LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
cpbrfs.f
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1 *> \brief \b CPBRFS
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
22 * LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
27 * ..
28 * .. Array Arguments ..
29 * REAL BERR( * ), FERR( * ), RWORK( * )
30 * COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
31 * $ WORK( * ), X( LDX, * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> CPBRFS improves the computed solution to a system of linear
41 *> equations when the coefficient matrix is Hermitian positive definite
42 *> and banded, and provides error bounds and backward error estimates
43 *> 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] KD
63 *> \verbatim
64 *> KD is INTEGER
65 *> The number of superdiagonals of the matrix A if UPLO = 'U',
66 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
67 *> \endverbatim
68 *>
69 *> \param[in] NRHS
70 *> \verbatim
71 *> NRHS is INTEGER
72 *> The number of right hand sides, i.e., the number of columns
73 *> of the matrices B and X. NRHS >= 0.
74 *> \endverbatim
75 *>
76 *> \param[in] AB
77 *> \verbatim
78 *> AB is COMPLEX array, dimension (LDAB,N)
79 *> The upper or lower triangle of the Hermitian band matrix A,
80 *> stored in the first KD+1 rows of the array. The j-th column
81 *> of A is stored in the j-th column of the array AB as follows:
82 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
83 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
84 *> \endverbatim
85 *>
86 *> \param[in] LDAB
87 *> \verbatim
88 *> LDAB is INTEGER
89 *> The leading dimension of the array AB. LDAB >= KD+1.
90 *> \endverbatim
91 *>
92 *> \param[in] AFB
93 *> \verbatim
94 *> AFB is COMPLEX array, dimension (LDAFB,N)
95 *> The triangular factor U or L from the Cholesky factorization
96 *> A = U**H*U or A = L*L**H of the band matrix A as computed by
97 *> CPBTRF, in the same storage format as A (see AB).
98 *> \endverbatim
99 *>
100 *> \param[in] LDAFB
101 *> \verbatim
102 *> LDAFB is INTEGER
103 *> The leading dimension of the array AFB. LDAFB >= KD+1.
104 *> \endverbatim
105 *>
106 *> \param[in] B
107 *> \verbatim
108 *> B is COMPLEX array, dimension (LDB,NRHS)
109 *> The right hand side matrix B.
110 *> \endverbatim
111 *>
112 *> \param[in] LDB
113 *> \verbatim
114 *> LDB is INTEGER
115 *> The leading dimension of the array B. LDB >= max(1,N).
116 *> \endverbatim
117 *>
118 *> \param[in,out] X
119 *> \verbatim
120 *> X is COMPLEX array, dimension (LDX,NRHS)
121 *> On entry, the solution matrix X, as computed by CPBTRS.
122 *> On exit, the improved solution matrix X.
123 *> \endverbatim
124 *>
125 *> \param[in] LDX
126 *> \verbatim
127 *> LDX is INTEGER
128 *> The leading dimension of the array X. LDX >= max(1,N).
129 *> \endverbatim
130 *>
131 *> \param[out] FERR
132 *> \verbatim
133 *> FERR is REAL array, dimension (NRHS)
134 *> The estimated forward error bound for each solution vector
135 *> X(j) (the j-th column of the solution matrix X).
136 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
137 *> is an estimated upper bound for the magnitude of the largest
138 *> element in (X(j) - XTRUE) divided by the magnitude of the
139 *> largest element in X(j). The estimate is as reliable as
140 *> the estimate for RCOND, and is almost always a slight
141 *> overestimate of the true error.
142 *> \endverbatim
143 *>
144 *> \param[out] BERR
145 *> \verbatim
146 *> BERR is REAL array, dimension (NRHS)
147 *> The componentwise relative backward error of each solution
148 *> vector X(j) (i.e., the smallest relative change in
149 *> any element of A or B that makes X(j) an exact solution).
150 *> \endverbatim
151 *>
152 *> \param[out] WORK
153 *> \verbatim
154 *> WORK is COMPLEX array, dimension (2*N)
155 *> \endverbatim
156 *>
157 *> \param[out] RWORK
158 *> \verbatim
159 *> RWORK is REAL array, dimension (N)
160 *> \endverbatim
161 *>
162 *> \param[out] INFO
163 *> \verbatim
164 *> INFO is INTEGER
165 *> = 0: successful exit
166 *> < 0: if INFO = -i, the i-th argument had an illegal value
167 *> \endverbatim
168 *
169 *> \par Internal Parameters:
170 * =========================
171 *>
172 *> \verbatim
173 *> ITMAX is the maximum number of steps of iterative refinement.
174 *> \endverbatim
175 *
176 * Authors:
177 * ========
178 *
179 *> \author Univ. of Tennessee
180 *> \author Univ. of California Berkeley
181 *> \author Univ. of Colorado Denver
182 *> \author NAG Ltd.
183 *
184 *> \ingroup complexOTHERcomputational
185 *
186 * =====================================================================
187  SUBROUTINE cpbrfs( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
188  $ LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
189 *
190 * -- LAPACK computational routine --
191 * -- LAPACK is a software package provided by Univ. of Tennessee, --
192 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
193 *
194 * .. Scalar Arguments ..
195  CHARACTER UPLO
196  INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
197 * ..
198 * .. Array Arguments ..
199  REAL BERR( * ), FERR( * ), RWORK( * )
200  COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
201  $ work( * ), x( ldx, * )
202 * ..
203 *
204 * =====================================================================
205 *
206 * .. Parameters ..
207  INTEGER ITMAX
208  parameter( itmax = 5 )
209  REAL ZERO
210  parameter( zero = 0.0e+0 )
211  COMPLEX ONE
212  parameter( one = ( 1.0e+0, 0.0e+0 ) )
213  REAL TWO
214  parameter( two = 2.0e+0 )
215  REAL THREE
216  parameter( three = 3.0e+0 )
217 * ..
218 * .. Local Scalars ..
219  LOGICAL UPPER
220  INTEGER COUNT, I, J, K, KASE, L, NZ
221  REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
222  COMPLEX ZDUM
223 * ..
224 * .. Local Arrays ..
225  INTEGER ISAVE( 3 )
226 * ..
227 * .. External Subroutines ..
228  EXTERNAL caxpy, ccopy, chbmv, clacn2, cpbtrs, xerbla
229 * ..
230 * .. Intrinsic Functions ..
231  INTRINSIC abs, aimag, max, min, real
232 * ..
233 * .. External Functions ..
234  LOGICAL LSAME
235  REAL SLAMCH
236  EXTERNAL lsame, slamch
237 * ..
238 * .. Statement Functions ..
239  REAL CABS1
240 * ..
241 * .. Statement Function definitions ..
242  cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
243 * ..
244 * .. Executable Statements ..
245 *
246 * Test the input parameters.
247 *
248  info = 0
249  upper = lsame( uplo, 'U' )
250  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
251  info = -1
252  ELSE IF( n.LT.0 ) THEN
253  info = -2
254  ELSE IF( kd.LT.0 ) THEN
255  info = -3
256  ELSE IF( nrhs.LT.0 ) THEN
257  info = -4
258  ELSE IF( ldab.LT.kd+1 ) THEN
259  info = -6
260  ELSE IF( ldafb.LT.kd+1 ) THEN
261  info = -8
262  ELSE IF( ldb.LT.max( 1, n ) ) THEN
263  info = -10
264  ELSE IF( ldx.LT.max( 1, n ) ) THEN
265  info = -12
266  END IF
267  IF( info.NE.0 ) THEN
268  CALL xerbla( 'CPBRFS', -info )
269  RETURN
270  END IF
271 *
272 * Quick return if possible
273 *
274  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
275  DO 10 j = 1, nrhs
276  ferr( j ) = zero
277  berr( j ) = zero
278  10 CONTINUE
279  RETURN
280  END IF
281 *
282 * NZ = maximum number of nonzero elements in each row of A, plus 1
283 *
284  nz = min( n+1, 2*kd+2 )
285  eps = slamch( 'Epsilon' )
286  safmin = slamch( 'Safe minimum' )
287  safe1 = nz*safmin
288  safe2 = safe1 / eps
289 *
290 * Do for each right hand side
291 *
292  DO 140 j = 1, nrhs
293 *
294  count = 1
295  lstres = three
296  20 CONTINUE
297 *
298 * Loop until stopping criterion is satisfied.
299 *
300 * Compute residual R = B - A * X
301 *
302  CALL ccopy( n, b( 1, j ), 1, work, 1 )
303  CALL chbmv( uplo, n, kd, -one, ab, ldab, x( 1, j ), 1, one,
304  $ work, 1 )
305 *
306 * Compute componentwise relative backward error from formula
307 *
308 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
309 *
310 * where abs(Z) is the componentwise absolute value of the matrix
311 * or vector Z. If the i-th component of the denominator is less
312 * than SAFE2, then SAFE1 is added to the i-th components of the
313 * numerator and denominator before dividing.
314 *
315  DO 30 i = 1, n
316  rwork( i ) = cabs1( b( i, j ) )
317  30 CONTINUE
318 *
319 * Compute abs(A)*abs(X) + abs(B).
