LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
zherfs.f
Go to the documentation of this file.
1 *> \brief \b ZHERFS
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download ZHERFS + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zherfs.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zherfs.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zherfs.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZHERFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
22 * X, LDX, FERR, BERR, WORK, RWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IPIV( * )
30 * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
31 * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
32 * $ WORK( * ), X( LDX, * )
33 * ..
34 *
35 *
36 *> \par Purpose:
37 * =============
38 *>
39 *> \verbatim
40 *>
41 *> ZHERFS improves the computed solution to a system of linear
42 *> equations when the coefficient matrix is Hermitian indefinite, and
43 *> provides error bounds and backward error estimates for the solution.
44 *> \endverbatim
45 *
46 * Arguments:
47 * ==========
48 *
49 *> \param[in] UPLO
50 *> \verbatim
51 *> UPLO is CHARACTER*1
52 *> = 'U': Upper triangle of A is stored;
53 *> = 'L': Lower triangle of A is stored.
54 *> \endverbatim
55 *>
56 *> \param[in] N
57 *> \verbatim
58 *> N is INTEGER
59 *> The order of the matrix A. N >= 0.
60 *> \endverbatim
61 *>
62 *> \param[in] NRHS
63 *> \verbatim
64 *> NRHS is INTEGER
65 *> The number of right hand sides, i.e., the number of columns
66 *> of the matrices B and X. NRHS >= 0.
67 *> \endverbatim
68 *>
69 *> \param[in] A
70 *> \verbatim
71 *> A is COMPLEX*16 array, dimension (LDA,N)
72 *> The Hermitian matrix A. If UPLO = 'U', the leading N-by-N
73 *> upper triangular part of A contains the upper triangular part
74 *> of the matrix A, and the strictly lower triangular part of A
75 *> is not referenced. If UPLO = 'L', the leading N-by-N lower
76 *> triangular part of A contains the lower triangular part of
77 *> the matrix A, and the strictly upper triangular part of A is
78 *> not referenced.
79 *> \endverbatim
80 *>
81 *> \param[in] LDA
82 *> \verbatim
83 *> LDA is INTEGER
84 *> The leading dimension of the array A. LDA >= max(1,N).
85 *> \endverbatim
86 *>
87 *> \param[in] AF
88 *> \verbatim
89 *> AF is COMPLEX*16 array, dimension (LDAF,N)
90 *> The factored form of the matrix A. AF contains the block
91 *> diagonal matrix D and the multipliers used to obtain the
92 *> factor U or L from the factorization A = U*D*U**H or
93 *> A = L*D*L**H as computed by ZHETRF.
94 *> \endverbatim
95 *>
96 *> \param[in] LDAF
97 *> \verbatim
98 *> LDAF is INTEGER
99 *> The leading dimension of the array AF. LDAF >= max(1,N).
100 *> \endverbatim
101 *>
102 *> \param[in] IPIV
103 *> \verbatim
104 *> IPIV is INTEGER array, dimension (N)
105 *> Details of the interchanges and the block structure of D
106 *> as determined by ZHETRF.
107 *> \endverbatim
108 *>
109 *> \param[in] B
110 *> \verbatim
111 *> B is COMPLEX*16 array, dimension (LDB,NRHS)
112 *> The right hand side matrix B.
113 *> \endverbatim
114 *>
115 *> \param[in] LDB
116 *> \verbatim
117 *> LDB is INTEGER
118 *> The leading dimension of the array B. LDB >= max(1,N).
119 *> \endverbatim
120 *>
121 *> \param[in,out] X
122 *> \verbatim
123 *> X is COMPLEX*16 array, dimension (LDX,NRHS)
124 *> On entry, the solution matrix X, as computed by ZHETRS.
125 *> On exit, the improved solution matrix X.
126 *> \endverbatim
127 *>
128 *> \param[in] LDX
129 *> \verbatim
130 *> LDX is INTEGER
131 *> The leading dimension of the array X. LDX >= max(1,N).
132 *> \endverbatim
133 *>
134 *> \param[out] FERR
135 *> \verbatim
136 *> FERR is DOUBLE PRECISION array, dimension (NRHS)
137 *> The estimated forward error bound for each solution vector
138 *> X(j) (the j-th column of the solution matrix X).
