1*> \brief \b DPPRFS
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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13*> [ZIP]</a>
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
22*                          BERR, WORK, IWORK, INFO )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          UPLO
26*       INTEGER            INFO, LDB, LDX, N, NRHS
27*       ..
28*       .. Array Arguments ..
29*       INTEGER            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*> DPPRFS improves the computed solution to a system of linear
41*> equations when the coefficient matrix is symmetric positive definite
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)*(2n-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 triangular factor U or L from the Cholesky factorization
83*>          A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF,
84*>          packed columnwise in a linear array in the same format as A
85*>          (see AP).
86*> \endverbatim
87*>
88*> \param[in] B
89*> \verbatim
90*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
91*>          The right hand side matrix B.
92*> \endverbatim
93*>
94*> \param[in] LDB
95*> \verbatim
96*>          LDB is INTEGER
97*>          The leading dimension of the array B.  LDB >= max(1,N).
98*> \endverbatim
99*>
100*> \param[in,out] X
101*> \verbatim
102*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
103*>          On entry, the solution matrix X, as computed by DPPTRS.
104*>          On exit, the improved solution matrix X.
105*> \endverbatim
106*>
107*> \param[in] LDX
108*> \verbatim
109*>          LDX is INTEGER
110*>          The leading dimension of the array X.  LDX >= max(1,N).
111*> \endverbatim
112*>
113*> \param[out] FERR
114*> \verbatim
115*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
116*>          The estimated forward error bound for each solution vector
117*>          X(j) (the j-th column of the solution matrix X).
118*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
119*>          is an estimated upper bound for the magnitude of the largest
120*>          element in (X(j) - XTRUE) divided by the magnitude of the
121*>          largest element in X(j).  The estimate is as reliable as
122*>          the estimate for RCOND, and is almost always a slight
123*>          overestimate of the true error.
124*> \endverbatim
125*>
126*> \param[out] BERR
127*> \verbatim
128*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
129*>          The componentwise relative backward error of each solution
130*>          vector X(j) (i.e., the smallest relative change in
131*>          any element of A or B that makes X(j) an exact solution).
132*> \endverbatim
133*>
134*> \param[out] WORK
135*> \verbatim
136*>          WORK is DOUBLE PRECISION array, dimension (3*N)
137*> \endverbatim
138*>
139*> \param[out] IWORK
140*> \verbatim
141*>          IWORK is INTEGER array, dimension (N)
142*> \endverbatim
143*>
144*> \param[out] INFO
145*> \verbatim
146*>          INFO is INTEGER
147*>          = 0:  successful exit
148*>          < 0:  if INFO = -i, the i-th argument had an illegal value
149*> \endverbatim
150*
151*> \par Internal Parameters:
152*  =========================
153*>
154*> \verbatim
155*>  ITMAX is the maximum number of steps of iterative refinement.
156*> \endverbatim
157*
158*  Authors:
159*  ========
160*
161*> \author Univ. of Tennessee
162*> \author Univ. of California Berkeley
163*> \author Univ. of Colorado Denver
164*> \author NAG Ltd.
165*
166*> \date November 2011
167*
168*> \ingroup doubleOTHERcomputational
169*
170*  =====================================================================
171      SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
172     $                   BERR, WORK, IWORK, INFO )
173*
174*  -- LAPACK computational routine (version 3.4.0) --
175*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
176*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177*     November 2011
178*
179*     .. Scalar Arguments ..
180      CHARACTER          UPLO
181      INTEGER            INFO, LDB, LDX, N, NRHS
182*     ..
183*     .. Array Arguments ..
184      INTEGER            IWORK( * )
185      DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
186     $                   FERR( * ), WORK( * ), X( LDX, * )
187*     ..
188*
189*  =====================================================================
190*
191*     .. Parameters ..
192      INTEGER            ITMAX
193      PARAMETER          ( ITMAX = 5 )
194      DOUBLE PRECISION   ZERO
195      PARAMETER          ( ZERO = 0.0D+0 )
196      DOUBLE PRECISION   ONE
197      PARAMETER          ( ONE = 1.0D+0 )
198      DOUBLE PRECISION   TWO
199      PARAMETER          ( TWO = 2.0D+0 )
200      DOUBLE PRECISION   THREE
201      PARAMETER          ( THREE = 3.0D+0 )
202*     ..
203*     .. Local Scalars ..
204      LOGICAL            UPPER
205      INTEGER            COUNT, I, IK, J, K, KASE, KK, NZ
206      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
207*     ..
208*     .. Local Arrays ..
209      INTEGER            ISAVE( 3 )
210*     ..
211*     .. External Subroutines ..
212      EXTERNAL           DAXPY, DCOPY, DLACN2, DPPTRS, DSPMV, XERBLA
213*     ..
214*     .. Intrinsic Functions ..
215      INTRINSIC          ABS, MAX
216*     ..
217*     .. External Functions ..
218      LOGICAL            LSAME
219      DOUBLE PRECISION   DLAMCH
220      EXTERNAL           LSAME, DLAMCH
221*     ..
222*     .. Executable Statements ..
223*
224*     Test the input parameters.
