 LAPACK  3.10.0 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

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.`

Definition at line 190 of file zsyrfs.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  DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
204  COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
205  \$ WORK( * ), X( LDX, * )
206 * ..
207 *
208 * =====================================================================
209 *
210 * .. Parameters ..
211  INTEGER ITMAX
212  parameter( itmax = 5 )
213  DOUBLE PRECISION ZERO
214  parameter( zero = 0.0d+0 )
215  COMPLEX*16 ONE
216  parameter( one = ( 1.0d+0, 0.0d+0 ) )
217  DOUBLE PRECISION TWO
218  parameter( two = 2.0d+0 )
219  DOUBLE PRECISION THREE
220  parameter( three = 3.0d+0 )
221 * ..
222 * .. Local Scalars ..
223  LOGICAL UPPER
224  INTEGER COUNT, I, J, K, KASE, NZ
225  DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
226  COMPLEX*16 ZDUM
227 * ..
228 * .. Local Arrays ..
229  INTEGER ISAVE( 3 )
230 * ..
231 * .. External Subroutines ..
232  EXTERNAL xerbla, zaxpy, zcopy, zlacn2, zsymv, zsytrs
233 * ..
234 * .. Intrinsic Functions ..
235  INTRINSIC abs, dble, dimag, max
236 * ..
237 * .. External Functions ..
238  LOGICAL LSAME
239  DOUBLE PRECISION DLAMCH
240  EXTERNAL lsame, dlamch
241 * ..
242 * .. Statement Functions ..
243  DOUBLE PRECISION CABS1
244 * ..
245 * .. Statement Function definitions ..
246  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( 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( 'ZSYRFS', -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 = dlamch( 'Epsilon' )
288  safmin = dlamch( '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 zcopy( n, b( 1, j ), 1, work, 1 )
305  CALL zsymv( 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 ) + cabs1( 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 ) + cabs1( 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 zsytrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
367  CALL zaxpy( 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 ZLACN2 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 zlacn2( 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**T).
411 *
412  CALL zsytrs( 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 zsytrs( 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 ZSYRFS
442 *
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 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 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:157
subroutine zsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZSYTRS
Definition: zsytrs.f:120
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