139 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
140 *> is an estimated upper bound for the magnitude of the largest
141 *> element in (X(j) - XTRUE) divided by the magnitude of the
142 *> largest element in X(j). The estimate is as reliable as
143 *> the estimate for RCOND, and is almost always a slight
144 *> overestimate of the true error.
145 *> \endverbatim
146 *>
147 *> \param[out] BERR
148 *> \verbatim
149 *> BERR is DOUBLE PRECISION array, dimension (NRHS)
150 *> The componentwise relative backward error of each solution
151 *> vector X(j) (i.e., the smallest relative change in
152 *> any element of A or B that makes X(j) an exact solution).
153 *> \endverbatim
154 *>
155 *> \param[out] WORK
156 *> \verbatim
157 *> WORK is COMPLEX*16 array, dimension (2*N)
158 *> \endverbatim
159 *>
160 *> \param[out] RWORK
161 *> \verbatim
162 *> RWORK is DOUBLE PRECISION array, dimension (N)
163 *> \endverbatim
164 *>
165 *> \param[out] INFO
166 *> \verbatim
167 *> INFO is INTEGER
168 *> = 0: successful exit
169 *> < 0: if INFO = -i, the i-th argument had an illegal value
170 *> \endverbatim
171 *
172 *> \par Internal Parameters:
173 * =========================
174 *>
175 *> \verbatim
176 *> ITMAX is the maximum number of steps of iterative refinement.
177 *> \endverbatim
178 *
179 * Authors:
180 * ========
181 *
182 *> \author Univ. of Tennessee
183 *> \author Univ. of California Berkeley
184 *> \author Univ. of Colorado Denver
185 *> \author NAG Ltd.
186 *
187 *> \ingroup complex16HEcomputational
188 *
189 * =====================================================================
190  SUBROUTINE zherfs( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
191  $ X, LDX, FERR, BERR, WORK, RWORK, INFO )
192 *
193 * -- LAPACK computational routine --
194 * -- LAPACK is a software package provided by Univ. of Tennessee, --
195 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
196 *
197 * .. Scalar Arguments ..
198  CHARACTER UPLO
199  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
200 * ..
201 * .. Array Arguments ..
202  INTEGER IPIV( * )
203  DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
204  COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
205  $ work( * ), x( ldx, * )
206 * ..
207 *
208 * =====================================================================
209 *
210 * .. Parameters ..
211  INTEGER ITMAX
212  parameter( itmax = 5 )
213  DOUBLE PRECISION ZERO
214  parameter( zero = 0.0d+0 )
215  COMPLEX*16 ONE
216  parameter( one = ( 1.0d+0, 0.0d+0 ) )
217  DOUBLE PRECISION TWO
218  parameter( two = 2.0d+0 )
219  DOUBLE PRECISION THREE
220  parameter( three = 3.0d+0 )
221 * ..
222 * .. Local Scalars ..
223  LOGICAL UPPER
224  INTEGER COUNT, I, J, K, KASE, NZ
225  DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
226  COMPLEX*16 ZDUM
227 * ..
228 * .. Local Arrays ..
229  INTEGER ISAVE( 3 )
230 * ..
231 * .. External Subroutines ..
232  EXTERNAL xerbla, zaxpy, zcopy, zhemv, zhetrs, zlacn2
233 * ..
234 * .. Intrinsic Functions ..
235  INTRINSIC abs, dble, dimag, max
236 * ..
237 * .. External Functions ..
238  LOGICAL LSAME
239  DOUBLE PRECISION DLAMCH
240  EXTERNAL lsame, dlamch
241 * ..
242 * .. Statement Functions ..
243  DOUBLE PRECISION CABS1
244 * ..
245 * .. Statement Function definitions ..
246  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
247 * ..
248 * .. Executable Statements ..
249 *
250 * Test the input parameters.
