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

 subroutine cherfs ( character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info )

CHERFS

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Purpose:
``` CHERFS improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian indefinite, 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 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 array, dimension (LDAF,N) The factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by CHETRF.``` [in] LDAF ``` LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by CHETRF.``` [in] B ``` B is COMPLEX 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 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by CHETRS. 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 REAL 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 REAL 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 array, dimension (2*N)` [out] RWORK ` RWORK is REAL 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 190 of file cherfs.f.

192*
193* -- LAPACK computational routine --
194* -- LAPACK is a software package provided by Univ. of Tennessee, --
195* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
196*
197* .. Scalar Arguments ..
198 CHARACTER UPLO
199 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
200* ..
201* .. Array Arguments ..
202 INTEGER IPIV( * )
203 REAL BERR( * ), FERR( * ), RWORK( * )
204 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
205 \$ WORK( * ), X( LDX, * )
206* ..
207*
208* =====================================================================
209*
210* .. Parameters ..
211 INTEGER ITMAX
212 parameter( itmax = 5 )
213 REAL ZERO
214 parameter( zero = 0.0e+0 )
215 COMPLEX ONE
216 parameter( one = ( 1.0e+0, 0.0e+0 ) )
217 REAL TWO
218 parameter( two = 2.0e+0 )
219 REAL THREE
220 parameter( three = 3.0e+0 )
221* ..
222* .. Local Scalars ..
223 LOGICAL UPPER
224 INTEGER COUNT, I, J, K, KASE, NZ
225 REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
226 COMPLEX ZDUM
227* ..
228* .. Local Arrays ..
229 INTEGER ISAVE( 3 )
230* ..
231* .. External Subroutines ..
232 EXTERNAL caxpy, ccopy, chemv, chetrs, clacn2, xerbla
233* ..
234* .. Intrinsic Functions ..
235 INTRINSIC abs, aimag, max, real
236* ..
237* .. External Functions ..
238 LOGICAL LSAME
239 REAL SLAMCH
240 EXTERNAL lsame, slamch
241* ..
242* .. Statement Functions ..
243 REAL CABS1
244* ..
245* .. Statement Function definitions ..
246 cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
247* ..
248* .. Executable Statements ..
249*
250* Test the input parameters.
251*
252 info = 0
253 upper = lsame( uplo, 'U' )
254 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
255 info = -1
256 ELSE IF( n.LT.0 ) THEN
257 info = -2
258 ELSE IF( nrhs.LT.0 ) THEN
259 info = -3
260 ELSE IF( lda.LT.max( 1, n ) ) THEN
261 info = -5
262 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
263 info = -7
264 ELSE IF( ldb.LT.max( 1, n ) ) THEN
265 info = -10
266 ELSE IF( ldx.LT.max( 1, n ) ) THEN
267 info = -12
268 END IF
269 IF( info.NE.0 ) THEN
270 CALL xerbla( 'CHERFS', -info )
271 RETURN
272 END IF
273*
274* Quick return if possible
275*
276 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
277 DO 10 j = 1, nrhs
278 ferr( j ) = zero
279 berr( j ) = zero
280 10 CONTINUE
281 RETURN
282 END IF
283*
284* NZ = maximum number of nonzero elements in each row of A, plus 1
285*
286 nz = n + 1
287 eps = slamch( 'Epsilon' )
288 safmin = slamch( 'Safe minimum' )
289 safe1 = nz*safmin
290 safe2 = safe1 / eps
291*
292* Do for each right hand side
293*
294 DO 140 j = 1, nrhs
295*
296 count = 1
297 lstres = three
298 20 CONTINUE
299*
300* Loop until stopping criterion is satisfied.
301*
302* Compute residual R = B - A * X
303*
304 CALL ccopy( n, b( 1, j ), 1, work, 1 )
305 CALL chemv( uplo, n, -one, a, lda, x( 1, j ), 1, one, work, 1 )
306*
307* Compute componentwise relative backward error from formula
308*
309* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
310*
311* where abs(Z) is the componentwise absolute value of the matrix
312* or vector Z. If the i-th component of the denominator is less
313* than SAFE2, then SAFE1 is added to the i-th components of the
314* numerator and denominator before dividing.
315*
316 DO 30 i = 1, n
317 rwork( i ) = cabs1( b( i, j ) )
318 30 CONTINUE
319*
320* Compute abs(A)*abs(X) + abs(B).
321*
322 IF( upper ) THEN
323 DO 50 k = 1, n
324 s = zero
325 xk = cabs1( x( k, j ) )
326 DO 40 i = 1, k - 1
327 rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
328 s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
329 40 CONTINUE
330 rwork( k ) = rwork( k ) + abs( real( a( k, k ) ) )*xk + s
331 50 CONTINUE
332 ELSE
333 DO 70 k = 1, n
334 s = zero
335 xk = cabs1( x( k, j ) )
336 rwork( k ) = rwork( k ) + abs( real( a( k, k ) ) )*xk
337 DO 60 i = k + 1, n
338 rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
339 s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
340 60 CONTINUE
341 rwork( k ) = rwork( k ) + s
342 70 CONTINUE
343 END IF
344 s = zero
345 DO 80 i = 1, n
346 IF( rwork( i ).GT.safe2 ) THEN
347 s = max( s, cabs1( work( i ) ) / rwork( i ) )
348 ELSE
349 s = max( s, ( cabs1( work( i ) )+safe1 ) /
350 \$ ( rwork( i )+safe1 ) )
351 END IF
352 80 CONTINUE
353 berr( j ) = s
354*
355* Test stopping criterion. Continue iterating if
356* 1) The residual BERR(J) is larger than machine epsilon, and
357* 2) BERR(J) decreased by at least a factor of 2 during the
358* last iteration, and
359* 3) At most ITMAX iterations tried.
360*
361 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
362 \$ count.LE.itmax ) THEN
363*
364* Update solution and try again.
365*
366 CALL chetrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
367 CALL caxpy( n, one, work, 1, x( 1, j ), 1 )
368 lstres = berr( j )
369 count = count + 1
370 GO TO 20
371 END IF
372*
373* Bound error from formula
374*
375* norm(X - XTRUE) / norm(X) .le. FERR =
376* norm( abs(inv(A))*
377* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
378*
379* where
380* norm(Z) is the magnitude of the largest component of Z
381* inv(A) is the inverse of A
382* abs(Z) is the componentwise absolute value of the matrix or
383* vector Z
384* NZ is the maximum number of nonzeros in any row of A, plus 1
385* EPS is machine epsilon
386*
387* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
388* is incremented by SAFE1 if the i-th component of
389* abs(A)*abs(X) + abs(B) is less than SAFE2.
390*
391* Use CLACN2 to estimate the infinity-norm of the matrix
392* inv(A) * diag(W),
393* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
394*
395 DO 90 i = 1, n
396 IF( rwork( i ).GT.safe2 ) THEN
397 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
398 ELSE
399 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
400 \$ safe1
401 END IF
402 90 CONTINUE
403*
404 kase = 0
405 100 CONTINUE
406 CALL clacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
407 IF( kase.NE.0 ) THEN
408 IF( kase.EQ.1 ) THEN
409*
410* Multiply by diag(W)*inv(A**H).
411*
412 CALL chetrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
413 DO 110 i = 1, n
414 work( i ) = rwork( i )*work( i )
415 110 CONTINUE
416 ELSE IF( kase.EQ.2 ) THEN
417*
418* Multiply by inv(A)*diag(W).
419*
420 DO 120 i = 1, n
421 work( i ) = rwork( i )*work( i )
422 120 CONTINUE
423 CALL chetrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
424 END IF
425 GO TO 100
426 END IF
427*
428* Normalize error.
429*
430 lstres = zero
431 DO 130 i = 1, n
432 lstres = max( lstres, cabs1( x( i, j ) ) )
433 130 CONTINUE
434 IF( lstres.NE.zero )
435 \$ ferr( j ) = ferr( j ) / lstres
436*
437 140 CONTINUE
438*
439 RETURN
440*
441* End of CHERFS
442*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine caxpy(n, ca, cx, incx, cy, incy)
CAXPY
Definition caxpy.f:88
subroutine ccopy(n, cx, incx, cy, incy)
CCOPY
Definition ccopy.f:81
subroutine chemv(uplo, n, alpha, a, lda, x, incx, beta, y, incy)
CHEMV
Definition chemv.f:154
subroutine chetrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
CHETRS
Definition chetrs.f:120
subroutine clacn2(n, v, x, est, kase, isave)
CLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition clacn2.f:133
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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