LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dsyrfs()

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

DSYRFS

Purpose:
``` DSYRFS 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DSYTRF.``` [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 DSYTRF.``` [in] B ``` B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by DSYTRS. 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 DOUBLE PRECISION 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 dsyrfs.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 DOUBLE PRECISION 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 DOUBLE PRECISION ZERO
212 parameter( zero = 0.0d+0 )
213 DOUBLE PRECISION ONE
214 parameter( one = 1.0d+0 )
215 DOUBLE PRECISION TWO
216 parameter( two = 2.0d+0 )
217 DOUBLE PRECISION THREE
218 parameter( three = 3.0d+0 )
219* ..
220* .. Local Scalars ..
221 LOGICAL UPPER
222 INTEGER COUNT, I, J, K, KASE, NZ
223 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
224* ..
225* .. Local Arrays ..
226 INTEGER ISAVE( 3 )
227* ..
228* .. External Subroutines ..
229 EXTERNAL daxpy, dcopy, dlacn2, dsymv, dsytrs, xerbla
230* ..
231* .. Intrinsic Functions ..
232 INTRINSIC abs, max
233* ..
234* .. External Functions ..
235 LOGICAL LSAME
236 DOUBLE PRECISION DLAMCH
237 EXTERNAL lsame, dlamch
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( 'DSYRFS', -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 = dlamch( 'Epsilon' )
279 safmin = dlamch( '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 dcopy( n, b( 1, j ), 1, work( n+1 ), 1 )
296 CALL dsymv( 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 dsytrs( uplo, n, 1, af, ldaf, ipiv, work( n+1 ), n,
359 \$ info )
360 CALL daxpy( 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 DLACN2 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 dlacn2( 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 dsytrs( 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 dsytrs( 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 DSYRFS
437*
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 dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DSYMV
Definition: dsymv.f:152
subroutine dlacn2(N, V, X, ISGN, EST, KASE, ISAVE)
DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: dlacn2.f:136
subroutine dsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DSYTRS
Definition: dsytrs.f:120
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