 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ csprfs()

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

CSPRFS

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Purpose:
``` CSPRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefinite
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 array, dimension (N*(N+1)/2) The upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.``` [in] AFP ``` AFP is COMPLEX array, dimension (N*(N+1)/2) The factored form of the matrix A. AFP contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as a packed triangular matrix.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by CSPTRF.``` [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 CSPTRS. 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.`
Date
December 2016

Definition at line 182 of file csprfs.f.

182 *
183 * -- LAPACK computational routine (version 3.7.0) --
184 * -- LAPACK is a software package provided by Univ. of Tennessee, --
185 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
186 * December 2016
187 *
188 * .. Scalar Arguments ..
189  CHARACTER uplo
190  INTEGER info, ldb, ldx, n, nrhs
191 * ..
192 * .. Array Arguments ..
193  INTEGER ipiv( * )
194  REAL berr( * ), ferr( * ), rwork( * )
195  COMPLEX afp( * ), ap( * ), b( ldb, * ), work( * ),
196  \$ x( ldx, * )
197 * ..
198 *
199 * =====================================================================
200 *
201 * .. Parameters ..
202  INTEGER itmax
203  parameter( itmax = 5 )
204  REAL zero
205  parameter( zero = 0.0e+0 )
206  COMPLEX one
207  parameter( one = ( 1.0e+0, 0.0e+0 ) )
208  REAL two
209  parameter( two = 2.0e+0 )
210  REAL three
211  parameter( three = 3.0e+0 )
212 * ..
213 * .. Local Scalars ..
214  LOGICAL upper
215  INTEGER count, i, ik, j, k, kase, kk, nz
216  REAL eps, lstres, s, safe1, safe2, safmin, xk
217  COMPLEX zdum
218 * ..
219 * .. Local Arrays ..
220  INTEGER isave( 3 )
221 * ..
222 * .. External Subroutines ..
223  EXTERNAL caxpy, ccopy, clacn2, cspmv, csptrs, xerbla
224 * ..
225 * .. Intrinsic Functions ..
226  INTRINSIC abs, aimag, max, real
227 * ..
228 * .. External Functions ..
229  LOGICAL lsame
230  REAL slamch
231  EXTERNAL lsame, slamch
232 * ..
233 * .. Statement Functions ..
234  REAL cabs1
235 * ..
236 * .. Statement Function definitions ..
237  cabs1( zdum ) = abs( REAL( ZDUM ) ) + abs( aimag( zdum ) )
238 * ..
239 * .. Executable Statements ..
240 *
241 * Test the input parameters.
242 *
243  info = 0
244  upper = lsame( uplo, 'U' )
245  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
246  info = -1
247  ELSE IF( n.LT.0 ) THEN
248  info = -2
249  ELSE IF( nrhs.LT.0 ) THEN
250  info = -3
251  ELSE IF( ldb.LT.max( 1, n ) ) THEN
252  info = -8
253  ELSE IF( ldx.LT.max( 1, n ) ) THEN
254  info = -10
255  END IF
256  IF( info.NE.0 ) THEN
257  CALL xerbla( 'CSPRFS', -info )
258  RETURN
259  END IF
260 *
261 * Quick return if possible
262 *
263  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
264  DO 10 j = 1, nrhs
265  ferr( j ) = zero
266  berr( j ) = zero
267  10 CONTINUE
268  RETURN
269  END IF
270 *
271 * NZ = maximum number of nonzero elements in each row of A, plus 1
272 *
273  nz = n + 1
274  eps = slamch( 'Epsilon' )
275  safmin = slamch( 'Safe minimum' )
276  safe1 = nz*safmin
277  safe2 = safe1 / eps
278 *
279 * Do for each right hand side
280 *
281  DO 140 j = 1, nrhs
282 *
283  count = 1
284  lstres = three
285  20 CONTINUE
286 *
287 * Loop until stopping criterion is satisfied.
288 *
289 * Compute residual R = B - A * X
290 *
291  CALL ccopy( n, b( 1, j ), 1, work, 1 )
292  CALL cspmv( uplo, n, -one, ap, x( 1, j ), 1, one, work, 1 )
293 *
294 * Compute componentwise relative backward error from formula
295 *
296 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
297 *
298 * where abs(Z) is the componentwise absolute value of the matrix
299 * or vector Z. If the i-th component of the denominator is less
300 * than SAFE2, then SAFE1 is added to the i-th components of the
301 * numerator and denominator before dividing.
302 *
303  DO 30 i = 1, n
304  rwork( i ) = cabs1( b( i, j ) )
305  30 CONTINUE
306 *
