001:       SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
002:      $                   BERR, WORK, IWORK, INFO )
003: *
004: *  -- LAPACK routine (version 3.2) --
005: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
006: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
007: *     November 2006
008: *
009: *     Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
010: *
011: *     .. Scalar Arguments ..
012:       CHARACTER          UPLO
013:       INTEGER            INFO, LDB, LDX, N, NRHS
014: *     ..
015: *     .. Array Arguments ..
016:       INTEGER            IWORK( * )
017:       DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
018:      $                   FERR( * ), WORK( * ), X( LDX, * )
019: *     ..
020: *
021: *  Purpose
022: *  =======
023: *
024: *  DPPRFS improves the computed solution to a system of linear
025: *  equations when the coefficient matrix is symmetric positive definite
026: *  and packed, and provides error bounds and backward error estimates
027: *  for the solution.
028: *
029: *  Arguments
030: *  =========
031: *
032: *  UPLO    (input) CHARACTER*1
033: *          = 'U':  Upper triangle of A is stored;
034: *          = 'L':  Lower triangle of A is stored.
035: *
036: *  N       (input) INTEGER
037: *          The order of the matrix A.  N >= 0.
038: *
039: *  NRHS    (input) INTEGER
040: *          The number of right hand sides, i.e., the number of columns
041: *          of the matrices B and X.  NRHS >= 0.
042: *
043: *  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
044: *          The upper or lower triangle of the symmetric matrix A, packed
045: *          columnwise in a linear array.  The j-th column of A is stored
046: *          in the array AP as follows:
047: *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
048: *          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
049: *
050: *  AFP     (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
051: *          The triangular factor U or L from the Cholesky factorization
052: *          A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF,
053: *          packed columnwise in a linear array in the same format as A
054: *          (see AP).
055: *
056: *  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
057: *          The right hand side matrix B.
058: *
059: *  LDB     (input) INTEGER
060: *          The leading dimension of the array B.  LDB >= max(1,N).
061: *
062: *  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
063: *          On entry, the solution matrix X, as computed by DPPTRS.
064: *          On exit, the improved solution matrix X.
065: *
066: *  LDX     (input) INTEGER
067: *          The leading dimension of the array X.  LDX >= max(1,N).
068: *
069: *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
070: *          The estimated forward error bound for each solution vector
071: *          X(j) (the j-th column of the solution matrix X).
072: *          If XTRUE is the true solution corresponding to X(j), FERR(j)
073: *          is an estimated upper bound for the magnitude of the largest
074: *          element in (X(j) - XTRUE) divided by the magnitude of the
075: *          largest element in X(j).  The estimate is as reliable as
076: *          the estimate for RCOND, and is almost always a slight
077: *          overestimate of the true error.
078: *
079: *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
080: *          The componentwise relative backward error of each solution
081: *          vector X(j) (i.e., the smallest relative change in
082: *          any element of A or B that makes X(j) an exact solution).
083: *
084: *  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)
085: *
086: *  IWORK   (workspace) INTEGER array, dimension (N)
087: *
088: *  INFO    (output) INTEGER
089: *          = 0:  successful exit
090: *          < 0:  if INFO = -i, the i-th argument had an illegal value
091: *
092: *  Internal Parameters
093: *  ===================
094: *
095: *  ITMAX is the maximum number of steps of iterative refinement.
096: *
097: *  =====================================================================
098: *
099: *     .. Parameters ..
100:       INTEGER            ITMAX
101:       PARAMETER          ( ITMAX = 5 )
102:       DOUBLE PRECISION   ZERO
103:       PARAMETER          ( ZERO = 0.0D+0 )
104:       DOUBLE PRECISION   ONE
105:       PARAMETER          ( ONE = 1.0D+0 )
106:       DOUBLE PRECISION   TWO
107:       PARAMETER          ( TWO = 2.0D+0 )
108:       DOUBLE PRECISION   THREE
109:       PARAMETER          ( THREE = 3.0D+0 )
110: *     ..
111: *     .. Local Scalars ..
112:       LOGICAL            UPPER
113:       INTEGER            COUNT, I, IK, J, K, KASE, KK, NZ
114:       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
115: *     ..
116: *     .. Local Arrays ..
117:       INTEGER            ISAVE( 3 )
118: *     ..
119: *     .. External Subroutines ..
120:       EXTERNAL           DAXPY, DCOPY, DLACN2, DPPTRS, DSPMV, XERBLA
121: *     ..
122: *     .. Intrinsic Functions ..
123:       INTRINSIC          ABS, MAX
124: *     ..
125: *     .. External Functions ..
126:       LOGICAL            LSAME
127:       DOUBLE PRECISION   DLAMCH
128:       EXTERNAL           LSAME, DLAMCH
129: *     ..
130: *     .. Executable Statements ..
131: *
132: *     Test the input parameters.
