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

 subroutine dsprfs ( character UPLO, integer N, integer NRHS, double precision, dimension( * ) AP, double precision, dimension( * ) AFP, 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 )

DSPRFS

Purpose:
``` DSPRFS 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DSPTRF, 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 DSPTRF.``` [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 DSPTRS. 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 177 of file dsprfs.f.

179*
180* -- LAPACK computational routine --
181* -- LAPACK is a software package provided by Univ. of Tennessee, --
182* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
183*
184* .. Scalar Arguments ..
185 CHARACTER UPLO
186 INTEGER INFO, LDB, LDX, N, NRHS
187* ..
188* .. Array Arguments ..
189 INTEGER IPIV( * ), IWORK( * )
190 DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
191 \$ FERR( * ), WORK( * ), X( LDX, * )
192* ..
193*
194* =====================================================================
195*
196* .. Parameters ..
197 INTEGER ITMAX
198 parameter( itmax = 5 )
199 DOUBLE PRECISION ZERO
200 parameter( zero = 0.0d+0 )
201 DOUBLE PRECISION ONE
202 parameter( one = 1.0d+0 )
203 DOUBLE PRECISION TWO
204 parameter( two = 2.0d+0 )
205 DOUBLE PRECISION THREE
206 parameter( three = 3.0d+0 )
207* ..
208* .. Local Scalars ..
209 LOGICAL UPPER
210 INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
211 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
212* ..
213* .. Local Arrays ..
214 INTEGER ISAVE( 3 )
215* ..
216* .. External Subroutines ..
217 EXTERNAL daxpy, dcopy, dlacn2, dspmv, dsptrs, xerbla
218* ..
219* .. Intrinsic Functions ..
220 INTRINSIC abs, max
221* ..
222* .. External Functions ..
223 LOGICAL LSAME
224 DOUBLE PRECISION DLAMCH
225 EXTERNAL lsame, dlamch
226* ..
227* .. Executable Statements ..
228*
229* Test the input parameters.
230*
231 info = 0
232 upper = lsame( uplo, 'U' )
233 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
234 info = -1
235 ELSE IF( n.LT.0 ) THEN
236 info = -2
237 ELSE IF( nrhs.LT.0 ) THEN
238 info = -3
239 ELSE IF( ldb.LT.max( 1, n ) ) THEN
240 info = -8
241 ELSE IF( ldx.LT.max( 1, n ) ) THEN
242 info = -10
243 END IF
244 IF( info.NE.0 ) THEN
245 CALL xerbla( 'DSPRFS', -info )
246 RETURN
247 END IF
248*
249* Quick return if possible
250*
251 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
252 DO 10 j = 1, nrhs
253 ferr( j ) = zero
254 berr( j ) = zero
255 10 CONTINUE
256 RETURN
257 END IF
258*
259* NZ = maximum number of nonzero elements in each row of A, plus 1
260*
261 nz = n + 1
262 eps = dlamch( 'Epsilon' )
263 safmin = dlamch( 'Safe minimum' )
264 safe1 = nz*safmin
265 safe2 = safe1 / eps
266*
267* Do for each right hand side
268*
269 DO 140 j = 1, nrhs
270*
271 count = 1
272 lstres = three
273 20 CONTINUE
274*
275* Loop until stopping criterion is satisfied.
276*
277* Compute residual R = B - A * X
278*
279 CALL dcopy( n, b( 1, j ), 1, work( n+1 ), 1 )
280 CALL dspmv( uplo, n, -one, ap, x( 1, j ), 1, one, work( n+1 ),
281 \$ 1 )
282*
283* Compute componentwise relative backward error from formula
284*
285* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
286*
287* where abs(Z) is the componentwise absolute value of the matrix
288* or vector Z. If the i-th component of the denominator is less
289* than SAFE2, then SAFE1 is added to the i-th components of the
290* numerator and denominator before dividing.
291*
292 DO 30 i = 1, n
293 work( i ) = abs( b( i, j ) )
294 30 CONTINUE
295*
296* Compute abs(A)*abs(X) + abs(B).
297*
298 kk = 1
299 IF( upper ) THEN
300 DO 50 k = 1, n
