 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zsyrfs()

 subroutine zsyrfs ( character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, 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 )

ZSYRFS

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Purpose:
``` ZSYRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric 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*16 array, dimension (LDA,N) The symmetric 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 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**T or A = L*D*L**T as computed by ZSYTRF.``` [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 ZSYTRF.``` [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 ZSYTRS. 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.`
Date
December 2016

Definition at line 194 of file zsyrfs.f.

194 *
195 * -- LAPACK computational routine (version 3.7.0) --
196 * -- LAPACK is a software package provided by Univ. of Tennessee, --
197 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
198 * December 2016
199 *
200 * .. Scalar Arguments ..
201  CHARACTER uplo
202  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs
203 * ..
204 * .. Array Arguments ..
205  INTEGER ipiv( * )
206  DOUBLE PRECISION berr( * ), ferr( * ), rwork( * )
207  COMPLEX*16 a( lda, * ), af( ldaf, * ), b( ldb, * ),
208  \$ work( * ), x( ldx, * )
209 * ..
210 *
211 * =====================================================================
212 *
213 * .. Parameters ..
214  INTEGER itmax
215  parameter( itmax = 5 )
216  DOUBLE PRECISION zero
217  parameter( zero = 0.0d+0 )
218  COMPLEX*16 one
219  parameter( one = ( 1.0d+0, 0.0d+0 ) )
220  DOUBLE PRECISION two
221  parameter( two = 2.0d+0 )
222  DOUBLE PRECISION three
223  parameter( three = 3.0d+0 )
224 * ..
225 * .. Local Scalars ..
226  LOGICAL upper
227  INTEGER count, i, j, k, kase, nz
228  DOUBLE PRECISION eps, lstres, s, safe1, safe2, safmin, xk
229  COMPLEX*16 zdum
230 * ..
231 * .. Local Arrays ..
232  INTEGER isave( 3 )
233 * ..
234 * .. External Subroutines ..
235  EXTERNAL xerbla, zaxpy, zcopy, zlacn2, zsymv, zsytrs
236 * ..
237 * .. Intrinsic Functions ..
238  INTRINSIC abs, dble, dimag, max
239 * ..
240 * .. External Functions ..
241  LOGICAL lsame
242  DOUBLE PRECISION dlamch
243  EXTERNAL lsame, dlamch
244 * ..
245 * .. Statement Functions ..
246  DOUBLE PRECISION cabs1
247 * ..
248 * .. Statement Function definitions ..
249  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
250 * ..
251 * .. Executable Statements ..
252 *
253 * Test the input parameters.
254 *
255  info = 0
256  upper = lsame( uplo, 'U' )
257  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
258  info = -1
259  ELSE IF( n.LT.0 ) THEN
260  info = -2
261  ELSE IF( nrhs.LT.0 ) THEN
262  info = -3
263  ELSE IF( lda.LT.max( 1, n ) ) THEN
264  info = -5
265  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
266  info = -7
267  ELSE IF( ldb.LT.max( 1, n ) ) THEN
268  info = -10
269  ELSE IF( ldx.LT.max( 1, n ) ) THEN
270  info = -12
271  END IF
272  IF( info.NE.0 ) THEN
273  CALL xerbla( 'ZSYRFS', -info )
274  RETURN
275  END IF
276 *
277 * Quick return if possible
278 *
279  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
280  DO 10 j = 1, nrhs
281  ferr( j ) = zero
282  berr( j ) = zero
283  10 CONTINUE
284  RETURN
285  END IF
286 *
287 * NZ = maximum number of nonzero elements in each row of A, plus 1
288 *
289  nz = n + 1
290  eps = dlamch( 'Epsilon' )
291  safmin = dlamch( 'Safe minimum' )
292  safe1 = nz*safmin
293  safe2 = safe1 / eps
294 *
295 * Do for each right hand side
296 *
297  DO 140 j = 1, nrhs
298 *
299  count = 1
300  lstres = three
301  20 CONTINUE
302 *
303 * Loop until stopping criterion is satisfied.
304 *
305 * Compute residual R = B - A * X
306 *
307  CALL zcopy( n, b( 1, j ), 1, work, 1 )
308  CALL zsymv( uplo, n, -one, a, lda, x( 1, j ), 1, one, work, 1 )
309 *
310 * Compute componentwise relative backward error from formula
311 *
312 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
313 *
314 * where abs(Z) is the componentwise absolute value of the matrix
315 * or vector Z. If the i-th component of the denominator is less
