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

 subroutine sgtrfs ( character trans, integer n, integer nrhs, real, dimension( * ) dl, real, dimension( * ) d, real, dimension( * ) du, real, dimension( * ) dlf, real, dimension( * ) df, real, dimension( * ) duf, real, dimension( * ) du2, 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 )

SGTRFS

Purpose:
``` SGTRFS improves the computed solution to a system of linear
equations when the coefficient matrix is tridiagonal, 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 matrix B. NRHS >= 0.``` [in] DL ``` DL is REAL array, dimension (N-1) The (n-1) subdiagonal elements of A.``` [in] D ``` D is REAL array, dimension (N) The diagonal elements of A.``` [in] DU ``` DU is REAL array, dimension (N-1) The (n-1) superdiagonal elements of A.``` [in] DLF ``` DLF is REAL array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by SGTTRF.``` [in] DF ``` DF is REAL array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A.``` [in] DUF ``` DUF is REAL array, dimension (N-1) The (n-1) elements of the first superdiagonal of U.``` [in] DU2 ``` DU2 is REAL array, dimension (N-2) The (n-2) elements of the second superdiagonal of U.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required.``` [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 SGTTRS. 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 206 of file sgtrfs.f.

209*
210* -- LAPACK computational routine --
211* -- LAPACK is a software package provided by Univ. of Tennessee, --
212* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
213*
214* .. Scalar Arguments ..
215 CHARACTER TRANS
216 INTEGER INFO, LDB, LDX, N, NRHS
217* ..
218* .. Array Arguments ..
219 INTEGER IPIV( * ), IWORK( * )
220 REAL B( LDB, * ), BERR( * ), D( * ), DF( * ),
221 \$ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
222 \$ FERR( * ), WORK( * ), X( LDX, * )
223* ..
224*
225* =====================================================================
226*
227* .. Parameters ..
228 INTEGER ITMAX
229 parameter( itmax = 5 )
230 REAL ZERO, ONE
231 parameter( zero = 0.0e+0, one = 1.0e+0 )
232 REAL TWO
233 parameter( two = 2.0e+0 )
234 REAL THREE
235 parameter( three = 3.0e+0 )
236* ..
237* .. Local Scalars ..
238 LOGICAL NOTRAN
239 CHARACTER TRANSN, TRANST
240 INTEGER COUNT, I, J, KASE, NZ
241 REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
242* ..
243* .. Local Arrays ..
244 INTEGER ISAVE( 3 )
245* ..
246* .. External Subroutines ..
247 EXTERNAL saxpy, scopy, sgttrs, slacn2, slagtm, xerbla
248* ..
249* .. Intrinsic Functions ..
250 INTRINSIC abs, max
251* ..
252* .. External Functions ..
253 LOGICAL LSAME
254 REAL SLAMCH
255 EXTERNAL lsame, slamch
256* ..
257* .. Executable Statements ..
258*
259* Test the input parameters.
260*
261 info = 0
262 notran = lsame( trans, 'N' )
263 IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
264 \$ lsame( trans, 'C' ) ) THEN
265 info = -1
266 ELSE IF( n.LT.0 ) THEN
267 info = -2
268 ELSE IF( nrhs.LT.0 ) THEN
269 info = -3
270 ELSE IF( ldb.LT.max( 1, n ) ) THEN
271 info = -13
272 ELSE IF( ldx.LT.max( 1, n ) ) THEN
273 info = -15
274 END IF
275 IF( info.NE.0 ) THEN
276 CALL xerbla( 'SGTRFS', -info )
277 RETURN
278 END IF
279*
280* Quick return if possible
281*
282 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
283 DO 10 j = 1, nrhs
284 ferr( j ) = zero
285 berr( j ) = zero
286 10 CONTINUE
287 RETURN
288 END IF
289*
290 IF( notran ) THEN
291 transn = 'N'
292 transt = 'T'
293 ELSE
294 transn = 'T'
295 transt = 'N'
296 END IF
297*
298* NZ = maximum number of nonzero elements in each row of A, plus 1
299*
300 nz = 4
301 eps = slamch( 'Epsilon' )
302 safmin = slamch( 'Safe minimum' )
303 safe1 = nz*safmin
304 safe2 = safe1 / eps
305*
306* Do for each right hand side
307*
308 DO 110 j = 1, nrhs
309*
310 count = 1
311 lstres = three
312 20 CONTINUE
313*
314* Loop until stopping criterion is satisfied.
315*
316* Compute residual R = B - op(A) * X,
317* where op(A) = A, A**T, or A**H, depending on TRANS.
318*
319 CALL scopy( n, b( 1, j ), 1, work( n+1 ), 1 )
320 CALL slagtm( trans, n, 1, -one, dl, d, du, x( 1, j ), ldx, one,
321 \$ work( n+1 ), n )
322*
323* Compute abs(op(A))*abs(x) + abs(b) for use in the backward
324* error bound.
325*
326 IF( notran ) THEN
327 IF( n.EQ.1 ) THEN
328 work( 1 ) = abs( b( 1, j ) ) + abs( d( 1 )*x( 1, j ) )
329 ELSE
330 work( 1 ) = abs( b( 1, j ) ) + abs( d( 1 )*x( 1, j ) ) +
331 \$ abs( du( 1 )*x( 2, j ) )
332 DO 30 i = 2, n - 1
