LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
dsprfs.f
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1 *> \brief \b DSPRFS
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
22 * FERR, BERR, WORK, IWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, LDB, LDX, N, NRHS
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IPIV( * ), IWORK( * )
30 * DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
31 * \$ FERR( * ), WORK( * ), X( LDX, * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> DSPRFS improves the computed solution to a system of linear
41 *> equations when the coefficient matrix is symmetric indefinite
42 *> and packed, 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] 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] AP
70 *> \verbatim
71 *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
72 *> The upper or lower triangle of the symmetric matrix A, packed
73 *> columnwise in a linear array. The j-th column of A is stored
74 *> in the array AP as follows:
75 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
76 *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
77 *> \endverbatim
78 *>
79 *> \param[in] AFP
80 *> \verbatim
81 *> AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
82 *> The factored form of the matrix A. AFP contains the block
83 *> diagonal matrix D and the multipliers used to obtain the
84 *> factor U or L from the factorization A = U*D*U**T or
85 *> A = L*D*L**T as computed by DSPTRF, stored as a packed
86 *> triangular matrix.
87 *> \endverbatim
88 *>
89 *> \param[in] IPIV
90 *> \verbatim
91 *> IPIV is INTEGER array, dimension (N)
92 *> Details of the interchanges and the block structure of D
93 *> as determined by DSPTRF.
94 *> \endverbatim
95 *>
96 *> \param[in] B
97 *> \verbatim
98 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
99 *> The right hand side matrix B.
100 *> \endverbatim
101 *>
102 *> \param[in] LDB
103 *> \verbatim
104 *> LDB is INTEGER
105 *> The leading dimension of the array B. LDB >= max(1,N).
106 *> \endverbatim
107 *>
108 *> \param[in,out] X
109 *> \verbatim
110 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
111 *> On entry, the solution matrix X, as computed by DSPTRS.
112 *> On exit, the improved solution matrix X.
113 *> \endverbatim
114 *>
115 *> \param[in] LDX
116 *> \verbatim
117 *> LDX is INTEGER
118 *> The leading dimension of the array X. LDX >= max(1,N).
119 *> \endverbatim
120 *>
121 *> \param[out] FERR
122 *> \verbatim
123 *> FERR is DOUBLE PRECISION array, dimension (NRHS)
124 *> The estimated forward error bound for each solution vector
125 *> X(j) (the j-th column of the solution matrix X).
126 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
127 *> is an estimated upper bound for the magnitude of the largest
128 *> element in (X(j) - XTRUE) divided by the magnitude of the
129 *> largest element in X(j). The estimate is as reliable as
130 *> the estimate for RCOND, and is almost always a slight
131 *> overestimate of the true error.
132 *> \endverbatim
133 *>
134 *> \param[out] BERR
135 *> \verbatim
136 *> BERR is DOUBLE PRECISION array, dimension (NRHS)
137 *> The componentwise relative backward error of each solution
138 *> vector X(j) (i.e., the smallest relative change in
139 *> any element of A or B that makes X(j) an exact solution).
140 *> \endverbatim
141 *>
142 *> \param[out] WORK
143 *> \verbatim
144 *> WORK is DOUBLE PRECISION array, dimension (3*N)
145 *> \endverbatim
146 *>
147 *> \param[out] IWORK
148 *> \verbatim
149 *> IWORK is INTEGER array, dimension (N)
150 *> \endverbatim
151 *>
152 *> \param[out] INFO
153 *> \verbatim
154 *> INFO is INTEGER
155 *> = 0: successful exit
156 *> < 0: if INFO = -i, the i-th argument had an illegal value
157 *> \endverbatim
158 *
159 *> \par Internal Parameters:
160 * =========================
161 *>
162 *> \verbatim
163 *> ITMAX is the maximum number of steps of iterative refinement.
164 *> \endverbatim
165 *
166 * Authors:
167 * ========
168 *
169 *> \author Univ. of Tennessee
170 *> \author Univ. of California Berkeley
171 *> \author Univ. of Colorado Denver
172 *> \author NAG Ltd.
173 *
174 *> \ingroup doubleOTHERcomputational
175 *
176 * =====================================================================
177  SUBROUTINE dsprfs( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
178  \$ FERR, BERR, WORK, IWORK, INFO )
179 *
180 * -- LAPACK computational routine --
181 * -- LAPACK is a software package provided by Univ. of Tennessee, --
182 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
183 *
184 * .. Scalar Arguments ..
