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

 subroutine sgerfs ( character TRANS, 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 )

SGERFS

Purpose:
``` SGERFS 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 = 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 REAL 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 REAL array, dimension (LDAF,N) The factors L and U from the factorization A = P*L*U as computed by SGETRF.``` [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 SGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).``` [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 SGETRS. 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 183 of file sgerfs.f.

185*
186* -- LAPACK computational routine --
187* -- LAPACK is a software package provided by Univ. of Tennessee, --
188* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189*
190* .. Scalar Arguments ..
191 CHARACTER TRANS
192 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
193* ..
194* .. Array Arguments ..
195 INTEGER IPIV( * ), IWORK( * )
196 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
197 \$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
198* ..
199*
200* =====================================================================
201*
202* .. Parameters ..
203 INTEGER ITMAX
204 parameter( itmax = 5 )
205 REAL ZERO
206 parameter( zero = 0.0e+0 )
207 REAL ONE
208 parameter( one = 1.0e+0 )
209 REAL TWO
210 parameter( two = 2.0e+0 )
211 REAL THREE
212 parameter( three = 3.0e+0 )
213* ..
214* .. Local Scalars ..
215 LOGICAL NOTRAN
216 CHARACTER TRANST
217 INTEGER COUNT, I, J, K, KASE, NZ
218 REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
219* ..
220* .. Local Arrays ..
221 INTEGER ISAVE( 3 )
222* ..
223* .. External Subroutines ..
224 EXTERNAL saxpy, scopy, sgemv, sgetrs, slacn2, xerbla
225* ..
226* .. Intrinsic Functions ..
227 INTRINSIC abs, max
228* ..
229* .. External Functions ..
230 LOGICAL LSAME
231 REAL SLAMCH
232 EXTERNAL lsame, slamch
233* ..
234* .. Executable Statements ..
235*
236* Test the input parameters.
237*
238 info = 0
239 notran = lsame( trans, 'N' )
240 IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
241 \$ lsame( trans, 'C' ) ) THEN
242 info = -1
243 ELSE IF( n.LT.0 ) THEN
244 info = -2
245 ELSE IF( nrhs.LT.0 ) THEN
246 info = -3
247 ELSE IF( lda.LT.max( 1, n ) ) THEN
248 info = -5
249 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
250 info = -7
251 ELSE IF( ldb.LT.max( 1, n ) ) THEN
252 info = -10
253 ELSE IF( ldx.LT.max( 1, n ) ) THEN
254 info = -12
255 END IF
256 IF( info.NE.0 ) THEN
257 CALL xerbla( 'SGERFS', -info )
258 RETURN
259 END IF
260*
261* Quick return if possible
262*
263 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
264 DO 10 j = 1, nrhs
265 ferr( j ) = zero
266 berr( j ) = zero
267 10 CONTINUE
268 RETURN
269 END IF
270*
271 IF( notran ) THEN
272 transt = 'T'
273 ELSE
274 transt = 'N'
275 END IF
276*
277* NZ = maximum number of nonzero elements in each row of A, plus 1
278*
279 nz = n + 1
280 eps = slamch( 'Epsilon' )
281 safmin = slamch( 'Safe minimum' )
282 safe1 = nz*safmin
283 safe2 = safe1 / eps
284*
285* Do for each right hand side
286*
287 DO 140 j = 1, nrhs
288*
289 count = 1
290 lstres = three
291 20 CONTINUE
292*
293* Loop until stopping criterion is satisfied.
294*
295* Compute residual R = B - op(A) * X,
296* where op(A) = A, A**T, or A**H, depending on TRANS.
297*
298 CALL scopy( n, b( 1, j ), 1, work( n+1 ), 1 )
299 CALL sgemv( trans, n, n, -one, a, lda, x( 1, j ), 1, one,
300 \$ work( n+1 ), 1 )
301*
302* Compute componentwise relative backward error from formula
303*
304* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
305*
306* where abs(Z) is the componentwise absolute value of the matrix