301 s = zero
302 xk = abs( x( k, j ) )
303 ik = kk
304 DO 40 i = 1, k - 1
305 work( i ) = work( i ) + abs( ap( ik ) )*xk
306 s = s + abs( ap( ik ) )*abs( x( i, j ) )
307 ik = ik + 1
308 40 CONTINUE
309 work( k ) = work( k ) + abs( ap( kk+k-1 ) )*xk + s
310 kk = kk + k
311 50 CONTINUE
312 ELSE
313 DO 70 k = 1, n
314 s = zero
315 xk = abs( x( k, j ) )
316 work( k ) = work( k ) + abs( ap( kk ) )*xk
317 ik = kk + 1
318 DO 60 i = k + 1, n
319 work( i ) = work( i ) + abs( ap( ik ) )*xk
320 s = s + abs( ap( ik ) )*abs( x( i, j ) )
321 ik = ik + 1
322 60 CONTINUE
323 work( k ) = work( k ) + s
324 kk = kk + ( n-k+1 )
325 70 CONTINUE
326 END IF
327 s = zero
328 DO 80 i = 1, n
329 IF( work( i ).GT.safe2 ) THEN
330 s = max( s, abs( work( n+i ) ) / work( i ) )
331 ELSE
332 s = max( s, ( abs( work( n+i ) )+safe1 ) /
333 \$ ( work( i )+safe1 ) )
334 END IF
335 80 CONTINUE
336 berr( j ) = s
337*
338* Test stopping criterion. Continue iterating if
339* 1) The residual BERR(J) is larger than machine epsilon, and
340* 2) BERR(J) decreased by at least a factor of 2 during the
341* last iteration, and
342* 3) At most ITMAX iterations tried.
343*
344 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
345 \$ count.LE.itmax ) THEN
346*
347* Update solution and try again.
348*
349 CALL dsptrs( uplo, n, 1, afp, ipiv, work( n+1 ), n, info )
350 CALL daxpy( n, one, work( n+1 ), 1, x( 1, j ), 1 )
351 lstres = berr( j )
352 count = count + 1
353 GO TO 20
354 END IF
355*
356* Bound error from formula
357*
358* norm(X - XTRUE) / norm(X) .le. FERR =
359* norm( abs(inv(A))*
360* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
361*
362* where
363* norm(Z) is the magnitude of the largest component of Z
364* inv(A) is the inverse of A
365* abs(Z) is the componentwise absolute value of the matrix or
366* vector Z
367* NZ is the maximum number of nonzeros in any row of A, plus 1
368* EPS is machine epsilon
369*
370* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
371* is incremented by SAFE1 if the i-th component of
372* abs(A)*abs(X) + abs(B) is less than SAFE2.
373*
374* Use DLACN2 to estimate the infinity-norm of the matrix
375* inv(A) * diag(W),
376* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
377*
378 DO 90 i = 1, n
379 IF( work( i ).GT.safe2 ) THEN
380 work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
381 ELSE
382 work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
383 END IF
384 90 CONTINUE
385*
386 kase = 0
387 100 CONTINUE
388 CALL dlacn2( n, work( 2*n+1 ), work( n+1 ), iwork, ferr( j ),
389 \$ kase, isave )
390 IF( kase.NE.0 ) THEN
391 IF( kase.EQ.1 ) THEN
392*
393* Multiply by diag(W)*inv(A**T).
394*
395 CALL dsptrs( uplo, n, 1, afp, ipiv, work( n+1 ), n,
396 \$ info )
397 DO 110 i = 1, n
398 work( n+i ) = work( i )*work( n+i )
399 110 CONTINUE
400 ELSE IF( kase.EQ.2 ) THEN
401*
402* Multiply by inv(A)*diag(W).
403*
404 DO 120 i = 1, n
405 work( n+i ) = work( i )*work( n+i )
406 120 CONTINUE
407 CALL dsptrs( uplo, n, 1, afp, ipiv, work( n+1 ), n,
408 \$ info )
409 END IF
410 GO TO 100
411 END IF
412*
413* Normalize error.
414*
415 lstres = zero
416 DO 130 i = 1, n
417 lstres = max( lstres, abs( x( i, j ) ) )
418 130 CONTINUE
419 IF( lstres.NE.zero )
420 \$ ferr( j ) = ferr( j ) / lstres
421*
422 140 CONTINUE
423*
424 RETURN
425*
426* End of DSPRFS
427*
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 dspmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
DSPMV
Definition: dspmv.f:147
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 dsptrs(UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
DSPTRS
Definition: dsptrs.f:115
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