LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
zptrfs.f
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1 *> \brief \b ZPTRFS
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZPTRFS( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
22 * FERR, BERR, WORK, RWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, LDB, LDX, N, NRHS
27 * ..
28 * .. Array Arguments ..
29 * DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
30 * $ RWORK( * )
31 * COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
32 * $ X( LDX, * )
33 * ..
34 *
35 *
36 *> \par Purpose:
37 * =============
38 *>
39 *> \verbatim
40 *>
41 *> ZPTRFS improves the computed solution to a system of linear
42 *> equations when the coefficient matrix is Hermitian positive definite
43 *> and tridiagonal, and provides error bounds and backward error
44 *> estimates for the solution.
45 *> \endverbatim
46 *
47 * Arguments:
48 * ==========
49 *
50 *> \param[in] UPLO
51 *> \verbatim
52 *> UPLO is CHARACTER*1
53 *> Specifies whether the superdiagonal or the subdiagonal of the
54 *> tridiagonal matrix A is stored and the form of the
55 *> factorization:
56 *> = 'U': E is the superdiagonal of A, and A = U**H*D*U;
57 *> = 'L': E is the subdiagonal of A, and A = L*D*L**H.
58 *> (The two forms are equivalent if A is real.)
59 *> \endverbatim
60 *>
61 *> \param[in] N
62 *> \verbatim
63 *> N is INTEGER
64 *> The order of the matrix A. N >= 0.
65 *> \endverbatim
66 *>
67 *> \param[in] NRHS
68 *> \verbatim
69 *> NRHS is INTEGER
70 *> The number of right hand sides, i.e., the number of columns
71 *> of the matrix B. NRHS >= 0.
72 *> \endverbatim
73 *>
74 *> \param[in] D
75 *> \verbatim
76 *> D is DOUBLE PRECISION array, dimension (N)
77 *> The n real diagonal elements of the tridiagonal matrix A.
78 *> \endverbatim
79 *>
80 *> \param[in] E
81 *> \verbatim
82 *> E is COMPLEX*16 array, dimension (N-1)
83 *> The (n-1) off-diagonal elements of the tridiagonal matrix A
84 *> (see UPLO).
85 *> \endverbatim
86 *>
87 *> \param[in] DF
88 *> \verbatim
89 *> DF is DOUBLE PRECISION array, dimension (N)
90 *> The n diagonal elements of the diagonal matrix D from
91 *> the factorization computed by ZPTTRF.
92 *> \endverbatim
93 *>
94 *> \param[in] EF
95 *> \verbatim
96 *> EF is COMPLEX*16 array, dimension (N-1)
97 *> The (n-1) off-diagonal elements of the unit bidiagonal
98 *> factor U or L from the factorization computed by ZPTTRF
99 *> (see UPLO).
100 *> \endverbatim
101 *>
102 *> \param[in] B
103 *> \verbatim
104 *> B is COMPLEX*16 array, dimension (LDB,NRHS)
105 *> The right hand side matrix B.
106 *> \endverbatim
107 *>
108 *> \param[in] LDB
109 *> \verbatim
110 *> LDB is INTEGER
111 *> The leading dimension of the array B. LDB >= max(1,N).
112 *> \endverbatim
113 *>
114 *> \param[in,out] X
115 *> \verbatim
116 *> X is COMPLEX*16 array, dimension (LDX,NRHS)
117 *> On entry, the solution matrix X, as computed by ZPTTRS.
118 *> On exit, the improved solution matrix X.
119 *> \endverbatim
120 *>
121 *> \param[in] LDX
122 *> \verbatim
123 *> LDX is INTEGER
124 *> The leading dimension of the array X. LDX >= max(1,N).
125 *> \endverbatim
126 *>
127 *> \param[out] FERR
128 *> \verbatim
129 *> FERR is DOUBLE PRECISION array, dimension (NRHS)
130 *> The forward error bound for each solution vector
131 *> X(j) (the j-th column of the solution matrix X).
132 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
133 *> is an estimated upper bound for the magnitude of the largest
134 *> element in (X(j) - XTRUE) divided by the magnitude of the
135 *> largest element in X(j).
