1*> \brief \b ZSPRFS
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE ZSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, 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*       INTEGER            IPIV( * )
30*       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
31*       COMPLEX*16         AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
32*      $                   X( LDX, * )
33*       ..
34*
35*
36*> \par Purpose:
37*  =============
38*>
39*> \verbatim
40*>
41*> ZSPRFS improves the computed solution to a system of linear
42*> equations when the coefficient matrix is symmetric indefinite
43*> and packed, and provides error bounds and backward error estimates
44*> for the solution.
45*> \endverbatim
46*
47*  Arguments:
48*  ==========
49*
50*> \param[in] UPLO
51*> \verbatim
52*>          UPLO is CHARACTER*1
53*>          = 'U':  Upper triangle of A is stored;
54*>          = 'L':  Lower triangle of A is stored.
55*> \endverbatim
56*>
57*> \param[in] N
58*> \verbatim
59*>          N is INTEGER
60*>          The order of the matrix A.  N >= 0.
61*> \endverbatim
62*>
63*> \param[in] NRHS
64*> \verbatim
65*>          NRHS is INTEGER
66*>          The number of right hand sides, i.e., the number of columns
67*>          of the matrices B and X.  NRHS >= 0.
68*> \endverbatim
69*>
70*> \param[in] AP
71*> \verbatim
72*>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
73*>          The upper or lower triangle of the symmetric matrix A, packed
74*>          columnwise in a linear array.  The j-th column of A is stored
75*>          in the array AP as follows:
76*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
77*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
78*> \endverbatim
79*>
80*> \param[in] AFP
81*> \verbatim
82*>          AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
83*>          The factored form of the matrix A.  AFP contains the block
84*>          diagonal matrix D and the multipliers used to obtain the
85*>          factor U or L from the factorization A = U*D*U**T or
86*>          A = L*D*L**T as computed by ZSPTRF, stored as a packed
87*>          triangular matrix.
88*> \endverbatim
89*>
90*> \param[in] IPIV
91*> \verbatim
92*>          IPIV is INTEGER array, dimension (N)
93*>          Details of the interchanges and the block structure of D
94*>          as determined by ZSPTRF.
95*> \endverbatim
96*>
97*> \param[in] B
98*> \verbatim
99*>          B is COMPLEX*16 array, dimension (LDB,NRHS)
100*>          The right hand side matrix B.
101*> \endverbatim
102*>
103*> \param[in] LDB
104*> \verbatim
105*>          LDB is INTEGER
106*>          The leading dimension of the array B.  LDB >= max(1,N).
107*> \endverbatim
108*>
109*> \param[in,out] X
110*> \verbatim
111*>          X is COMPLEX*16 array, dimension (LDX,NRHS)
112*>          On entry, the solution matrix X, as computed by ZSPTRS.
113*>          On exit, the improved solution matrix X.
114*> \endverbatim
115*>
116*> \param[in] LDX
117*> \verbatim
118*>          LDX is INTEGER
119*>          The leading dimension of the array X.  LDX >= max(1,N).
120*> \endverbatim
121*>
122*> \param[out] FERR
123*> \verbatim
124*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
125*>          The estimated forward error bound for each solution vector
126*>          X(j) (the j-th column of the solution matrix X).
127*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
128*>          is an estimated upper bound for the magnitude of the largest
129*>          element in (X(j) - XTRUE) divided by the magnitude of the
130*>          largest element in X(j).  The estimate is as reliable as
131*>          the estimate for RCOND, and is almost always a slight
132*>          overestimate of the true error.
133*> \endverbatim
134*>
135*> \param[out] BERR
136*> \verbatim
137*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
138*>          The componentwise relative backward error of each solution
139*>          vector X(j) (i.e., the smallest relative change in
140*>          any element of A or B that makes X(j) an exact solution).
