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

 subroutine zsprfs ( character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( * ) AFP, 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 )

ZSPRFS

Purpose:
``` ZSPRFS 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 COMPLEX*16 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 COMPLEX*16 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 ZSPTRF, 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 ZSPTRF.``` [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 ZSPTRS. 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 178 of file zsprfs.f.

180*
181* -- LAPACK computational routine --
182* -- LAPACK is a software package provided by Univ. of Tennessee, --
183* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
184*
185* .. Scalar Arguments ..
186 CHARACTER UPLO
187 INTEGER INFO, LDB, LDX, N, NRHS
188* ..
189* .. Array Arguments ..
190 INTEGER IPIV( * )
191 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
192 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
193 \$ X( LDX, * )
194* ..
195*
196* =====================================================================
197*
198* .. Parameters ..
199 INTEGER ITMAX
200 parameter( itmax = 5 )
201 DOUBLE PRECISION ZERO
202 parameter( zero = 0.0d+0 )
203 COMPLEX*16 ONE
204 parameter( one = ( 1.0d+0, 0.0d+0 ) )
205 DOUBLE PRECISION TWO
206 parameter( two = 2.0d+0 )
207 DOUBLE PRECISION THREE
208 parameter( three = 3.0d+0 )
209* ..
210* .. Local Scalars ..
211 LOGICAL UPPER
212 INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
213 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
214 COMPLEX*16 ZDUM
215* ..
216* .. Local Arrays ..
217 INTEGER ISAVE( 3 )
218* ..
219* .. External Subroutines ..
220 EXTERNAL xerbla, zaxpy, zcopy, zlacn2, zspmv, zsptrs
221* ..
222* .. Intrinsic Functions ..
223 INTRINSIC abs, dble, dimag, max
224* ..
225* .. External Functions ..
226 LOGICAL LSAME
227 DOUBLE PRECISION DLAMCH
228 EXTERNAL lsame, dlamch
229* ..
230* .. Statement Functions ..
231 DOUBLE PRECISION CABS1
232* ..
233* .. Statement Function definitions ..
234 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
235* ..
236* .. Executable Statements ..
237*
238* Test the input parameters.
239*
240 info = 0
241 upper = lsame( uplo, 'U' )
242 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
243 info = -1
244 ELSE IF( n.LT.0 ) THEN
245 info = -2
246 ELSE IF( nrhs.LT.0 ) THEN
247 info = -3
248 ELSE IF( ldb.LT.max( 1, n ) ) THEN
249 info = -8
250 ELSE IF( ldx.LT.max( 1, n ) ) THEN
251 info = -10
252 END IF
253 IF( info.NE.0 ) THEN
254 CALL xerbla( 'ZSPRFS', -info )
255 RETURN
256 END IF
257*
258* Quick return if possible
259*
260 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
261 DO 10 j = 1, nrhs
262 ferr( j ) = zero
263 berr( j ) = zero
264 10 CONTINUE
265 RETURN
266 END IF
267*
268* NZ = maximum number of nonzero elements in each row of A, plus 1
269*
270 nz = n + 1
271 eps = dlamch( 'Epsilon' )
272 safmin = dlamch( 'Safe minimum' )
273 safe1 = nz*safmin
274 safe2 = safe1 / eps
275*
276* Do for each right hand side
277*
278 DO 140 j = 1, nrhs
279*
280 count = 1
281 lstres = three
282 20 CONTINUE
283*
284* Loop until stopping criterion is satisfied.
285*
286* Compute residual R = B - A * X
287*
288 CALL zcopy( n, b( 1, j ), 1, work, 1 )
289 CALL zspmv( uplo, n, -one, ap, x( 1, j ), 1, one, work, 1 )
290*
291* Compute componentwise relative backward error from formula
292*
293* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
294*
295* where abs(Z) is the componentwise absolute value of the matrix
296* or vector Z. If the i-th component of the denominator is less
297* than SAFE2, then SAFE1 is added to the i-th components of the
298* numerator and denominator before dividing.
299*
300 DO 30 i = 1, n