316 * than SAFE2, then SAFE1 is added to the i-th components of the
317 * numerator and denominator before dividing.
318 *
319  DO 30 i = 1, n
320  rwork( i ) = cabs1( b( i, j ) )
321  30 CONTINUE
322 *
323 * Compute abs(A)*abs(X) + abs(B).
324 *
325  IF( upper ) THEN
326  DO 50 k = 1, n
327  s = zero
328  xk = cabs1( x( k, j ) )
329  DO 40 i = 1, k - 1
330  rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
331  s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
332  40 CONTINUE
333  rwork( k ) = rwork( k ) + cabs1( a( k, k ) )*xk + s
334  50 CONTINUE
335  ELSE
336  DO 70 k = 1, n
337  s = zero
338  xk = cabs1( x( k, j ) )
339  rwork( k ) = rwork( k ) + cabs1( a( k, k ) )*xk
340  DO 60 i = k + 1, n
341  rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
342  s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
343  60 CONTINUE
344  rwork( k ) = rwork( k ) + s
345  70 CONTINUE
346  END IF
347  s = zero
348  DO 80 i = 1, n
349  IF( rwork( i ).GT.safe2 ) THEN
350  s = max( s, cabs1( work( i ) ) / rwork( i ) )
351  ELSE
352  s = max( s, ( cabs1( work( i ) )+safe1 ) /
353  \$ ( rwork( i )+safe1 ) )
354  END IF
355  80 CONTINUE
356  berr( j ) = s
357 *
358 * Test stopping criterion. Continue iterating if
359 * 1) The residual BERR(J) is larger than machine epsilon, and
360 * 2) BERR(J) decreased by at least a factor of 2 during the
361 * last iteration, and
362 * 3) At most ITMAX iterations tried.
363 *
364  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
365  \$ count.LE.itmax ) THEN
366 *
367 * Update solution and try again.
368 *
369  CALL zsytrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
370  CALL zaxpy( n, one, work, 1, x( 1, j ), 1 )
371  lstres = berr( j )
372  count = count + 1
373  GO TO 20
374  END IF
375 *
376 * Bound error from formula
377 *
378 * norm(X - XTRUE) / norm(X) .le. FERR =
379 * norm( abs(inv(A))*
380 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
381 *
382 * where
383 * norm(Z) is the magnitude of the largest component of Z
384 * inv(A) is the inverse of A
385 * abs(Z) is the componentwise absolute value of the matrix or
386 * vector Z
387 * NZ is the maximum number of nonzeros in any row of A, plus 1
388 * EPS is machine epsilon
389 *
390 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
391 * is incremented by SAFE1 if the i-th component of
392 * abs(A)*abs(X) + abs(B) is less than SAFE2.
393 *
394 * Use ZLACN2 to estimate the infinity-norm of the matrix
395 * inv(A) * diag(W),
396 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
397 *
398  DO 90 i = 1, n
399  IF( rwork( i ).GT.safe2 ) THEN
400  rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
401  ELSE
402  rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
403  \$ safe1
404  END IF
405  90 CONTINUE
406 *
407  kase = 0
408  100 CONTINUE
409  CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
410  IF( kase.NE.0 ) THEN
411  IF( kase.EQ.1 ) THEN
412 *
413 * Multiply by diag(W)*inv(A**T).
414 *
415  CALL zsytrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
416  DO 110 i = 1, n
417  work( i ) = rwork( i )*work( i )
418  110 CONTINUE
419  ELSE IF( kase.EQ.2 ) THEN
420 *
421 * Multiply by inv(A)*diag(W).
422 *
423  DO 120 i = 1, n
424  work( i ) = rwork( i )*work( i )
425  120 CONTINUE
426  CALL zsytrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
427  END IF
428  GO TO 100
429  END IF
430 *
431 * Normalize error.
432 *
433  lstres = zero
434  DO 130 i = 1, n
435  lstres = max( lstres, cabs1( x( i, j ) ) )
436  130 CONTINUE
437  IF( lstres.NE.zero )
438  \$ ferr( j ) = ferr( j ) / lstres
439 *
440  140 CONTINUE
441 *
442  RETURN
443 *
444 * End of ZSYRFS
445 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:83
subroutine zsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZSYMV computes a matrix-vector product for a complex symmetric matrix.
Definition: zsymv.f:159
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:135
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZSYTRS
Definition: zsytrs.f:122
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:90
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