LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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zgtrfs.f
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1*> \brief \b ZGTRFS
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download ZGTRFS + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgtrfs.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgtrfs.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgtrfs.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
22* IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
23* INFO )
24*
25* .. Scalar Arguments ..
26* CHARACTER TRANS
27* INTEGER INFO, LDB, LDX, N, NRHS
28* ..
29* .. Array Arguments ..
30* INTEGER IPIV( * )
31* DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
32* COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
33* $ DLF( * ), DU( * ), DU2( * ), DUF( * ),
34* $ WORK( * ), X( LDX, * )
35* ..
36*
37*
38*> \par Purpose:
39* =============
40*>
41*> \verbatim
42*>
43*> ZGTRFS improves the computed solution to a system of linear
44*> equations when the coefficient matrix is tridiagonal, and provides
45*> error bounds and backward error estimates for the solution.
46*> \endverbatim
47*
48* Arguments:
49* ==========
50*
51*> \param[in] TRANS
52*> \verbatim
53*> TRANS is CHARACTER*1
54*> Specifies the form of the system of equations:
55*> = 'N': A * X = B (No transpose)
56*> = 'T': A**T * X = B (Transpose)
57*> = 'C': A**H * X = B (Conjugate transpose)
58*> \endverbatim
59*>
60*> \param[in] N
61*> \verbatim
62*> N is INTEGER
63*> The order of the matrix A. N >= 0.
64*> \endverbatim
65*>
66*> \param[in] NRHS
67*> \verbatim
68*> NRHS is INTEGER
69*> The number of right hand sides, i.e., the number of columns
70*> of the matrix B. NRHS >= 0.
71*> \endverbatim
72*>
73*> \param[in] DL
74*> \verbatim
75*> DL is COMPLEX*16 array, dimension (N-1)
76*> The (n-1) subdiagonal elements of A.
77*> \endverbatim
78*>
79*> \param[in] D
80*> \verbatim
81*> D is COMPLEX*16 array, dimension (N)
82*> The diagonal elements of A.
83*> \endverbatim
84*>
85*> \param[in] DU
86*> \verbatim
87*> DU is COMPLEX*16 array, dimension (N-1)
88*> The (n-1) superdiagonal elements of A.
89*> \endverbatim
90*>
91*> \param[in] DLF
92*> \verbatim
93*> DLF is COMPLEX*16 array, dimension (N-1)
94*> The (n-1) multipliers that define the matrix L from the
95*> LU factorization of A as computed by ZGTTRF.
96*> \endverbatim
97*>
98*> \param[in] DF
99*> \verbatim
100*> DF is COMPLEX*16 array, dimension (N)
101*> The n diagonal elements of the upper triangular matrix U from
102*> the LU factorization of A.
103*> \endverbatim
104*>
105*> \param[in] DUF
106*> \verbatim
107*> DUF is COMPLEX*16 array, dimension (N-1)
108*> The (n-1) elements of the first superdiagonal of U.
109*> \endverbatim
110*>
111*> \param[in] DU2
112*> \verbatim
113*> DU2 is COMPLEX*16 array, dimension (N-2)
114*> The (n-2) elements of the second superdiagonal of U.
115*> \endverbatim
116*>
117*> \param[in] IPIV
118*> \verbatim
119*> IPIV is INTEGER array, dimension (N)
120*> The pivot indices; for 1 <= i <= n, row i of the matrix was
121*> interchanged with row IPIV(i). IPIV(i) will always be either
122*> i or i+1; IPIV(i) = i indicates a row interchange was not
123*> required.
124*> \endverbatim
125*>
126*> \param[in] B
127*> \verbatim
128*> B is COMPLEX*16 array, dimension (LDB,NRHS)
129*> The right hand side matrix B.
130*> \endverbatim
131*>
132*> \param[in] LDB
133*> \verbatim
134*> LDB is INTEGER
135*> The leading dimension of the array B. LDB >= max(1,N).
136*> \endverbatim
137*>
138*> \param[in,out] X
139*> \verbatim
140*> X is COMPLEX*16 array, dimension (LDX,NRHS)
141*> On entry, the solution matrix X, as computed by ZGTTRS.
142*> On exit, the improved solution matrix X.
143*> \endverbatim
144*>
145*> \param[in] LDX
146*> \verbatim
147*> LDX is INTEGER
148*> The leading dimension of the array X. LDX >= max(1,N).
149*> \endverbatim
150*>
151*> \param[out] FERR
152*> \verbatim
153*> FERR is DOUBLE PRECISION array, dimension (NRHS)
154*> The estimated forward error bound for each solution vector
155*> X(j) (the j-th column of the solution matrix X).
156*> If XTRUE is the true solution corresponding to X(j), FERR(j)
157*> is an estimated upper bound for the magnitude of the largest
158*> element in (X(j) - XTRUE) divided by the magnitude of the
159*> largest element in X(j). The estimate is as reliable as
160*> the estimate for RCOND, and is almost always a slight
161*> overestimate of the true error.
162*> \endverbatim
163*>
164*> \param[out] BERR
165*> \verbatim
166*> BERR is DOUBLE PRECISION array, dimension (NRHS)
167*> The componentwise relative backward error of each solution
168*> vector X(j) (i.e., the smallest relative change in
169*> any element of A or B that makes X(j) an exact solution).
170*> \endverbatim
171*>
172*> \param[out] WORK
173*> \verbatim
174*> WORK is COMPLEX*16 array, dimension (2*N)
175*> \endverbatim
176*>
177*> \param[out] RWORK
178*> \verbatim
179*> RWORK is DOUBLE PRECISION array, dimension (N)
180*> \endverbatim
181*>
182*> \param[out] INFO
183*> \verbatim
184*> INFO is INTEGER
185*> = 0: successful exit
186*> < 0: if INFO = -i, the i-th argument had an illegal value
187*> \endverbatim
188*
189*> \par Internal Parameters:
190* =========================
191*>
192*> \verbatim
193*> ITMAX is the maximum number of steps of iterative refinement.
194*> \endverbatim
195*
196* Authors:
197* ========
198*
199*> \author Univ. of Tennessee
200*> \author Univ. of California Berkeley
201*> \author Univ. of Colorado Denver
202*> \author NAG Ltd.
203*
204*> \ingroup gtrfs
205*
206* =====================================================================
207 SUBROUTINE zgtrfs( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
208 $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
209 $ INFO )
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*
484 END
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
zaxpy
subroutine zaxpy(n, za, zx, incx, zy, incy)
ZAXPY
Definition zaxpy.f:88
zcopy
subroutine zcopy(n, zx, incx, zy, incy)
ZCOPY
Definition zcopy.f:81
zgtrfs
subroutine zgtrfs(trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZGTRFS
Definition zgtrfs.f:210
zgttrs
subroutine zgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info)
ZGTTRS
Definition zgttrs.f:138
zlacn2
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
zlagtm
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