301 rwork( i ) = cabs1( b( i, j ) )
302 30 CONTINUE
303*
304* Compute abs(A)*abs(X) + abs(B).
305*
306 kk = 1
307 IF( upper ) THEN
308 DO 50 k = 1, n
309 s = zero
310 xk = cabs1( x( k, j ) )
311 ik = kk
312 DO 40 i = 1, k - 1
313 rwork( i ) = rwork( i ) + cabs1( ap( ik ) )*xk
314 s = s + cabs1( ap( ik ) )*cabs1( x( i, j ) )
315 ik = ik + 1
316 40 CONTINUE
317 rwork( k ) = rwork( k ) + cabs1( ap( kk+k-1 ) )*xk + s
318 kk = kk + k
319 50 CONTINUE
320 ELSE
321 DO 70 k = 1, n
322 s = zero
323 xk = cabs1( x( k, j ) )
324 rwork( k ) = rwork( k ) + cabs1( ap( kk ) )*xk
325 ik = kk + 1
326 DO 60 i = k + 1, n
327 rwork( i ) = rwork( i ) + cabs1( ap( ik ) )*xk
328 s = s + cabs1( ap( ik ) )*cabs1( x( i, j ) )
329 ik = ik + 1
330 60 CONTINUE
331 rwork( k ) = rwork( k ) + s
332 kk = kk + ( n-k+1 )
333 70 CONTINUE
334 END IF
335 s = zero
336 DO 80 i = 1, n
337 IF( rwork( i ).GT.safe2 ) THEN
338 s = max( s, cabs1( work( i ) ) / rwork( i ) )
339 ELSE
340 s = max( s, ( cabs1( work( i ) )+safe1 ) /
341 \$ ( rwork( i )+safe1 ) )
342 END IF
343 80 CONTINUE
344 berr( j ) = s
345*
346* Test stopping criterion. Continue iterating if
347* 1) The residual BERR(J) is larger than machine epsilon, and
348* 2) BERR(J) decreased by at least a factor of 2 during the
349* last iteration, and
350* 3) At most ITMAX iterations tried.
351*
352 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
353 \$ count.LE.itmax ) THEN
354*
355* Update solution and try again.
356*
357 CALL zsptrs( uplo, n, 1, afp, ipiv, work, n, info )
358 CALL zaxpy( n, one, work, 1, x( 1, j ), 1 )
359 lstres = berr( j )
360 count = count + 1
361 GO TO 20
362 END IF
363*
364* Bound error from formula
365*
366* norm(X - XTRUE) / norm(X) .le. FERR =
367* norm( abs(inv(A))*
368* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
369*
370* where
371* norm(Z) is the magnitude of the largest component of Z
372* inv(A) is the inverse of A
373* abs(Z) is the componentwise absolute value of the matrix or
374* vector Z
375* NZ is the maximum number of nonzeros in any row of A, plus 1
376* EPS is machine epsilon
377*
378* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
379* is incremented by SAFE1 if the i-th component of
380* abs(A)*abs(X) + abs(B) is less than SAFE2.
381*
382* Use ZLACN2 to estimate the infinity-norm of the matrix
383* inv(A) * diag(W),
384* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
385*
386 DO 90 i = 1, n
387 IF( rwork( i ).GT.safe2 ) THEN
388 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
389 ELSE
390 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
391 \$ safe1
392 END IF
393 90 CONTINUE
394*
395 kase = 0
396 100 CONTINUE
397 CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
398 IF( kase.NE.0 ) THEN
399 IF( kase.EQ.1 ) THEN
400*
401* Multiply by diag(W)*inv(A**T).
402*
403 CALL zsptrs( uplo, n, 1, afp, ipiv, work, n, info )
404 DO 110 i = 1, n
405 work( i ) = rwork( i )*work( i )
406 110 CONTINUE
407 ELSE IF( kase.EQ.2 ) THEN
408*
409* Multiply by inv(A)*diag(W).
410*
411 DO 120 i = 1, n
412 work( i ) = rwork( i )*work( i )
413 120 CONTINUE
414 CALL zsptrs( uplo, n, 1, afp, ipiv, work, n, info )
415 END IF
416 GO TO 100
417 END IF
418*
419* Normalize error.
420*
421 lstres = zero
422 DO 130 i = 1, n
423 lstres = max( lstres, cabs1( x( i, j ) ) )
424 130 CONTINUE
425 IF( lstres.NE.zero )
426 \$ ferr( j ) = ferr( j ) / lstres
427*
428 140 CONTINUE
429*
430 RETURN
431*
432* End of ZSPRFS
433*
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 zspmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
ZSPMV computes a matrix-vector product for complex vectors using a complex symmetric packed matrix
Definition: zspmv.f:151
subroutine zsptrs(UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
ZSPTRS
Definition: zsptrs.f:115
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