo, n, 1, af, ldaf, work, n, info )
414 END IF
415 GO TO 100
416 END IF
417*
418* Normalize error.
419*
420 lstres = zero
421 DO 130 i = 1, n
422 lstres = max( lstres, cabs1( x( i, j ) ) )
423 130 CONTINUE
424 IF( lstres.NE.zero )
425 \$ ferr( j ) = ferr( j ) / lstres
426*
427 140 CONTINUE
428*
429 RETURN
430*
431* End of ZPORFS
432*
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 zhemv(uplo, n, alpha, a, lda, x, incx, beta, y, incy)
ZHEMV
Definition zhemv.f:154
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 zpotrs(uplo, n, nrhs, a, lda, b, ldb, info)
ZPOTRS
Definition zpotrs.f:110
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