307* or vector Z. If the i-th component of the denominator is less
308* than SAFE2, then SAFE1 is added to the i-th components of the
309* numerator and denominator before dividing.
310*
311 DO 30 i = 1, n
312 work( i ) = abs( b( i, j ) )
313 30 CONTINUE
314*
315* Compute abs(op(A))*abs(X) + abs(B).
316*
317 IF( notran ) THEN
318 DO 50 k = 1, n
319 xk = abs( x( k, j ) )
320 DO 40 i = 1, n
321 work( i ) = work( i ) + abs( a( i, k ) )*xk
322 40 CONTINUE
323 50 CONTINUE
324 ELSE
325 DO 70 k = 1, n
326 s = zero
327 DO 60 i = 1, n
328 s = s + abs( a( i, k ) )*abs( x( i, j ) )
329 60 CONTINUE
330 work( k ) = work( k ) + s
331 70 CONTINUE
332 END IF
333 s = zero
334 DO 80 i = 1, n
335 IF( work( i ).GT.safe2 ) THEN
336 s = max( s, abs( work( n+i ) ) / work( i ) )
337 ELSE
338 s = max( s, ( abs( work( n+i ) )+safe1 ) /
339 \$ ( work( i )+safe1 ) )
340 END IF
341 80 CONTINUE
342 berr( j ) = s
343*
344* Test stopping criterion. Continue iterating if
345* 1) The residual BERR(J) is larger than machine epsilon, and
346* 2) BERR(J) decreased by at least a factor of 2 during the
347* last iteration, and
348* 3) At most ITMAX iterations tried.
349*
350 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
351 \$ count.LE.itmax ) THEN
352*
353* Update solution and try again.
354*
355 CALL sgetrs( trans, n, 1, af, ldaf, ipiv, work( n+1 ), n,
356 \$ info )
357 CALL saxpy( n, one, work( n+1 ), 1, x( 1, j ), 1 )
358 lstres = berr( j )
359 count = count + 1
360 GO TO 20
361 END IF
362*
363* Bound error from formula
364*
365* norm(X - XTRUE) / norm(X) .le. FERR =
366* norm( abs(inv(op(A)))*
367* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
368*
369* where
370* norm(Z) is the magnitude of the largest component of Z
371* inv(op(A)) is the inverse of op(A)
372* abs(Z) is the componentwise absolute value of the matrix or
373* vector Z
374* NZ is the maximum number of nonzeros in any row of A, plus 1
375* EPS is machine epsilon
376*
377* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
378* is incremented by SAFE1 if the i-th component of
379* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
380*
381* Use SLACN2 to estimate the infinity-norm of the matrix
382* inv(op(A)) * diag(W),
383* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
384*
385 DO 90 i = 1, n
386 IF( work( i ).GT.safe2 ) THEN
387 work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
388 ELSE
389 work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
390 END IF
391 90 CONTINUE
392*
393 kase = 0
394 100 CONTINUE
395 CALL slacn2( n, work( 2*n+1 ), work( n+1 ), iwork, ferr( j ),
396 \$ kase, isave )
397 IF( kase.NE.0 ) THEN
398 IF( kase.EQ.1 ) THEN
399*
400* Multiply by diag(W)*inv(op(A)**T).
401*
402 CALL sgetrs( transt, n, 1, af, ldaf, ipiv, work( n+1 ),
403 \$ n, info )
404 DO 110 i = 1, n
405 work( n+i ) = work( i )*work( n+i )
406 110 CONTINUE
407 ELSE
408*
409* Multiply by inv(op(A))*diag(W).
410*
411 DO 120 i = 1, n
412 work( n+i ) = work( i )*work( n+i )
413 120 CONTINUE
414 CALL sgetrs( trans, n, 1, af, ldaf, ipiv, work( n+1 ), n,
415 \$ info )
416 END IF
417 GO TO 100
418 END IF
419*
420* Normalize error.
421*
422 lstres = zero
423 DO 130 i = 1, n
424 lstres = max( lstres, abs( x( i, j ) ) )
425 130 CONTINUE
426 IF( lstres.NE.zero )
427 \$ ferr( j ) = ferr( j ) / lstres
428*
429 140 CONTINUE
430*
431 RETURN
432*
433* End of SGERFS
434*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine sgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS
Definition: sgetrs.f:121
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 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 sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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