LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ zgerfs()

 subroutine zgerfs ( character trans, 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 )

ZGERFS

Purpose:
``` ZGERFS improves the computed solution to a system of linear
equations and provides error bounds and backward error estimates for
the solution.```
Parameters
 [in] TRANS ``` TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose)``` [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 original N-by-N matrix A.``` [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 factors L and U from the factorization A = P*L*U as computed by ZGETRF.``` [in] LDAF ``` LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) The pivot indices from ZGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).``` [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 ZGETRS. 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 184 of file zgerfs.f.

186*
187* -- LAPACK computational routine --
188* -- LAPACK is a software package provided by Univ. of Tennessee, --
189* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
190*
191* .. Scalar Arguments ..
192 CHARACTER TRANS
193 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
194* ..
195* .. Array Arguments ..
196 INTEGER IPIV( * )
197 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
198 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
199 \$ WORK( * ), X( LDX, * )
200* ..
201*
202* =====================================================================
203*
204* .. Parameters ..
205 INTEGER ITMAX
206 parameter( itmax = 5 )
207 DOUBLE PRECISION ZERO
208 parameter( zero = 0.0d+0 )
209 COMPLEX*16 ONE
210 parameter( one = ( 1.0d+0, 0.0d+0 ) )
211 DOUBLE PRECISION TWO
212 parameter( two = 2.0d+0 )
213 DOUBLE PRECISION THREE
214 parameter( three = 3.0d+0 )
215* ..
216* .. Local Scalars ..
217 LOGICAL NOTRAN
218 CHARACTER TRANSN, TRANST
219 INTEGER COUNT, I, J, K, KASE, NZ
220 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
221 COMPLEX*16 ZDUM
222* ..
223* .. Local Arrays ..
224 INTEGER ISAVE( 3 )
225* ..
226* .. External Functions ..
227 LOGICAL LSAME
228 DOUBLE PRECISION DLAMCH
229 EXTERNAL lsame, dlamch
230* ..
231* .. External Subroutines ..
232 EXTERNAL xerbla, zaxpy, zcopy, zgemv, zgetrs, zlacn2
233* ..
234* .. Intrinsic Functions ..
235 INTRINSIC abs, dble, dimag, max
236* ..
237* .. Statement Functions ..
238 DOUBLE PRECISION CABS1
239* ..
240* .. Statement Function definitions ..
241 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
242* ..
243* .. Executable Statements ..
244*
245* Test the input parameters.
246*
247 info = 0
248 notran = lsame( trans, 'N' )
249 IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
250 \$ lsame( trans, 'C' ) ) THEN
251 info = -1
252 ELSE IF( n.LT.0 ) THEN
253 info = -2
254 ELSE IF( nrhs.LT.0 ) THEN
255 info = -3
256 ELSE IF( lda.LT.max( 1, n ) ) THEN
257 info = -5
258 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
259 info = -7
260 ELSE IF( ldb.LT.max( 1, n ) ) THEN
261 info = -10
262 ELSE IF( ldx.LT.max( 1, n ) ) THEN
263 info = -12
264 END IF
265 IF( info.NE.0 ) THEN
266 CALL xerbla( 'ZGERFS', -info )
267 RETURN
268 END IF
269*
270* Quick return if possible
271*
272 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
273 DO 10 j = 1, nrhs
274 ferr( j ) = zero
275 berr( j ) = zero
276 10 CONTINUE
277 RETURN
278 END IF
279*
280 IF( notran ) THEN
281 transn = 'N'
282 transt = 'C'
283 ELSE
284 transn = 'C'
285 transt = 'N'
286 END IF
287*
288* NZ = maximum number of nonzero elements in each row of A, plus 1
289*
290 nz = n + 1
291 eps = dlamch( 'Epsilon' )
292 safmin = dlamch( 'Safe minimum' )
293 safe1 = nz*safmin
294 safe2 = safe1 / eps
295*
296* Do for each right hand side
297*
298 DO 140 j = 1, nrhs
299*
300 count = 1
301 lstres = three
302 20 CONTINUE
303*
304* Loop until stopping criterion is satisfied.
305*
306* Compute residual R = B - op(A) * X,
307* where op(A) = A, A**T, or A**H, depending on TRANS.
308*
309 CALL zcopy( n, b( 1, j ), 1, work, 1 )
310 CALL zgemv( trans, n, n, -one, a, lda, x( 1, j ), 1, one, work,
311 \$ 1 )
312*
313* Compute componentwise relative backward error from formula
314*
315* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
316*
317* where abs(Z) is the componentwise absolute value of the matrix
318* or vector Z. If the i-th component of the denominator is less
319* than SAFE2, then SAFE1 is added to the i-th components of the
320* numerator and denominator before dividing.
321*
322 DO 30 i = 1, n
323 rwork( i ) = cabs1( b( i, j ) )
324 30 CONTINUE
325*
326* Compute abs(op(A))*abs(X) + abs(B).
327*
328 IF( notran ) THEN
329 DO 50 k = 1, n
330 xk = cabs1( x( k, j ) )
331 DO 40 i = 1, n
332 rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
333 40 CONTINUE
334 50 CONTINUE
335 ELSE
336 DO 70 k = 1, n
337 s = zero
338 DO 60 i = 1, n
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 zgetrs( trans, 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(op(A)))*
377* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
378*
379* where
380* norm(Z) is the magnitude of the largest component of Z
381* inv(op(A)) is the inverse of op(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(op(A))*abs(X)+abs(B))
388* is incremented by SAFE1 if the i-th component of
389* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
390*
391* Use ZLACN2 to estimate the infinity-norm of the matrix
392* inv(op(A)) * diag(W),
393* where W = abs(R) + NZ*EPS*( abs(op(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(op(A)**H).
411*
412 CALL zgetrs( transt, n, 1, af, ldaf, ipiv, work, n,
413 \$ info )
414 DO 110 i = 1, n
415 work( i ) = rwork( i )*work( i )
416 110 CONTINUE
417 ELSE
418*
419* Multiply by inv(op(A))*diag(W).
420*
421 DO 120 i = 1, n
422 work( i ) = rwork( i )*work( i )
423 120 CONTINUE
424 CALL zgetrs( transn, n, 1, af, ldaf, ipiv, work, n,
425 \$ info )
426 END IF
427 GO TO 100
428 END IF
429*
430* Normalize error.
431*
432 lstres = zero
433 DO 130 i = 1, n
434 lstres = max( lstres, cabs1( x( i, j ) ) )
435 130 CONTINUE
436 IF( lstres.NE.zero )
437 \$ ferr( j ) = ferr( j ) / lstres
438*
439 140 CONTINUE
440*
441 RETURN
442*
443* End of ZGERFS
444*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
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 zgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
ZGEMV
Definition zgemv.f:160
subroutine zgetrs(trans, n, nrhs, a, lda, ipiv, b, ldb, info)
ZGETRS
Definition zgetrs.f:121
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
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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