 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

◆ ssyrfs()

 subroutine ssyrfs ( character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

SSYRFS

Purpose:
SSYRFS 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 REAL 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 REAL 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 SSYTRF. [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 SSYTRF. [in] B B is REAL 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 REAL array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by SSYTRS. 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 REAL array, dimension (3*N) [out] IWORK IWORK is INTEGER 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 189 of file ssyrfs.f.

191 *
192 * -- LAPACK computational routine --
193 * -- LAPACK is a software package provided by Univ. of Tennessee, --
194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
195 *
196 * .. Scalar Arguments ..
197  CHARACTER UPLO
198  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
199 * ..
200 * .. Array Arguments ..
201  INTEGER IPIV( * ), IWORK( * )
202  REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
203  \$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
204 * ..
205 *
206 * =====================================================================
207 *
208 * .. Parameters ..
209  INTEGER ITMAX
210  parameter( itmax = 5 )
211  REAL ZERO
212  parameter( zero = 0.0e+0 )
213  REAL ONE
214  parameter( one = 1.0e+0 )
215  REAL TWO
216  parameter( two = 2.0e+0 )
217  REAL THREE
218  parameter( three = 3.0e+0 )
219 * ..
220 * .. Local Scalars ..
221  LOGICAL UPPER
222  INTEGER COUNT, I, J, K, KASE, NZ
223  REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
224 * ..
225 * .. Local Arrays ..
226  INTEGER ISAVE( 3 )
227 * ..
228 * .. External Subroutines ..
229  EXTERNAL saxpy, scopy, slacn2, ssymv, ssytrs, xerbla
230 * ..
231 * .. Intrinsic Functions ..
232  INTRINSIC abs, max
233 * ..
234 * .. External Functions ..
235  LOGICAL LSAME
236  REAL SLAMCH
237  EXTERNAL lsame, slamch
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( lda.LT.max( 1, n ) ) THEN
252  info = -5
253  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
254  info = -7
255  ELSE IF( ldb.LT.max( 1, n ) ) THEN
256  info = -10
257  ELSE IF( ldx.LT.max( 1, n ) ) THEN
258  info = -12
259  END IF
260  IF( info.NE.0 ) THEN
261  CALL xerbla( 'SSYRFS', -info )
262  RETURN
263  END IF
264 *
265 * Quick return if possible
266 *
267  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
268  DO 10 j = 1, nrhs
269  ferr( j ) = zero
270  berr( j ) = zero
271  10 CONTINUE
272  RETURN
273  END IF
274 *
275 * NZ = maximum number of nonzero elements in each row of A, plus 1
276 *
277  nz = n + 1
278  eps = slamch( 'Epsilon' )
279  safmin = slamch( 'Safe minimum' )
280  safe1 = nz*safmin
281  safe2 = safe1 / eps
282 *
283 * Do for each right hand side
284 *
285  DO 140 j = 1, nrhs
286 *
287  count = 1
288  lstres = three
289  20 CONTINUE
290 *
291 * Loop until stopping criterion is satisfied.
292 *
293 * Compute residual R = B - A * X
294 *
295  CALL scopy( n, b( 1, j ), 1, work( n+1 ), 1 )
296  CALL ssymv( uplo, n, -one, a, lda, x( 1, j ), 1, one,
297  \$ work( n+1 ), 1 )
298 *
299 * Compute componentwise relative backward error from formula
300 *
301 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
302 *
303 * where abs(Z) is the componentwise absolute value of the matrix
304 * or vector Z. If the i-th component of the denominator is less
305 * than SAFE2, then SAFE1 is added to the i-th components of the
306 * numerator and denominator before dividing.
307 *
308  DO 30 i = 1, n
309  work( i ) = abs( b( i, j ) )
310  30 CONTINUE
311 *
312 * Compute abs(A)*abs(X) + abs(B).
313 *
314  IF( upper ) THEN
315  DO 50 k = 1, n
316  s = zero