225*
226      INFO = 0
227      UPPER = LSAME( UPLO, 'U' )
228      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
229         INFO = -1
230      ELSE IF( N.LT.0 ) THEN
231         INFO = -2
232      ELSE IF( NRHS.LT.0 ) THEN
233         INFO = -3
234      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
235         INFO = -7
236      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
237         INFO = -9
238      END IF
239      IF( INFO.NE.0 ) THEN
240         CALL XERBLA( 'DPPRFS', -INFO )
241         RETURN
242      END IF
243*
244*     Quick return if possible
245*
246      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
247         DO 10 J = 1, NRHS
248            FERR( J ) = ZERO
249            BERR( J ) = ZERO
250   10    CONTINUE
251         RETURN
252      END IF
253*
254*     NZ = maximum number of nonzero elements in each row of A, plus 1
255*
256      NZ = N + 1
257      EPS = DLAMCH( 'Epsilon' )
258      SAFMIN = DLAMCH( 'Safe minimum' )
259      SAFE1 = NZ*SAFMIN
260      SAFE2 = SAFE1 / EPS
261*
262*     Do for each right hand side
263*
264      DO 140 J = 1, NRHS
265*
266         COUNT = 1
267         LSTRES = THREE
268   20    CONTINUE
269*
270*        Loop until stopping criterion is satisfied.
271*
272*        Compute residual R = B - A * X
273*
274         CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
275         CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
276     $               1 )
277*
278*        Compute componentwise relative backward error from formula
279*
280*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
281*
282*        where abs(Z) is the componentwise absolute value of the matrix
283*        or vector Z.  If the i-th component of the denominator is less
284*        than SAFE2, then SAFE1 is added to the i-th components of the
285*        numerator and denominator before dividing.
286*
287         DO 30 I = 1, N
288            WORK( I ) = ABS( B( I, J ) )
289   30    CONTINUE
290*
291*        Compute abs(A)*abs(X) + abs(B).
292*
293         KK = 1
294         IF( UPPER ) THEN
295            DO 50 K = 1, N
296               S = ZERO
297               XK = ABS( X( K, J ) )
298               IK = KK
299               DO 40 I = 1, K - 1
300                  WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
301                  S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
302                  IK = IK + 1
303   40          CONTINUE
304               WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
305               KK = KK + K
306   50       CONTINUE
307         ELSE
308            DO 70 K = 1, N
309               S = ZERO
310               XK = ABS( X( K, J ) )
311               WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
312               IK = KK + 1
313               DO 60 I = K + 1, N
314                  WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
315                  S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
316                  IK = IK + 1
317   60          CONTINUE
318               WORK( K ) = WORK( K ) + S
319               KK = KK + ( N-K+1 )
320   70       CONTINUE
321         END IF
322         S = ZERO
323         DO 80 I = 1, N
324            IF( WORK( I ).GT.SAFE2 ) THEN
325               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
326            ELSE
327               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
328     $             ( WORK( I )+SAFE1 ) )
329            END IF
330   80    CONTINUE
331         BERR( J ) = S
332*
333*        Test stopping criterion. Continue iterating if
334*           1) The residual BERR(J) is larger than machine epsilon, and
335*           2) BERR(J) decreased by at least a factor of 2 during the
336*              last iteration, and
337*           3) At most ITMAX iterations tried.
338*
339         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
340     $       COUNT.LE.ITMAX ) THEN
341*
342*           Update solution and try again.
343*
344            CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
345            CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
346            LSTRES = BERR( J )
347            COUNT = COUNT + 1
348            GO TO 20
349         END IF
350*
351*        Bound error from formula
352*
353*        norm(X - XTRUE) / norm(X) .le. FERR =
354*        norm( abs(inv(A))*
355*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
356*
357*        where
358*          norm(Z) is the magnitude of the largest component of Z
359*          inv(A) is the inverse of A
360*          abs(Z) is the componentwise absolute value of the matrix or
361*             vector Z
362*          NZ is the maximum number of nonzeros in any row of A, plus 1
363*          EPS is machine epsilon
364*
365*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
366*        is incremented by SAFE1 if the i-th component of
367*        abs(A)*abs(X) + abs(B) is less than SAFE2.
368*
369*        Use DLACN2 to estimate the infinity-norm of the matrix
370*           inv(A) * diag(W),
371*        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
372*
373         DO 90 I = 1, N
374            IF( WORK( I ).GT.SAFE2 ) THEN
375               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
376            ELSE
377               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
378            END IF
379   90    CONTINUE
380*
381         KASE = 0
382  100    CONTINUE
383         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
384     $                KASE, ISAVE )
385         IF( KASE.NE.0 ) THEN
386            IF( KASE.EQ.1 ) THEN
387*
388*              Multiply by diag(W)*inv(A**T).
389*
390               CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
391               DO 110 I = 1, N
392                  WORK( N+I ) = WORK( I )*WORK( N+I )
393  110          CONTINUE
394            ELSE IF( KASE.EQ.2 ) THEN
395*
396*              Multiply by inv(A)*diag(W).
397*
398               DO 120 I = 1, N
399                  WORK( N+I ) = WORK( I )*WORK( N+I )
400  120          CONTINUE
401               CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
402            END IF
403            GO TO 100
404         END IF
405*
406*        Normalize error.
407*
408         LSTRES = ZERO
409         DO 130 I = 1, N
410            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
411  130    CONTINUE
412         IF( LSTRES.NE.ZERO )
413     $      FERR( J ) = FERR( J ) / LSTRES
414*
415  140 CONTINUE
416*
417      RETURN
418*
419*     End of DPPRFS
420*
421      END
422