185  CHARACTER UPLO
186  INTEGER INFO, LDB, LDX, N, NRHS
187 * ..
188 * .. Array Arguments ..
189  INTEGER IPIV( * ), IWORK( * )
190  DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
191  \$ ferr( * ), work( * ), x( ldx, * )
192 * ..
193 *
194 * =====================================================================
195 *
196 * .. Parameters ..
197  INTEGER ITMAX
198  parameter( itmax = 5 )
199  DOUBLE PRECISION ZERO
200  parameter( zero = 0.0d+0 )
201  DOUBLE PRECISION ONE
202  parameter( one = 1.0d+0 )
203  DOUBLE PRECISION TWO
204  parameter( two = 2.0d+0 )
205  DOUBLE PRECISION THREE
206  parameter( three = 3.0d+0 )
207 * ..
208 * .. Local Scalars ..
209  LOGICAL UPPER
210  INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
211  DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
212 * ..
213 * .. Local Arrays ..
214  INTEGER ISAVE( 3 )
215 * ..
216 * .. External Subroutines ..
217  EXTERNAL daxpy, dcopy, dlacn2, dspmv, dsptrs, xerbla
218 * ..
219 * .. Intrinsic Functions ..
220  INTRINSIC abs, max
221 * ..
222 * .. External Functions ..
223  LOGICAL LSAME
224  DOUBLE PRECISION DLAMCH
225  EXTERNAL lsame, dlamch
226 * ..
227 * .. Executable Statements ..
228 *
229 * Test the input parameters.
230 *
231  info = 0
232  upper = lsame( uplo, 'U' )
233  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
234  info = -1
235  ELSE IF( n.LT.0 ) THEN
236  info = -2
237  ELSE IF( nrhs.LT.0 ) THEN
238  info = -3
239  ELSE IF( ldb.LT.max( 1, n ) ) THEN
240  info = -8
241  ELSE IF( ldx.LT.max( 1, n ) ) THEN
242  info = -10
243  END IF
244  IF( info.NE.0 ) THEN
245  CALL xerbla( 'DSPRFS', -info )
246  RETURN
247  END IF
248 *
249 * Quick return if possible
250 *
251  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
252  DO 10 j = 1, nrhs
253  ferr( j ) = zero
254  berr( j ) = zero
255  10 CONTINUE
256  RETURN
257  END IF
258 *
259 * NZ = maximum number of nonzero elements in each row of A, plus 1
260 *
261  nz = n + 1
262  eps = dlamch( 'Epsilon' )
263  safmin = dlamch( 'Safe minimum' )
264  safe1 = nz*safmin
265  safe2 = safe1 / eps
266 *
267 * Do for each right hand side
268 *
269  DO 140 j = 1, nrhs
270 *
271  count = 1
272  lstres = three
273  20 CONTINUE
274 *
275 * Loop until stopping criterion is satisfied.
276 *
277 * Compute residual R = B - A * X
278 *
279  CALL dcopy( n, b( 1, j ), 1, work( n+1 ), 1 )
280  CALL dspmv( uplo, n, -one, ap, x( 1, j ), 1, one, work( n+1 ),
281  \$ 1 )
282 *
283 * Compute componentwise relative backward error from formula
284 *
285 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
286 *
287 * where abs(Z) is the componentwise absolute value of the matrix
288 * or vector Z. If the i-th component of the denominator is less
289 * than SAFE2, then SAFE1 is added to the i-th components of the
290 * numerator and denominator before dividing.
291 *
292  DO 30 i = 1, n
293  work( i ) = abs( b( i, j ) )
294  30 CONTINUE
295 *
296 * Compute abs(A)*abs(X) + abs(B).