317  xk = abs( x( k, j ) )
318  DO 40 i = 1, k - 1
319  work( i ) = work( i ) + abs( a( i, k ) )*xk
320  s = s + abs( a( i, k ) )*abs( x( i, j ) )
321  40 CONTINUE
322  work( k ) = work( k ) + abs( a( k, k ) )*xk + s
323  50 CONTINUE
324  ELSE
325  DO 70 k = 1, n
326  s = zero
327  xk = abs( x( k, j ) )
328  work( k ) = work( k ) + abs( a( k, k ) )*xk
329  DO 60 i = k + 1, n
330  work( i ) = work( i ) + abs( a( i, k ) )*xk
331  s = s + abs( a( i, k ) )*abs( x( i, j ) )
332  60 CONTINUE
333  work( k ) = work( k ) + s
334  70 CONTINUE
335  END IF
336  s = zero
337  DO 80 i = 1, n
338  IF( work( i ).GT.safe2 ) THEN
339  s = max( s, abs( work( n+i ) ) / work( i ) )
340  ELSE
341  s = max( s, ( abs( work( n+i ) )+safe1 ) /
342  \$ ( work( i )+safe1 ) )
343  END IF
344  80 CONTINUE
345  berr( j ) = s
346 *
347 * Test stopping criterion. Continue iterating if
348 * 1) The residual BERR(J) is larger than machine epsilon, and
349 * 2) BERR(J) decreased by at least a factor of 2 during the
350 * last iteration, and
351 * 3) At most ITMAX iterations tried.
352 *
353  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
354  \$ count.LE.itmax ) THEN
355 *
356 * Update solution and try again.
357 *
358  CALL ssytrs( uplo, n, 1, af, ldaf, ipiv, work( n+1 ), n,
359  \$ info )
360  CALL saxpy( n, one, work( n+1 ), 1, x( 1, j ), 1 )
361  lstres = berr( j )
362  count = count + 1
363  GO TO 20
364  END IF
365 *
366 * Bound error from formula
367 *
368 * norm(X - XTRUE) / norm(X) .le. FERR =
369 * norm( abs(inv(A))*
370 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
371 *
372 * where
373 * norm(Z) is the magnitude of the largest component of Z
374 * inv(A) is the inverse of A
375 * abs(Z) is the componentwise absolute value of the matrix or
376 * vector Z
377 * NZ is the maximum number of nonzeros in any row of A, plus 1
378 * EPS is machine epsilon
379 *
380 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
381 * is incremented by SAFE1 if the i-th component of
382 * abs(A)*abs(X) + abs(B) is less than SAFE2.
383 *
384 * Use SLACN2 to estimate the infinity-norm of the matrix
385 * inv(A) * diag(W),
386 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
387 *
388  DO 90 i = 1, n
389  IF( work( i ).GT.safe2 ) THEN
390  work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
391  ELSE
392  work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
393  END IF
394  90 CONTINUE
395 *
396  kase = 0
397  100 CONTINUE
398  CALL slacn2( n, work( 2*n+1 ), work( n+1 ), iwork, ferr( j ),
399  \$ kase, isave )
400  IF( kase.NE.0 ) THEN
401  IF( kase.EQ.1 ) THEN
402 *
403 * Multiply by diag(W)*inv(A**T).
404 *
405  CALL ssytrs( uplo, n, 1, af, ldaf, ipiv, work( n+1 ), n,
406  \$ info )
407  DO 110 i = 1, n
408  work( n+i ) = work( i )*work( n+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( n+i ) = work( i )*work( n+i )
416  120 CONTINUE
417  CALL ssytrs( uplo, n, 1, af, ldaf, ipiv, work( n+1 ), n,
418  \$ info )
419  END IF
420  GO TO 100
421  END IF
422 *
423 * Normalize error.
424 *
425  lstres = zero
426  DO 130 i = 1, n
427  lstres = max( lstres, abs( x( i, j ) ) )
428  130 CONTINUE
429  IF( lstres.NE.zero )
430  \$ ferr( j ) = ferr( j ) / lstres
431 *
432  140 CONTINUE
433 *
434  RETURN
435 *
436 * End of SSYRFS
437 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine slacn2(N, V, X, ISGN, EST, KASE, ISAVE)
SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: slacn2.f:136
subroutine ssytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SSYTRS
Definition: ssytrs.f:120
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine ssymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SSYMV
Definition: ssymv.f:152
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
Here is the call graph for this function:
Here is the caller graph for this function: