 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ 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

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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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