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

 subroutine zgtrfs ( character trans, integer n, integer nrhs, complex*16, dimension( * ) dl, complex*16, dimension( * ) d, complex*16, dimension( * ) du, complex*16, dimension( * ) dlf, complex*16, dimension( * ) df, complex*16, dimension( * ) duf, complex*16, dimension( * ) du2, 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 )

ZGTRFS

Purpose:
``` ZGTRFS 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)``` [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 COMPLEX*16 array, dimension (N-1) The (n-1) subdiagonal elements of A.``` [in] D ``` D is COMPLEX*16 array, dimension (N) The diagonal elements of A.``` [in] DU ``` DU is COMPLEX*16 array, dimension (N-1) The (n-1) superdiagonal elements of A.``` [in] DLF ``` DLF is COMPLEX*16 array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by ZGTTRF.``` [in] DF ``` DF is COMPLEX*16 array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A.``` [in] DUF ``` DUF is COMPLEX*16 array, dimension (N-1) The (n-1) elements of the first superdiagonal of U.``` [in] DU2 ``` DU2 is COMPLEX*16 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 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 ZGTTRS. 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 207 of file zgtrfs.f.

210*
211* -- LAPACK computational routine --
212* -- LAPACK is a software package provided by Univ. of Tennessee, --
213* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
214*
215* .. Scalar Arguments ..
216 CHARACTER TRANS
217 INTEGER INFO, LDB, LDX, N, NRHS
218* ..
219* .. Array Arguments ..
220 INTEGER IPIV( * )
221 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
222 COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
223 \$ DLF( * ), DU( * ), DU2( * ), DUF( * ),
224 \$ WORK( * ), X( LDX, * )
225* ..
226*
227* =====================================================================
228*
229* .. Parameters ..
230 INTEGER ITMAX
231 parameter( itmax = 5 )
232 DOUBLE PRECISION ZERO, ONE
233 parameter( zero = 0.0d+0, one = 1.0d+0 )
234 DOUBLE PRECISION TWO
235 parameter( two = 2.0d+0 )
236 DOUBLE PRECISION THREE
237 parameter( three = 3.0d+0 )
238* ..
239* .. Local Scalars ..
240 LOGICAL NOTRAN
241 CHARACTER TRANSN, TRANST
242 INTEGER COUNT, I, J, KASE, NZ
243 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
244 COMPLEX*16 ZDUM
245* ..
246* .. Local Arrays ..
247 INTEGER ISAVE( 3 )
248* ..
249* .. External Subroutines ..
250 EXTERNAL xerbla, zaxpy, zcopy, zgttrs, zlacn2, zlagtm
251* ..
252* .. Intrinsic Functions ..
253 INTRINSIC abs, dble, dcmplx, dimag, max
254* ..
255* .. External Functions ..
256 LOGICAL LSAME
257 DOUBLE PRECISION DLAMCH
258 EXTERNAL lsame, dlamch
259* ..
260* .. Statement Functions ..
261 DOUBLE PRECISION CABS1
262* ..
263* .. Statement Function definitions ..
264 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
265* ..
266* .. Executable Statements ..
267*
268* Test the input parameters.
269*
270 info = 0
271 notran = lsame( trans, 'N' )
272 IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
273 \$ lsame( trans, 'C' ) ) THEN
274 info = -1
275 ELSE IF( n.LT.0 ) THEN
276 info = -2
277 ELSE IF( nrhs.LT.0 ) THEN
278 info = -3
279 ELSE IF( ldb.LT.max( 1, n ) ) THEN
280 info = -13
281 ELSE IF( ldx.LT.max( 1, n ) ) THEN
282 info = -15
283 END IF
284 IF( info.NE.0 ) THEN
285 CALL xerbla( 'ZGTRFS', -info )
286 RETURN
287 END IF
288*
289* Quick return if possible
290*
291 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
292 DO 10 j = 1, nrhs
293 ferr( j ) = zero
294 berr( j ) = zero
295 10 CONTINUE
296 RETURN
297 END IF
298*
299 IF( notran ) THEN
300 transn = 'N'
301 transt = 'C'
302 ELSE
303 transn = 'C'
304 transt = 'N'
305 END IF
306*
307* NZ = maximum number of nonzero elements in each row of A, plus 1
308*
309 nz = 4
310 eps = dlamch( 'Epsilon' )
311 safmin = dlamch( 'Safe minimum' )
312 safe1 = nz*safmin
313 safe2 = safe1 / eps
314*
315* Do for each right hand side
316*
317 DO 110 j = 1, nrhs
318*
319 count = 1
320 lstres = three
321 20 CONTINUE
322*
323* Loop until stopping criterion is satisfied.
324*
325* Compute residual R = B - op(A) * X,
326* where op(A) = A, A**T, or A**H, depending on TRANS.
327*
328 CALL zcopy( n, b( 1, j ), 1, work, 1 )
329 CALL zlagtm( trans, n, 1, -one, dl, d, du, x( 1, j ), ldx, one,
330 \$ work, n )
331*
332* Compute abs(op(A))*abs(x) + abs(b) for use in the backward
333* error bound.
334*
335 IF( notran ) THEN
336 IF( n.EQ.1 ) THEN
337 rwork( 1 ) = cabs1( b( 1, j ) ) +
338 \$ cabs1( d( 1 ) )*cabs1( x( 1, j ) )
339 ELSE
340 rwork( 1 ) = cabs1( b( 1, j ) ) +