141*> \endverbatim
142*>
143*> \param[out] WORK
144*> \verbatim
145*>          WORK is COMPLEX*16 array, dimension (2*N)
146*> \endverbatim
147*>
148*> \param[out] RWORK
149*> \verbatim
150*>          RWORK is DOUBLE PRECISION array, dimension (N)
151*> \endverbatim
152*>
153*> \param[out] INFO
154*> \verbatim
155*>          INFO is INTEGER
156*>          = 0:  successful exit
157*>          < 0:  if INFO = -i, the i-th argument had an illegal value
158*> \endverbatim
159*
160*> \par Internal Parameters:
161*  =========================
162*>
163*> \verbatim
164*>  ITMAX is the maximum number of steps of iterative refinement.
165*> \endverbatim
166*
167*  Authors:
168*  ========
169*
170*> \author Univ. of Tennessee
171*> \author Univ. of California Berkeley
172*> \author Univ. of Colorado Denver
173*> \author NAG Ltd.
174*
175*> \date November 2011
176*
177*> \ingroup complex16OTHERcomputational
178*
179*  =====================================================================
180      SUBROUTINE ZSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
181     $                   FERR, BERR, WORK, RWORK, INFO )
182*
183*  -- LAPACK computational routine (version 3.4.0) --
184*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
185*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
186*     November 2011
187*
188*     .. Scalar Arguments ..
189      CHARACTER          UPLO
190      INTEGER            INFO, LDB, LDX, N, NRHS
191*     ..
192*     .. Array Arguments ..
193      INTEGER            IPIV( * )
194      DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
195      COMPLEX*16         AFP( * ), AP( * ), B( LDB, * ), 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      COMPLEX*16         ONE
207      PARAMETER          ( ONE = ( 1.0D+0, 0.0D+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, IK, J, K, KASE, KK, NZ
216      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
217      COMPLEX*16         ZDUM
218*     ..
219*     .. Local Arrays ..
220      INTEGER            ISAVE( 3 )
221*     ..
222*     .. External Subroutines ..
223      EXTERNAL           XERBLA, ZAXPY, ZCOPY, ZLACN2, ZSPMV, ZSPTRS
224*     ..
225*     .. Intrinsic Functions ..
226      INTRINSIC          ABS, DBLE, DIMAG, MAX
227*     ..
228*     .. External Functions ..
229      LOGICAL            LSAME
230      DOUBLE PRECISION   DLAMCH
231      EXTERNAL           LSAME, DLAMCH
232*     ..
233*     .. Statement Functions ..
234      DOUBLE PRECISION   CABS1
235*     ..
236*     .. Statement Function definitions ..
237      CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
238*     ..
239*     .. Executable Statements ..
240*
241*     Test the input parameters.
242*
243      INFO = 0
244      UPPER = LSAME( UPLO, 'U' )
245      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
246         INFO = -1
247      ELSE IF( N.LT.0 ) THEN
248         INFO = -2
249      ELSE IF( NRHS.LT.0 ) THEN
250         INFO = -3
251      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
252         INFO = -8
253      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
254         INFO = -10
255      END IF
256      IF( INFO.NE.0 ) THEN
257         CALL XERBLA( 'ZSPRFS', -INFO )
258         RETURN
259      END IF
260*
261*     Quick return if possible
262*
263      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
264         DO 10 J = 1, NRHS
265            FERR( J ) = ZERO
266            BERR( J ) = ZERO
267   10    CONTINUE
268         RETURN
269      END IF
270*
271*     NZ = maximum number of nonzero elements in each row of A, plus 1
272*
273      NZ = N + 1
274      EPS = DLAMCH( 'Epsilon' )
275      SAFMIN = DLAMCH( 'Safe minimum' )
276      SAFE1 = NZ*SAFMIN
277      SAFE2 = SAFE1 / EPS
278*
279*     Do for each right hand side
280*
281      DO 140 J = 1, NRHS
282*
283         COUNT = 1
284         LSTRES = THREE
285   20    CONTINUE
286*
287*        Loop until stopping criterion is satisfied.
288*
289*        Compute residual R = B - A * X
290*
291         CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
292         CALL ZSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK, 1 )
293*
294*        Compute componentwise relative backward error from formula
295*
296*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
297*
298*        where abs(Z) is the componentwise absolute value of the matrix
299*        or vector Z.  If the i-th component of the denominator is less
300*        than SAFE2, then SAFE1 is added to the i-th components of the
301*        numerator and denominator before dividing.