251 *
252  info = 0
253  upper = lsame( uplo, 'U' )
254  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
255  info = -1
256  ELSE IF( n.LT.0 ) THEN
257  info = -2
258  ELSE IF( nrhs.LT.0 ) THEN
259  info = -3
260  ELSE IF( lda.LT.max( 1, n ) ) THEN
261  info = -5
262  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
263  info = -7
264  ELSE IF( ldb.LT.max( 1, n ) ) THEN
265  info = -10
266  ELSE IF( ldx.LT.max( 1, n ) ) THEN
267  info = -12
268  END IF
269  IF( info.NE.0 ) THEN
270  CALL xerbla( 'ZHERFS', -info )
271  RETURN
272  END IF
273 *
274 * Quick return if possible
275 *
276  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
277  DO 10 j = 1, nrhs
278  ferr( j ) = zero
279  berr( j ) = zero
280  10 CONTINUE
281  RETURN
282  END IF
283 *
284 * NZ = maximum number of nonzero elements in each row of A, plus 1
285 *
286  nz = n + 1
287  eps = dlamch( 'Epsilon' )
288  safmin = dlamch( 'Safe minimum' )
289  safe1 = nz*safmin
290  safe2 = safe1 / eps
291 *
292 * Do for each right hand side
293 *
294  DO 140 j = 1, nrhs
295 *
296  count = 1
297  lstres = three
298  20 CONTINUE
299 *
300 * Loop until stopping criterion is satisfied.
301 *
302 * Compute residual R = B - A * X
303 *
304  CALL zcopy( n, b( 1, j ), 1, work, 1 )
305  CALL zhemv( uplo, n, -one, a, lda, x( 1, j ), 1, one, work, 1 )
306 *
307 * Compute componentwise relative backward error from formula
308 *
309 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
310 *
311 * where abs(Z) is the componentwise absolute value of the matrix
312 * or vector Z. If the i-th component of the denominator is less
313 * than SAFE2, then SAFE1 is added to the i-th components of the
314 * numerator and denominator before dividing.
315 *
316  DO 30 i = 1, n
317  rwork( i ) = cabs1( b( i, j ) )
318  30 CONTINUE
319 *
320 * Compute abs(A)*abs(X) + abs(B).
321 *
322  IF( upper ) THEN
323  DO 50 k = 1, n
324  s = zero
325  xk = cabs1( x( k, j ) )
326  DO 40 i = 1, k - 1
327  rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
328  s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
329  40 CONTINUE
330  rwork( k ) = rwork( k ) + abs( dble( a( k, k ) ) )*xk + s
331  50 CONTINUE
332  ELSE
333  DO 70 k = 1, n
334  s = zero
335  xk = cabs1( x( k, j ) )
336  rwork( k ) = rwork( k ) + abs( dble( a( k, k ) ) )*xk
337  DO 60 i = k + 1, n
338  rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
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 zhetrs( uplo, 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(A))*
377 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
378 *
379 * where
380 * norm(Z) is the magnitude of the largest component of Z
381 * inv(A) is the inverse of 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(A)*abs(X)+abs(B))
388 * is incremented by SAFE1 if the i-th component of
389 * abs(A)*abs(X) + abs(B) is less than SAFE2.
390 *
391 * Use ZLACN2 to estimate the infinity-norm of the matrix
392 * inv(A) * diag(W),
393 * where W = abs(R) + NZ*EPS*( abs(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(A**H).
411 *
412  CALL zhetrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
413  DO 110 i = 1, n
414  work( i ) = rwork( i )*work( i )
415  110 CONTINUE
416  ELSE IF( kase.EQ.2 ) THEN
417 *
418 * Multiply by inv(A)*diag(W).
419 *
420  DO 120 i = 1, n
421  work( i ) = rwork( i )*work( i )
422  120 CONTINUE
423  CALL zhetrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
424  END IF
425  GO TO 100
426  END IF
427 *
428 * Normalize error.
429 *
430  lstres = zero
431  DO 130 i = 1, n
432  lstres = max( lstres, cabs1( x( i, j ) ) )
433  130 CONTINUE
434  IF( lstres.NE.zero )
435  $ ferr( j ) = ferr( j ) / lstres
436 *
437  140 CONTINUE
438 *
439  RETURN
440 *
441 * End of ZHERFS
442 *
443  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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 zhemv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZHEMV
Definition: zhemv.f:154
subroutine zherfs(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZHERFS
Definition: zherfs.f:192
subroutine zhetrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZHETRS
Definition: zhetrs.f:120
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