297 *
298  kk = 1
299  IF( upper ) THEN
300  DO 50 k = 1, n
301  s = zero
302  xk = abs( x( k, j ) )
303  ik = kk
304  DO 40 i = 1, k - 1
305  work( i ) = work( i ) + abs( ap( ik ) )*xk
306  s = s + abs( ap( ik ) )*abs( x( i, j ) )
307  ik = ik + 1
308  40 CONTINUE
309  work( k ) = work( k ) + abs( ap( kk+k-1 ) )*xk + s
310  kk = kk + k
311  50 CONTINUE
312  ELSE
313  DO 70 k = 1, n
314  s = zero
315  xk = abs( x( k, j ) )
316  work( k ) = work( k ) + abs( ap( kk ) )*xk
317  ik = kk + 1
318  DO 60 i = k + 1, n
319  work( i ) = work( i ) + abs( ap( ik ) )*xk
320  s = s + abs( ap( ik ) )*abs( x( i, j ) )
321  ik = ik + 1
322  60 CONTINUE
323  work( k ) = work( k ) + s
324  kk = kk + ( n-k+1 )
325  70 CONTINUE
326  END IF
327  s = zero
328  DO 80 i = 1, n
329  IF( work( i ).GT.safe2 ) THEN
330  s = max( s, abs( work( n+i ) ) / work( i ) )
331  ELSE
332  s = max( s, ( abs( work( n+i ) )+safe1 ) /
333  \$ ( work( i )+safe1 ) )
334  END IF
335  80 CONTINUE
336  berr( j ) = s
337 *
338 * Test stopping criterion. Continue iterating if
339 * 1) The residual BERR(J) is larger than machine epsilon, and
340 * 2) BERR(J) decreased by at least a factor of 2 during the
341 * last iteration, and
342 * 3) At most ITMAX iterations tried.
343 *
344  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
345  \$ count.LE.itmax ) THEN
346 *
347 * Update solution and try again.
348 *
349  CALL dsptrs( uplo, n, 1, afp, ipiv, work( n+1 ), n, info )
350  CALL daxpy( n, one, work( n+1 ), 1, x( 1, j ), 1 )
351  lstres = berr( j )
352  count = count + 1
353  GO TO 20
354  END IF
355 *
356 * Bound error from formula
357 *
358 * norm(X - XTRUE) / norm(X) .le. FERR =
359 * norm( abs(inv(A))*
360 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
361 *
362 * where
363 * norm(Z) is the magnitude of the largest component of Z
364 * inv(A) is the inverse of A
365 * abs(Z) is the componentwise absolute value of the matrix or
366 * vector Z
367 * NZ is the maximum number of nonzeros in any row of A, plus 1
368 * EPS is machine epsilon
369 *
370 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
371 * is incremented by SAFE1 if the i-th component of
372 * abs(A)*abs(X) + abs(B) is less than SAFE2.
373 *
374 * Use DLACN2 to estimate the infinity-norm of the matrix
375 * inv(A) * diag(W),
376 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
377 *
378  DO 90 i = 1, n
379  IF( work( i ).GT.safe2 ) THEN
380  work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
381  ELSE
382  work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
383  END IF
384  90 CONTINUE
385 *
386  kase = 0
387  100 CONTINUE
388  CALL dlacn2( n, work( 2*n+1 ), work( n+1 ), iwork, ferr( j ),
389  \$ kase, isave )
390  IF( kase.NE.0 ) THEN
391  IF( kase.EQ.1 ) THEN
392 *
393 * Multiply by diag(W)*inv(A**T).
394 *
395  CALL dsptrs( uplo, n, 1, afp, ipiv, work( n+1 ), n,
396  \$ info )
397  DO 110 i = 1, n
398  work( n+i ) = work( i )*work( n+i )
399  110 CONTINUE
400  ELSE IF( kase.EQ.2 ) THEN
401 *
402 * Multiply by inv(A)*diag(W).
403 *
404  DO 120 i = 1, n
405  work( n+i ) = work( i )*work( n+i )
406  120 CONTINUE
407  CALL dsptrs( uplo, n, 1, afp, ipiv, work( n+1 ), n,
408  \$ info )
409  END IF
410  GO TO 100
411  END IF
412 *
413 * Normalize error.
414 *
415  lstres = zero
416  DO 130 i = 1, n
417  lstres = max( lstres, abs( x( i, j ) ) )
418  130 CONTINUE
419  IF( lstres.NE.zero )
420  \$ ferr( j ) = ferr( j ) / lstres
421 *
422  140 CONTINUE
423 *
424  RETURN
425 *
426 * End of DSPRFS
427 *
428  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dspmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
DSPMV
Definition: dspmv.f:147
subroutine dlacn2(N, V, X, ISGN, EST, KASE, ISAVE)
DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: dlacn2.f:136
subroutine dsprfs(UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DSPRFS
Definition: dsprfs.f:179
subroutine dsptrs(UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
DSPTRS
Definition: dsptrs.f:115