320 *
321  IF( upper ) THEN
322  DO 50 k = 1, n
323  s = zero
324  xk = cabs1( x( k, j ) )
325  l = kd + 1 - k
326  DO 40 i = max( 1, k-kd ), k - 1
327  rwork( i ) = rwork( i ) + cabs1( ab( l+i, k ) )*xk
328  s = s + cabs1( ab( l+i, k ) )*cabs1( x( i, j ) )
329  40 CONTINUE
330  rwork( k ) = rwork( k ) + abs( real( ab( kd+1, k ) ) )*
331  $ xk + s
332  50 CONTINUE
333  ELSE
334  DO 70 k = 1, n
335  s = zero
336  xk = cabs1( x( k, j ) )
337  rwork( k ) = rwork( k ) + abs( real( ab( 1, k ) ) )*xk
338  l = 1 - k
339  DO 60 i = k + 1, min( n, k+kd )
340  rwork( i ) = rwork( i ) + cabs1( ab( l+i, k ) )*xk
341  s = s + cabs1( ab( l+i, k ) )*cabs1( x( i, j ) )
342  60 CONTINUE
343  rwork( k ) = rwork( k ) + s
344  70 CONTINUE
345  END IF
346  s = zero
347  DO 80 i = 1, n
348  IF( rwork( i ).GT.safe2 ) THEN
349  s = max( s, cabs1( work( i ) ) / rwork( i ) )
350  ELSE
351  s = max( s, ( cabs1( work( i ) )+safe1 ) /
352  $ ( rwork( i )+safe1 ) )
353  END IF
354  80 CONTINUE
355  berr( j ) = s
356 *
357 * Test stopping criterion. Continue iterating if
358 * 1) The residual BERR(J) is larger than machine epsilon, and
359 * 2) BERR(J) decreased by at least a factor of 2 during the
360 * last iteration, and
361 * 3) At most ITMAX iterations tried.
362 *
363  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
364  $ count.LE.itmax ) THEN
365 *
366 * Update solution and try again.
367 *
368  CALL cpbtrs( uplo, n, kd, 1, afb, ldafb, work, n, info )
369  CALL caxpy( n, one, work, 1, x( 1, j ), 1 )
370  lstres = berr( j )
371  count = count + 1
372  GO TO 20
373  END IF
374 *
375 * Bound error from formula
376 *
377 * norm(X - XTRUE) / norm(X) .le. FERR =
378 * norm( abs(inv(A))*
379 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
380 *
381 * where
382 * norm(Z) is the magnitude of the largest component of Z
383 * inv(A) is the inverse of A
384 * abs(Z) is the componentwise absolute value of the matrix or
385 * vector Z
386 * NZ is the maximum number of nonzeros in any row of A, plus 1
387 * EPS is machine epsilon
388 *
389 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
390 * is incremented by SAFE1 if the i-th component of
391 * abs(A)*abs(X) + abs(B) is less than SAFE2.
392 *
393 * Use CLACN2 to estimate the infinity-norm of the matrix
394 * inv(A) * diag(W),
395 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
396 *
397  DO 90 i = 1, n
398  IF( rwork( i ).GT.safe2 ) THEN
399  rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
400  ELSE
401  rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
402  $ safe1
403  END IF
404  90 CONTINUE
405 *
406  kase = 0
407  100 CONTINUE
408  CALL clacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
409  IF( kase.NE.0 ) THEN
410  IF( kase.EQ.1 ) THEN
411 *
412 * Multiply by diag(W)*inv(A**H).
413 *
414  CALL cpbtrs( uplo, n, kd, 1, afb, ldafb, work, n, info )
415  DO 110 i = 1, n
416  work( i ) = rwork( i )*work( i )
417  110 CONTINUE
418  ELSE IF( kase.EQ.2 ) THEN
419 *
420 * Multiply by inv(A)*diag(W).
421 *
422  DO 120 i = 1, n
423  work( i ) = rwork( i )*work( i )
424  120 CONTINUE
425  CALL cpbtrs( uplo, n, kd, 1, afb, ldafb, work, n, info )
426  END IF
427  GO TO 100
428  END IF
429 *
430 * Normalize error.
431 *
432  lstres = zero
433  DO 130 i = 1, n
434  lstres = max( lstres, cabs1( x( i, j ) ) )
435  130 CONTINUE
436  IF( lstres.NE.zero )
437  $ ferr( j ) = ferr( j ) / lstres
438 *
439  140 CONTINUE
440 *
441  RETURN
442 *
443 * End of CPBRFS
444 *
445  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:88
subroutine chbmv(UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CHBMV
Definition: chbmv.f:187
subroutine clacn2(N, V, X, EST, KASE, ISAVE)
CLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: clacn2.f:133
subroutine cpbrfs(UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPBRFS
Definition: cpbrfs.f:189
subroutine cpbtrs(UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
CPBTRS
Definition: cpbtrs.f:121