136 *> \endverbatim
137 *>
138 *> \param[out] BERR
139 *> \verbatim
140 *> BERR is DOUBLE PRECISION array, dimension (NRHS)
141 *> The componentwise relative backward error of each solution
142 *> vector X(j) (i.e., the smallest relative change in
143 *> any element of A or B that makes X(j) an exact solution).
144 *> \endverbatim
145 *>
146 *> \param[out] WORK
147 *> \verbatim
148 *> WORK is COMPLEX*16 array, dimension (N)
149 *> \endverbatim
150 *>
151 *> \param[out] RWORK
152 *> \verbatim
153 *> RWORK is DOUBLE PRECISION array, dimension (N)
154 *> \endverbatim
155 *>
156 *> \param[out] INFO
157 *> \verbatim
158 *> INFO is INTEGER
159 *> = 0: successful exit
160 *> < 0: if INFO = -i, the i-th argument had an illegal value
161 *> \endverbatim
162 *
163 *> \par Internal Parameters:
164 * =========================
165 *>
166 *> \verbatim
167 *> ITMAX is the maximum number of steps of iterative refinement.
168 *> \endverbatim
169 *
170 * Authors:
171 * ========
172 *
173 *> \author Univ. of Tennessee
174 *> \author Univ. of California Berkeley
175 *> \author Univ. of Colorado Denver
176 *> \author NAG Ltd.
177 *
178 *> \ingroup complex16PTcomputational
179 *
180 * =====================================================================
181  SUBROUTINE zptrfs( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
182  $ FERR, BERR, WORK, RWORK, INFO )
183 *
184 * -- LAPACK computational routine --
185 * -- LAPACK is a software package provided by Univ. of Tennessee, --
186 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
187 *
188 * .. Scalar Arguments ..
189  CHARACTER UPLO
190  INTEGER INFO, LDB, LDX, N, NRHS
191 * ..
192 * .. Array Arguments ..
193  DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
194  $ rwork( * )
195  COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
196  $ x( ldx, * )
197 * ..
198 *
199 * =====================================================================
200 *
201 * .. Parameters ..
202  INTEGER ITMAX
203  parameter( itmax = 5 )
204  DOUBLE PRECISION ZERO
205  parameter( zero = 0.0d+0 )
206  DOUBLE PRECISION ONE
207  parameter( one = 1.0d+0 )
208  DOUBLE PRECISION TWO
209  parameter( two = 2.0d+0 )
210  DOUBLE PRECISION THREE
211  parameter( three = 3.0d+0 )
212 * ..
213 * .. Local Scalars ..
214  LOGICAL UPPER
215  INTEGER COUNT, I, IX, J, NZ
216  DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
217  COMPLEX*16 BI, CX, DX, EX, ZDUM
218 * ..
219 * .. External Functions ..
220  LOGICAL LSAME
221  INTEGER IDAMAX
222  DOUBLE PRECISION DLAMCH
223  EXTERNAL lsame, idamax, dlamch
224 * ..
225 * .. External Subroutines ..
226  EXTERNAL xerbla, zaxpy, zpttrs
227 * ..
228 * .. Intrinsic Functions ..
229  INTRINSIC abs, dble, dcmplx, dconjg, dimag, max
230 * ..
231 * .. Statement Functions ..
232  DOUBLE PRECISION CABS1
233 * ..
234 * .. Statement Function definitions ..
235  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
236 * ..
237 * .. Executable Statements ..
238 *
239 * Test the input parameters.
