LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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dsprfs.f
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1*> \brief \b DSPRFS
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 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 hprfs
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)
Definition cblat2.f:3285
subroutine daxpy(n, da, dx, incx, dy, incy)
DAXPY
Definition daxpy.f:89
subroutine dcopy(n, dx, incx, dy, incy)
DCOPY
Definition dcopy.f:82
subroutine dspmv(uplo, n, alpha, ap, x, incx, beta, y, incy)
DSPMV
Definition dspmv.f:147
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
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