333 work( i ) = abs( b( i, j ) ) +
334 \$ abs( dl( i-1 )*x( i-1, j ) ) +
335 \$ abs( d( i )*x( i, j ) ) +
336 \$ abs( du( i )*x( i+1, j ) )
337 30 CONTINUE
338 work( n ) = abs( b( n, j ) ) +
339 \$ abs( dl( n-1 )*x( n-1, j ) ) +
340 \$ abs( d( n )*x( n, j ) )
341 END IF
342 ELSE
343 IF( n.EQ.1 ) THEN
344 work( 1 ) = abs( b( 1, j ) ) + abs( d( 1 )*x( 1, j ) )
345 ELSE
346 work( 1 ) = abs( b( 1, j ) ) + abs( d( 1 )*x( 1, j ) ) +
347 \$ abs( dl( 1 )*x( 2, j ) )
348 DO 40 i = 2, n - 1
349 work( i ) = abs( b( i, j ) ) +
350 \$ abs( du( i-1 )*x( i-1, j ) ) +
351 \$ abs( d( i )*x( i, j ) ) +
352 \$ abs( dl( i )*x( i+1, j ) )
353 40 CONTINUE
354 work( n ) = abs( b( n, j ) ) +
355 \$ abs( du( n-1 )*x( n-1, j ) ) +
356 \$ abs( d( n )*x( n, j ) )
357 END IF
358 END IF
359*
360* Compute componentwise relative backward error from formula
361*
362* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
363*
364* where abs(Z) is the componentwise absolute value of the matrix
365* or vector Z. If the i-th component of the denominator is less
366* than SAFE2, then SAFE1 is added to the i-th components of the
367* numerator and denominator before dividing.
368*
369 s = zero
370 DO 50 i = 1, n
371 IF( work( i ).GT.safe2 ) THEN
372 s = max( s, abs( work( n+i ) ) / work( i ) )
373 ELSE
374 s = max( s, ( abs( work( n+i ) )+safe1 ) /
375 \$ ( work( i )+safe1 ) )
376 END IF
377 50 CONTINUE
378 berr( j ) = s
379*
380* Test stopping criterion. Continue iterating if
381* 1) The residual BERR(J) is larger than machine epsilon, and
382* 2) BERR(J) decreased by at least a factor of 2 during the
383* last iteration, and
384* 3) At most ITMAX iterations tried.
385*
386 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
387 \$ count.LE.itmax ) THEN
388*
389* Update solution and try again.
390*
391 CALL sgttrs( trans, n, 1, dlf, df, duf, du2, ipiv,
392 \$ work( n+1 ), n, info )
393 CALL saxpy( n, one, work( n+1 ), 1, x( 1, j ), 1 )
394 lstres = berr( j )
395 count = count + 1
396 GO TO 20
397 END IF
398*
399* Bound error from formula
400*
401* norm(X - XTRUE) / norm(X) .le. FERR =
402* norm( abs(inv(op(A)))*
403* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
404*
405* where
406* norm(Z) is the magnitude of the largest component of Z
407* inv(op(A)) is the inverse of op(A)
408* abs(Z) is the componentwise absolute value of the matrix or
409* vector Z
410* NZ is the maximum number of nonzeros in any row of A, plus 1
411* EPS is machine epsilon
412*
413* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
414* is incremented by SAFE1 if the i-th component of
415* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
416*
417* Use SLACN2 to estimate the infinity-norm of the matrix
418* inv(op(A)) * diag(W),
419* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
420*
421 DO 60 i = 1, n
422 IF( work( i ).GT.safe2 ) THEN
423 work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
424 ELSE
425 work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
426 END IF
427 60 CONTINUE
428*
429 kase = 0
430 70 CONTINUE
431 CALL slacn2( n, work( 2*n+1 ), work( n+1 ), iwork, ferr( j ),
432 \$ kase, isave )
433 IF( kase.NE.0 ) THEN
434 IF( kase.EQ.1 ) THEN
435*
436* Multiply by diag(W)*inv(op(A)**T).
437*
438 CALL sgttrs( transt, n, 1, dlf, df, duf, du2, ipiv,
439 \$ work( n+1 ), n, info )
440 DO 80 i = 1, n
441 work( n+i ) = work( i )*work( n+i )
442 80 CONTINUE
443 ELSE
444*
445* Multiply by inv(op(A))*diag(W).
446*
447 DO 90 i = 1, n
448 work( n+i ) = work( i )*work( n+i )
449 90 CONTINUE
450 CALL sgttrs( transn, n, 1, dlf, df, duf, du2, ipiv,
451 \$ work( n+1 ), n, info )
452 END IF
453 GO TO 70
454 END IF
455*
456* Normalize error.
457*
458 lstres = zero
459 DO 100 i = 1, n
460 lstres = max( lstres, abs( x( i, j ) ) )
461 100 CONTINUE
462 IF( lstres.NE.zero )
463 \$ ferr( j ) = ferr( j ) / lstres
464*
465 110 CONTINUE
466*
467 RETURN
468*
469* End of SGTRFS
470*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine saxpy(n, sa, sx, incx, sy, incy)
SAXPY
Definition saxpy.f:89
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
subroutine sgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info)
SGTTRS
Definition sgttrs.f:138
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 slagtm(trans, n, nrhs, alpha, dl, d, du, x, ldx, beta, b, ldb)
SLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix,...
Definition slagtm.f:145
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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