240 *
241  info = 0
242  upper = lsame( uplo, 'U' )
243  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
244  info = -1
245  ELSE IF( n.LT.0 ) THEN
246  info = -2
247  ELSE IF( nrhs.LT.0 ) THEN
248  info = -3
249  ELSE IF( ldb.LT.max( 1, n ) ) THEN
250  info = -9
251  ELSE IF( ldx.LT.max( 1, n ) ) THEN
252  info = -11
253  END IF
254  IF( info.NE.0 ) THEN
255  CALL xerbla( 'ZPTRFS', -info )
256  RETURN
257  END IF
258 *
259 * Quick return if possible
260 *
261  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
262  DO 10 j = 1, nrhs
263  ferr( j ) = zero
264  berr( j ) = zero
265  10 CONTINUE
266  RETURN
267  END IF
268 *
269 * NZ = maximum number of nonzero elements in each row of A, plus 1
270 *
271  nz = 4
272  eps = dlamch( 'Epsilon' )
273  safmin = dlamch( 'Safe minimum' )
274  safe1 = nz*safmin
275  safe2 = safe1 / eps
276 *
277 * Do for each right hand side
278 *
279  DO 100 j = 1, nrhs
280 *
281  count = 1
282  lstres = three
283  20 CONTINUE
284 *
285 * Loop until stopping criterion is satisfied.
286 *
287 * Compute residual R = B - A * X. Also compute
288 * abs(A)*abs(x) + abs(b) for use in the backward error bound.
289 *
290  IF( upper ) THEN
291  IF( n.EQ.1 ) THEN
292  bi = b( 1, j )
293  dx = d( 1 )*x( 1, j )
294  work( 1 ) = bi - dx
295  rwork( 1 ) = cabs1( bi ) + cabs1( dx )
296  ELSE
297  bi = b( 1, j )
298  dx = d( 1 )*x( 1, j )
299  ex = e( 1 )*x( 2, j )
300  work( 1 ) = bi - dx - ex
301  rwork( 1 ) = cabs1( bi ) + cabs1( dx ) +
302  $ cabs1( e( 1 ) )*cabs1( x( 2, j ) )
303  DO 30 i = 2, n - 1
304  bi = b( i, j )
305  cx = dconjg( e( i-1 ) )*x( i-1, j )
306  dx = d( i )*x( i, j )
307  ex = e( i )*x( i+1, j )
308  work( i ) = bi - cx - dx - ex
309  rwork( i ) = cabs1( bi ) +
310  $ cabs1( e( i-1 ) )*cabs1( x( i-1, j ) ) +
311  $ cabs1( dx ) + cabs1( e( i ) )*
312  $ cabs1( x( i+1, j ) )
313  30 CONTINUE
314  bi = b( n, j )
315  cx = dconjg( e( n-1 ) )*x( n-1, j )
316  dx = d( n )*x( n, j )
317  work( n ) = bi - cx - dx
318  rwork( n ) = cabs1( bi ) + cabs1( e( n-1 ) )*
319  $ cabs1( x( n-1, j ) ) + cabs1( dx )
320  END IF
321  ELSE
322  IF( n.EQ.1 ) THEN
323  bi = b( 1, j )
324  dx = d( 1 )*x( 1, j )
325  work( 1 ) = bi - dx
326  rwork( 1 ) = cabs1( bi ) + cabs1( dx )
327  ELSE
328  bi = b( 1, j )
329  dx = d( 1 )*x( 1, j )
330  ex = dconjg( e( 1 ) )*x( 2, j )
331  work( 1 ) = bi - dx - ex
332  rwork( 1 ) = cabs1( bi ) + cabs1( dx ) +
333  $ cabs1( e( 1 ) )*cabs1( x( 2, j ) )
334  DO 40 i = 2, n - 1
335  bi = b( i, j )
336  cx = e( i-1 )*x( i-1, j )
337  dx = d( i )*x( i, j )
338  ex = dconjg( e( i ) )*x( i+1, j )
339  work( i ) = bi - cx - dx - ex
340  rwork( i ) = cabs1( bi ) +
341  $ cabs1( e( i-1 ) )*cabs1( x( i-1, j ) ) +
342  $ cabs1( dx ) + cabs1( e( i ) )*
343  $ cabs1( x( i+1, j ) )
344  40 CONTINUE
345  bi = b( n, j )
346  cx = e( n-1 )*x( n-1, j )
347  dx = d( n )*x( n, j )
348  work( n ) = bi - cx - dx
349  rwork( n ) = cabs1( bi ) + cabs1( e( n-1 ) )*
350  $ cabs1( x( n-1, j ) ) + cabs1( dx )
351  END IF
352  END IF
353 *
354 * Compute componentwise relative backward error from formula
355 *
356 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
357 *
358 * where abs(Z) is the componentwise absolute value of the matrix
359 * or vector Z. If the i-th component of the denominator is less
360 * than SAFE2, then SAFE1 is added to the i-th components of the
361 * numerator and denominator before dividing.