307 * Compute abs(A)*abs(X) + abs(B).
308 *
309  kk = 1
310  IF( upper ) THEN
311  DO 50 k = 1, n
312  s = zero
313  xk = cabs1( x( k, j ) )
314  ik = kk
315  DO 40 i = 1, k - 1
316  rwork( i ) = rwork( i ) + cabs1( ap( ik ) )*xk
317  s = s + cabs1( ap( ik ) )*cabs1( x( i, j ) )
318  ik = ik + 1
319  40 CONTINUE
320  rwork( k ) = rwork( k ) + cabs1( ap( kk+k-1 ) )*xk + s
321  kk = kk + k
322  50 CONTINUE
323  ELSE
324  DO 70 k = 1, n
325  s = zero
326  xk = cabs1( x( k, j ) )
327  rwork( k ) = rwork( k ) + cabs1( ap( kk ) )*xk
328  ik = kk + 1
329  DO 60 i = k + 1, n
330  rwork( i ) = rwork( i ) + cabs1( ap( ik ) )*xk
331  s = s + cabs1( ap( ik ) )*cabs1( x( i, j ) )
332  ik = ik + 1
333  60 CONTINUE
334  rwork( k ) = rwork( k ) + s
335  kk = kk + ( n-k+1 )
336  70 CONTINUE
337  END IF
338  s = zero
339  DO 80 i = 1, n
340  IF( rwork( i ).GT.safe2 ) THEN
341  s = max( s, cabs1( work( i ) ) / rwork( i ) )
342  ELSE
343  s = max( s, ( cabs1( work( i ) )+safe1 ) /
344  \$ ( rwork( i )+safe1 ) )
345  END IF
346  80 CONTINUE
347  berr( j ) = s
348 *
349 * Test stopping criterion. Continue iterating if
350 * 1) The residual BERR(J) is larger than machine epsilon, and
351 * 2) BERR(J) decreased by at least a factor of 2 during the
352 * last iteration, and
353 * 3) At most ITMAX iterations tried.
354 *
355  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
356  \$ count.LE.itmax ) THEN
357 *
358 * Update solution and try again.
359 *
360  CALL csptrs( uplo, n, 1, afp, ipiv, work, n, info )
361  CALL caxpy( n, one, work, 1, x( 1, j ), 1 )
362  lstres = berr( j )
363  count = count + 1
364  GO TO 20
365  END IF
366 *
367 * Bound error from formula
368 *
369 * norm(X - XTRUE) / norm(X) .le. FERR =
370 * norm( abs(inv(A))*
371 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
372 *
373 * where
374 * norm(Z) is the magnitude of the largest component of Z
375 * inv(A) is the inverse of A
376 * abs(Z) is the componentwise absolute value of the matrix or
377 * vector Z
378 * NZ is the maximum number of nonzeros in any row of A, plus 1
379 * EPS is machine epsilon
380 *
381 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
382 * is incremented by SAFE1 if the i-th component of
383 * abs(A)*abs(X) + abs(B) is less than SAFE2.
384 *
385 * Use CLACN2 to estimate the infinity-norm of the matrix
386 * inv(A) * diag(W),
387 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
388 *
389  DO 90 i = 1, n
390  IF( rwork( i ).GT.safe2 ) THEN
391  rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
392  ELSE
393  rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
394  \$ safe1
395  END IF
396  90 CONTINUE
397 *
398  kase = 0
399  100 CONTINUE
400  CALL clacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
401  IF( kase.NE.0 ) THEN
402  IF( kase.EQ.1 ) THEN
403 *
404 * Multiply by diag(W)*inv(A**T).
405 *
406  CALL csptrs( uplo, n, 1, afp, ipiv, work, n, info )
407  DO 110 i = 1, n
408  work( i ) = rwork( i )*work( i )
409  110 CONTINUE
410  ELSE IF( kase.EQ.2 ) THEN
411 *
412 * Multiply by inv(A)*diag(W).
413 *
414  DO 120 i = 1, n
415  work( i ) = rwork( i )*work( i )
416  120 CONTINUE
417  CALL csptrs( uplo, n, 1, afp, ipiv, work, n, info )
418  END IF
419  GO TO 100
420  END IF
421 *
422 * Normalize error.
423 *
424  lstres = zero
425  DO 130 i = 1, n
426  lstres = max( lstres, cabs1( x( i, j ) ) )
427  130 CONTINUE
428  IF( lstres.NE.zero )
429  \$ ferr( j ) = ferr( j ) / lstres
430 *
431  140 CONTINUE
432 *
433  RETURN
434 *
435 * End of CSPRFS
436 *
subroutine csptrs(UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
CSPTRS
Definition: csptrs.f:117
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine cspmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
CSPMV computes a matrix-vector product for complex vectors using a complex symmetric packed matrix ...
Definition: cspmv.f:153
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
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:135
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:83
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:90
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