302*
303         DO 30 I = 1, N
304            RWORK( I ) = CABS1( B( I, J ) )
305   30    CONTINUE
306*
307*        Compute abs(A)*abs(X) + abs(B).
308*
309         KK = 1
310         IF( UPPER ) THEN
311            DO 50 K = 1, N
312               S = ZERO
313               XK = CABS1( X( K, J ) )
314               IK = KK
315               DO 40 I = 1, K - 1
316                  RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
317                  S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
318                  IK = IK + 1
319   40          CONTINUE
320               RWORK( K ) = RWORK( K ) + CABS1( AP( KK+K-1 ) )*XK + S
321               KK = KK + K
322   50       CONTINUE
323         ELSE
324            DO 70 K = 1, N
325               S = ZERO
326               XK = CABS1( X( K, J ) )
327               RWORK( K ) = RWORK( K ) + CABS1( AP( KK ) )*XK
328               IK = KK + 1
329               DO 60 I = K + 1, N
330                  RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
331                  S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
332                  IK = IK + 1
333   60          CONTINUE
334               RWORK( K ) = RWORK( K ) + S
335               KK = KK + ( N-K+1 )
336   70       CONTINUE
337         END IF
338         S = ZERO
339         DO 80 I = 1, N
340            IF( RWORK( I ).GT.SAFE2 ) THEN
341               S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
342            ELSE
343               S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
344     $             ( RWORK( I )+SAFE1 ) )
345            END IF
346   80    CONTINUE
347         BERR( J ) = S
348*
349*        Test stopping criterion. Continue iterating if
350*           1) The residual BERR(J) is larger than machine epsilon, and
351*           2) BERR(J) decreased by at least a factor of 2 during the
352*              last iteration, and
353*           3) At most ITMAX iterations tried.
354*
355         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
356     $       COUNT.LE.ITMAX ) THEN
357*
358*           Update solution and try again.
359*
360            CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
361            CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
362            LSTRES = BERR( J )
363            COUNT = COUNT + 1
364            GO TO 20
365         END IF
366*
367*        Bound error from formula
368*
369*        norm(X - XTRUE) / norm(X) .le. FERR =
370*        norm( abs(inv(A))*
371*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
372*
373*        where
374*          norm(Z) is the magnitude of the largest component of Z
375*          inv(A) is the inverse of A
376*          abs(Z) is the componentwise absolute value of the matrix or
377*             vector Z
378*          NZ is the maximum number of nonzeros in any row of A, plus 1
379*          EPS is machine epsilon
380*
381*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
382*        is incremented by SAFE1 if the i-th component of
383*        abs(A)*abs(X) + abs(B) is less than SAFE2.
384*
385*        Use ZLACN2 to estimate the infinity-norm of the matrix
386*           inv(A) * diag(W),
387*        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
388*
389         DO 90 I = 1, N
390            IF( RWORK( I ).GT.SAFE2 ) THEN
391               RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
392            ELSE
393               RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
394     $                      SAFE1
395            END IF
396   90    CONTINUE
397*
398         KASE = 0
399  100    CONTINUE
400         CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
401         IF( KASE.NE.0 ) THEN
402            IF( KASE.EQ.1 ) THEN
403*
404*              Multiply by diag(W)*inv(A**T).
405*
406               CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
407               DO 110 I = 1, N
408                  WORK( I ) = RWORK( I )*WORK( I )
409  110          CONTINUE
410            ELSE IF( KASE.EQ.2 ) THEN
411*
412*              Multiply by inv(A)*diag(W).
413*
414               DO 120 I = 1, N
415                  WORK( I ) = RWORK( I )*WORK( I )
416  120          CONTINUE
417               CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
418            END IF
419            GO TO 100
420         END IF
421*
422*        Normalize error.
423*
424         LSTRES = ZERO
425         DO 130 I = 1, N
426            LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
427  130    CONTINUE
428         IF( LSTRES.NE.ZERO )
429     $      FERR( J ) = FERR( J ) / LSTRES
430*
431  140 CONTINUE
432*
433      RETURN
434*
435*     End of ZSPRFS
436*
437      END
438