341 \$ cabs1( d( 1 ) )*cabs1( x( 1, j ) ) +
342 \$ cabs1( du( 1 ) )*cabs1( x( 2, j ) )
343 DO 30 i = 2, n - 1
344 rwork( i ) = cabs1( b( i, j ) ) +
345 \$ cabs1( dl( i-1 ) )*cabs1( x( i-1, j ) ) +
346 \$ cabs1( d( i ) )*cabs1( x( i, j ) ) +
347 \$ cabs1( du( i ) )*cabs1( x( i+1, j ) )
348 30 CONTINUE
349 rwork( n ) = cabs1( b( n, j ) ) +
350 \$ cabs1( dl( n-1 ) )*cabs1( x( n-1, j ) ) +
351 \$ cabs1( d( n ) )*cabs1( x( n, j ) )
352 END IF
353 ELSE
354 IF( n.EQ.1 ) THEN
355 rwork( 1 ) = cabs1( b( 1, j ) ) +
356 \$ cabs1( d( 1 ) )*cabs1( x( 1, j ) )
357 ELSE
358 rwork( 1 ) = cabs1( b( 1, j ) ) +
359 \$ cabs1( d( 1 ) )*cabs1( x( 1, j ) ) +
360 \$ cabs1( dl( 1 ) )*cabs1( x( 2, j ) )
361 DO 40 i = 2, n - 1
362 rwork( i ) = cabs1( b( i, j ) ) +
363 \$ cabs1( du( i-1 ) )*cabs1( x( i-1, j ) ) +
364 \$ cabs1( d( i ) )*cabs1( x( i, j ) ) +
365 \$ cabs1( dl( i ) )*cabs1( x( i+1, j ) )
366 40 CONTINUE
367 rwork( n ) = cabs1( b( n, j ) ) +
368 \$ cabs1( du( n-1 ) )*cabs1( x( n-1, j ) ) +
369 \$ cabs1( d( n ) )*cabs1( x( n, j ) )
370 END IF
371 END IF
372*
373* Compute componentwise relative backward error from formula
374*
375* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
376*
377* where abs(Z) is the componentwise absolute value of the matrix
378* or vector Z. If the i-th component of the denominator is less
379* than SAFE2, then SAFE1 is added to the i-th components of the
380* numerator and denominator before dividing.
381*
382 s = zero
383 DO 50 i = 1, n
384 IF( rwork( i ).GT.safe2 ) THEN
385 s = max( s, cabs1( work( i ) ) / rwork( i ) )
386 ELSE
387 s = max( s, ( cabs1( work( i ) )+safe1 ) /
388 \$ ( rwork( i )+safe1 ) )
389 END IF
390 50 CONTINUE
391 berr( j ) = s
392*
393* Test stopping criterion. Continue iterating if
394* 1) The residual BERR(J) is larger than machine epsilon, and
395* 2) BERR(J) decreased by at least a factor of 2 during the
396* last iteration, and
397* 3) At most ITMAX iterations tried.
398*
399 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
400 \$ count.LE.itmax ) THEN
401*
402* Update solution and try again.
403*
404 CALL zgttrs( trans, n, 1, dlf, df, duf, du2, ipiv, work, n,
405 \$ info )
406 CALL zaxpy( n, dcmplx( one ), work, 1, x( 1, j ), 1 )
407 lstres = berr( j )
408 count = count + 1
409 GO TO 20
410 END IF
411*
412* Bound error from formula
413*
414* norm(X - XTRUE) / norm(X) .le. FERR =
415* norm( abs(inv(op(A)))*
416* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
417*
418* where
419* norm(Z) is the magnitude of the largest component of Z
420* inv(op(A)) is the inverse of op(A)
421* abs(Z) is the componentwise absolute value of the matrix or
422* vector Z
423* NZ is the maximum number of nonzeros in any row of A, plus 1
424* EPS is machine epsilon
425*
426* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
427* is incremented by SAFE1 if the i-th component of
428* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
429*
430* Use ZLACN2 to estimate the infinity-norm of the matrix
431* inv(op(A)) * diag(W),
432* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
433*
434 DO 60 i = 1, n
435 IF( rwork( i ).GT.safe2 ) THEN
436 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
437 ELSE
438 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
439 \$ safe1
440 END IF
441 60 CONTINUE
442*
443 kase = 0
444 70 CONTINUE
445 CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
446 IF( kase.NE.0 ) THEN
447 IF( kase.EQ.1 ) THEN
448*
449* Multiply by diag(W)*inv(op(A)**H).
450*
451 CALL zgttrs( transt, n, 1, dlf, df, duf, du2, ipiv, work,
452 \$ n, info )
453 DO 80 i = 1, n
454 work( i ) = rwork( i )*work( i )
455 80 CONTINUE
456 ELSE
457*
458* Multiply by inv(op(A))*diag(W).
459*
460 DO 90 i = 1, n
461 work( i ) = rwork( i )*work( i )
462 90 CONTINUE
463 CALL zgttrs( transn, n, 1, dlf, df, duf, du2, ipiv, work,
464 \$ n, info )
465 END IF
466 GO TO 70
467 END IF
468*
469* Normalize error.
470*
471 lstres = zero
472 DO 100 i = 1, n
473 lstres = max( lstres, cabs1( x( i, j ) ) )
474 100 CONTINUE
475 IF( lstres.NE.zero )
476 \$ ferr( j ) = ferr( j ) / lstres
477*
478 110 CONTINUE
479*
480 RETURN
481*
482* End of ZGTRFS
483*
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 zgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info)
ZGTTRS
Definition zgttrs.f:138
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 zlagtm(trans, n, nrhs, alpha, dl, d, du, x, ldx, beta, b, ldb)
ZLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix,...
Definition zlagtm.f:145
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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