133: *
134:       INFO = 0
135:       UPPER = LSAME( UPLO, 'U' )
136:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
137:          INFO = -1
138:       ELSE IF( N.LT.0 ) THEN
139:          INFO = -2
140:       ELSE IF( NRHS.LT.0 ) THEN
141:          INFO = -3
142:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
143:          INFO = -7
144:       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
145:          INFO = -9
146:       END IF
147:       IF( INFO.NE.0 ) THEN
148:          CALL XERBLA( 'DPPRFS', -INFO )
149:          RETURN
150:       END IF
151: *
152: *     Quick return if possible
153: *
154:       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
155:          DO 10 J = 1, NRHS
156:             FERR( J ) = ZERO
157:             BERR( J ) = ZERO
158:    10    CONTINUE
159:          RETURN
160:       END IF
161: *
162: *     NZ = maximum number of nonzero elements in each row of A, plus 1
163: *
164:       NZ = N + 1
165:       EPS = DLAMCH( 'Epsilon' )
166:       SAFMIN = DLAMCH( 'Safe minimum' )
167:       SAFE1 = NZ*SAFMIN
168:       SAFE2 = SAFE1 / EPS
169: *
170: *     Do for each right hand side
171: *
172:       DO 140 J = 1, NRHS
173: *
174:          COUNT = 1
175:          LSTRES = THREE
176:    20    CONTINUE
177: *
178: *        Loop until stopping criterion is satisfied.
179: *
180: *        Compute residual R = B - A * X
181: *
182:          CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
183:          CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
184:      $               1 )
185: *
186: *        Compute componentwise relative backward error from formula
187: *
188: *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
189: *
190: *        where abs(Z) is the componentwise absolute value of the matrix
191: *        or vector Z.  If the i-th component of the denominator is less
192: *        than SAFE2, then SAFE1 is added to the i-th components of the
193: *        numerator and denominator before dividing.
194: *
195:          DO 30 I = 1, N
196:             WORK( I ) = ABS( B( I, J ) )
197:    30    CONTINUE
198: *
199: *        Compute abs(A)*abs(X) + abs(B).
200: *
201:          KK = 1
202:          IF( UPPER ) THEN
203:             DO 50 K = 1, N
204:                S = ZERO
205:                XK = ABS( X( K, J ) )
206:                IK = KK
207:                DO 40 I = 1, K - 1
208:                   WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
209:                   S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
210:                   IK = IK + 1
211:    40          CONTINUE
212:                WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
213:                KK = KK + K
214:    50       CONTINUE
215:          ELSE
216:             DO 70 K = 1, N
217:                S = ZERO
218:                XK = ABS( X( K, J ) )
219:                WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
220:                IK = KK + 1
221:                DO 60 I = K + 1, N
222:                   WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
223:                   S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
224:                   IK = IK + 1
225:    60          CONTINUE
226:                WORK( K ) = WORK( K ) + S
227:                KK = KK + ( N-K+1 )
228:    70       CONTINUE
229:          END IF
230:          S = ZERO
231:          DO 80 I = 1, N
232:             IF( WORK( I ).GT.SAFE2 ) THEN
233:                S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
234:             ELSE
235:                S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
236:      $             ( WORK( I )+SAFE1 ) )
237:             END IF
238:    80    CONTINUE
239:          BERR( J ) = S
240: *
241: *        Test stopping criterion. Continue iterating if
242: *           1) The residual BERR(J) is larger than machine epsilon, and
243: *           2) BERR(J) decreased by at least a factor of 2 during the
244: *              last iteration, and
245: *           3) At most ITMAX iterations tried.
246: *
247:          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
248:      $       COUNT.LE.ITMAX ) THEN
249: *
250: *           Update solution and try again.
251: *
252:             CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
253:             CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
254:             LSTRES = BERR( J )
255:             COUNT = COUNT + 1
256:             GO TO 20
257:          END IF
258: *
259: *        Bound error from formula
260: *
261: *        norm(X - XTRUE) / norm(X) .le. FERR =
262: *        norm( abs(inv(A))*
263: *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
264: *
265: *        where
266: *          norm(Z) is the magnitude of the largest component of Z
267: *          inv(A) is the inverse of A
268: *          abs(Z) is the componentwise absolute value of the matrix or
269: *             vector Z
270: *          NZ is the maximum number of nonzeros in any row of A, plus 1
271: *          EPS is machine epsilon
272: *
273: *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
274: *        is incremented by SAFE1 if the i-th component of
275: *        abs(A)*abs(X) + abs(B) is less than SAFE2.
276: *
277: *        Use DLACN2 to estimate the infinity-norm of the matrix
278: *           inv(A) * diag(W),
279: *        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
280: *
281:          DO 90 I = 1, N
282:             IF( WORK( I ).GT.SAFE2 ) THEN
283:                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
284:             ELSE
285:                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
286:             END IF
287:    90    CONTINUE
288: *
289:          KASE = 0
290:   100    CONTINUE
291:          CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
292:      $                KASE, ISAVE )
293:          IF( KASE.NE.0 ) THEN
294:             IF( KASE.EQ.1 ) THEN
295: *
296: *              Multiply by diag(W)*inv(A').
297: *
298:                CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
299:                DO 110 I = 1, N
300:                   WORK( N+I ) = WORK( I )*WORK( N+I )
301:   110          CONTINUE
302:             ELSE IF( KASE.EQ.2 ) THEN
303: *
304: *              Multiply by inv(A)*diag(W).
305: *
306:                DO 120 I = 1, N
307:                   WORK( N+I ) = WORK( I )*WORK( N+I )
308:   120          CONTINUE
309:                CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
310:             END IF
311:             GO TO 100
312:          END IF
313: *
314: *        Normalize error.
315: *
316:          LSTRES = ZERO
317:          DO 130 I = 1, N
318:             LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
319:   130    CONTINUE
320:          IF( LSTRES.NE.ZERO )
321:      $      FERR( J ) = FERR( J ) / LSTRES
322: *
323:   140 CONTINUE
324: *
325:       RETURN
326: *
327: *     End of DPPRFS
328: *
329:       END
330: