 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zpprfs()

 subroutine zpprfs ( character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( * ) AFP, 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 )

ZPPRFS

Purpose:
``` ZPPRFS improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian positive definite
and packed, 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] AP ``` AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.``` [in] AFP ``` AFP is COMPLEX*16 array, dimension (N*(N+1)/2) The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by DPPTRF/ZPPTRF, packed columnwise in a linear array in the same format as A (see AP).``` [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 ZPPTRS. 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 169 of file zpprfs.f.

171*
172* -- LAPACK computational 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 INFO, LDB, LDX, N, NRHS
179* ..
180* .. Array Arguments ..
181 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
182 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
183 \$ X( LDX, * )
184* ..
185*
186* ====================================================================
187*
188* .. Parameters ..
189 INTEGER ITMAX
190 parameter( itmax = 5 )
191 DOUBLE PRECISION ZERO
192 parameter( zero = 0.0d+0 )
193 COMPLEX*16 CONE
194 parameter( cone = ( 1.0d+0, 0.0d+0 ) )
195 DOUBLE PRECISION TWO
196 parameter( two = 2.0d+0 )
197 DOUBLE PRECISION THREE
198 parameter( three = 3.0d+0 )
199* ..
200* .. Local Scalars ..
201 LOGICAL UPPER
202 INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
203 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
204 COMPLEX*16 ZDUM
205* ..
206* .. Local Arrays ..
207 INTEGER ISAVE( 3 )
208* ..
209* .. External Subroutines ..
210 EXTERNAL xerbla, zaxpy, zcopy, zhpmv, zlacn2, zpptrs
211* ..
212* .. Intrinsic Functions ..
213 INTRINSIC abs, dble, dimag, max
214* ..
215* .. External Functions ..
216 LOGICAL LSAME
217 DOUBLE PRECISION DLAMCH
218 EXTERNAL lsame, dlamch
219* ..
220* .. Statement Functions ..
221 DOUBLE PRECISION CABS1
222* ..
223* .. Statement Function definitions ..
224 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
225* ..
226* .. Executable Statements ..
227*
228* Test the input parameters.
229*
230 info = 0
231 upper = lsame( uplo, 'U' )
232 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
233 info = -1
234 ELSE IF( n.LT.0 ) THEN
235 info = -2
236 ELSE IF( nrhs.LT.0 ) THEN
237 info = -3
238 ELSE IF( ldb.LT.max( 1, n ) ) THEN
239 info = -7
240 ELSE IF( ldx.LT.max( 1, n ) ) THEN
241 info = -9
242 END IF
243 IF( info.NE.0 ) THEN
244 CALL xerbla( 'ZPPRFS', -info )
245 RETURN
246 END IF
247*
248* Quick return if possible
249*
250 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
251 DO 10 j = 1, nrhs
252 ferr( j ) = zero
253 berr( j ) = zero
254 10 CONTINUE
255 RETURN
256 END IF
257*
258* NZ = maximum number of nonzero elements in each row of A, plus 1
259*
260 nz = n + 1
261 eps = dlamch( 'Epsilon' )
262 safmin = dlamch( 'Safe minimum' )
263 safe1 = nz*safmin
264 safe2 = safe1 / eps
265*
266* Do for each right hand side
267*
268 DO 140 j = 1, nrhs
269*
270 count = 1
271 lstres = three
272 20 CONTINUE
273*
274* Loop until stopping criterion is satisfied.
275*
276* Compute residual R = B - A * X
277*
278 CALL zcopy( n, b( 1, j ), 1, work, 1 )
279 CALL zhpmv( uplo, n, -cone, ap, x( 1, j ), 1, cone, work, 1 )
280*
281* Compute componentwise relative backward error from formula
282*
283* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
284*
285* where abs(Z) is the componentwise absolute value of the matrix
286* or vector Z. If the i-th component of the denominator is less
287* than SAFE2, then SAFE1 is added to the i-th components of the
288* numerator and denominator before dividing.
289*
290 DO 30 i = 1, n
291 rwork( i ) = cabs1( b( i, j ) )
292 30 CONTINUE
293*
294* Compute abs(A)*abs(X) + abs(B).
295*
296 kk = 1
297 IF( upper ) THEN
298 DO 50 k = 1, n
299 s = zero
300 xk = cabs1( x( k, j ) )
301 ik = kk
302 DO 40 i = 1, k - 1
303 rwork( i ) = rwork( i ) + cabs1( ap( ik ) )*xk
304 s = s + cabs1( ap( ik ) )*cabs1( x( i, j ) )
305 ik = ik + 1
306 40 CONTINUE
307 rwork( k ) = rwork( k ) + abs( dble( ap( kk+k-1 ) ) )*
308 \$ xk + s
309 kk = kk + k
310 50 CONTINUE
311 ELSE
312 DO 70 k = 1, n
313 s = zero
314 xk = cabs1( x( k, j ) )
315 rwork( k ) = rwork( k ) + abs( dble( ap( kk ) ) )*xk
316 ik = kk + 1
317 DO 60 i = k + 1, n
318 rwork( i ) = rwork( i ) + cabs1( ap( ik ) )*xk
319 s = s + cabs1( ap( ik ) )*cabs1( x( i, j ) )
320 ik = ik + 1
321 60 CONTINUE
322 rwork( k ) = rwork( k ) + s
323 kk = kk + ( n-k+1 )
324 70 CONTINUE
325 END IF
326 s = zero
327 DO 80 i = 1, n
328 IF( rwork( i ).GT.safe2 ) THEN
329 s = max( s, cabs1( work( i ) ) / rwork( i ) )
330 ELSE
331 s = max( s, ( cabs1( work( i ) )+safe1 ) /
332 \$ ( rwork( i )+safe1 ) )
333 END IF
334 80 CONTINUE
335 berr( j ) = s
336*
337* Test stopping criterion. Continue iterating if
338* 1) The residual BERR(J) is larger than machine epsilon, and
339* 2) BERR(J) decreased by at least a factor of 2 during the
340* last iteration, and
341* 3) At most ITMAX iterations tried.
342*
343 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
344 \$ count.LE.itmax ) THEN
345*
346* Update solution and try again.
347*
348 CALL zpptrs( uplo, n, 1, afp, work, n, info )
349 CALL zaxpy( n, cone, work, 1, x( 1, j ), 1 )
350 lstres = berr( j )
351 count = count + 1
352 GO TO 20
353 END IF
354*
355* Bound error from formula
356*
357* norm(X - XTRUE) / norm(X) .le. FERR =
358* norm( abs(inv(A))*
359* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
360*
361* where
362* norm(Z) is the magnitude of the largest component of Z
363* inv(A) is the inverse of A
364* abs(Z) is the componentwise absolute value of the matrix or
365* vector Z
366* NZ is the maximum number of nonzeros in any row of A, plus 1
367* EPS is machine epsilon
368*
369* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
370* is incremented by SAFE1 if the i-th component of
371* abs(A)*abs(X) + abs(B) is less than SAFE2.
372*
373* Use ZLACN2 to estimate the infinity-norm of the matrix
374* inv(A) * diag(W),
375* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
376*
377 DO 90 i = 1, n
378 IF( rwork( i ).GT.safe2 ) THEN
379 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
380 ELSE
381 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
382 \$ safe1
383 END IF
384 90 CONTINUE
385*
386 kase = 0
387 100 CONTINUE
388 CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
389 IF( kase.NE.0 ) THEN
390 IF( kase.EQ.1 ) THEN
391*
392* Multiply by diag(W)*inv(A**H).
393*
394 CALL zpptrs( uplo, n, 1, afp, work, n, info )
395 DO 110 i = 1, n
396 work( i ) = rwork( i )*work( i )
397 110 CONTINUE
398 ELSE IF( kase.EQ.2 ) THEN
399*
400* Multiply by inv(A)*diag(W).
401*
402 DO 120 i = 1, n
403 work( i ) = rwork( i )*work( i )
404 120 CONTINUE
405 CALL zpptrs( uplo, n, 1, afp, work, n, info )
406 END IF
407 GO TO 100
408 END IF
409*
410* Normalize error.
411*
412 lstres = zero
413 DO 130 i = 1, n
414 lstres = max( lstres, cabs1( x( i, j ) ) )
415 130 CONTINUE
416 IF( lstres.NE.zero )
417 \$ ferr( j ) = ferr( j ) / lstres
418*
419 140 CONTINUE
420*
421 RETURN
422*
423* End of ZPPRFS
424*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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 zhpmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
ZHPMV
Definition: zhpmv.f:149
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
subroutine zpptrs(UPLO, N, NRHS, AP, B, LDB, INFO)
ZPPTRS
Definition: zpptrs.f:108
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