362 *
363  s = zero
364  DO 50 i = 1, n
365  IF( rwork( i ).GT.safe2 ) THEN
366  s = max( s, cabs1( work( i ) ) / rwork( i ) )
367  ELSE
368  s = max( s, ( cabs1( work( i ) )+safe1 ) /
369  $ ( rwork( i )+safe1 ) )
370  END IF
371  50 CONTINUE
372  berr( j ) = s
373 *
374 * Test stopping criterion. Continue iterating if
375 * 1) The residual BERR(J) is larger than machine epsilon, and
376 * 2) BERR(J) decreased by at least a factor of 2 during the
377 * last iteration, and
378 * 3) At most ITMAX iterations tried.
379 *
380  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
381  $ count.LE.itmax ) THEN
382 *
383 * Update solution and try again.
384 *
385  CALL zpttrs( uplo, n, 1, df, ef, work, n, info )
386  CALL zaxpy( n, dcmplx( one ), work, 1, x( 1, j ), 1 )
387  lstres = berr( j )
388  count = count + 1
389  GO TO 20
390  END IF
391 *
392 * Bound error from formula
393 *
394 * norm(X - XTRUE) / norm(X) .le. FERR =
395 * norm( abs(inv(A))*
396 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
397 *
398 * where
399 * norm(Z) is the magnitude of the largest component of Z
400 * inv(A) is the inverse of A
401 * abs(Z) is the componentwise absolute value of the matrix or
402 * vector Z
403 * NZ is the maximum number of nonzeros in any row of A, plus 1
404 * EPS is machine epsilon
405 *
406 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
407 * is incremented by SAFE1 if the i-th component of
408 * abs(A)*abs(X) + abs(B) is less than SAFE2.
409 *
410  DO 60 i = 1, n
411  IF( rwork( i ).GT.safe2 ) THEN
412  rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
413  ELSE
414  rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
415  $ safe1
416  END IF
417  60 CONTINUE
418  ix = idamax( n, rwork, 1 )
419  ferr( j ) = rwork( ix )
420 *
421 * Estimate the norm of inv(A).
422 *
423 * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
424 *
425 * m(i,j) = abs(A(i,j)), i = j,
426 * m(i,j) = -abs(A(i,j)), i .ne. j,
427 *
428 * and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**H.
429 *
430 * Solve M(L) * x = e.
431 *
432  rwork( 1 ) = one
433  DO 70 i = 2, n
434  rwork( i ) = one + rwork( i-1 )*abs( ef( i-1 ) )
435  70 CONTINUE
436 *
437 * Solve D * M(L)**H * x = b.
438 *
439  rwork( n ) = rwork( n ) / df( n )
440  DO 80 i = n - 1, 1, -1
441  rwork( i ) = rwork( i ) / df( i ) +
442  $ rwork( i+1 )*abs( ef( i ) )
443  80 CONTINUE
444 *
445 * Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
446 *
447  ix = idamax( n, rwork, 1 )
448  ferr( j ) = ferr( j )*abs( rwork( ix ) )
449 *
450 * Normalize error.
451 *
452  lstres = zero
453  DO 90 i = 1, n
454  lstres = max( lstres, abs( x( i, j ) ) )
455  90 CONTINUE
456  IF( lstres.NE.zero )
457  $ ferr( j ) = ferr( j ) / lstres
458 *
459  100 CONTINUE
460 *
461  RETURN
462 *
463 * End of ZPTRFS
464 *
465  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zpttrs(UPLO, N, NRHS, D, E, B, LDB, INFO)
ZPTTRS
Definition: zpttrs.f:121
subroutine zptrfs(UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZPTRFS
Definition: zptrfs.f:183