001:       SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
002:      $                   LDB, X, LDX, FERR, 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, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
014: *     ..
015: *     .. Array Arguments ..
016:       INTEGER            IWORK( * )
017:       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
018:      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
019: *     ..
020: *
021: *  Purpose
022: *  =======
023: *
024: *  DPBRFS improves the computed solution to a system of linear
025: *  equations when the coefficient matrix is symmetric positive definite
026: *  and banded, 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: *  KD      (input) INTEGER
040: *          The number of superdiagonals of the matrix A if UPLO = 'U',
041: *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
042: *
043: *  NRHS    (input) INTEGER
044: *          The number of right hand sides, i.e., the number of columns
045: *          of the matrices B and X.  NRHS >= 0.
046: *
047: *  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
048: *          The upper or lower triangle of the symmetric band matrix A,
049: *          stored in the first KD+1 rows of the array.  The j-th column
050: *          of A is stored in the j-th column of the array AB as follows:
051: *          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
052: *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
053: *
054: *  LDAB    (input) INTEGER
055: *          The leading dimension of the array AB.  LDAB >= KD+1.
056: *
057: *  AFB     (input) DOUBLE PRECISION array, dimension (LDAFB,N)
058: *          The triangular factor U or L from the Cholesky factorization
059: *          A = U**T*U or A = L*L**T of the band matrix A as computed by
060: *          DPBTRF, in the same storage format as A (see AB).
061: *
062: *  LDAFB   (input) INTEGER
063: *          The leading dimension of the array AFB.  LDAFB >= KD+1.
064: *
065: *  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
066: *          The right hand side matrix B.
067: *
068: *  LDB     (input) INTEGER
069: *          The leading dimension of the array B.  LDB >= max(1,N).
070: *
071: *  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
072: *          On entry, the solution matrix X, as computed by DPBTRS.
073: *          On exit, the improved solution matrix X.
074: *
075: *  LDX     (input) INTEGER
076: *          The leading dimension of the array X.  LDX >= max(1,N).
077: *
078: *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
079: *          The estimated forward error bound for each solution vector
080: *          X(j) (the j-th column of the solution matrix X).
081: *          If XTRUE is the true solution corresponding to X(j), FERR(j)
082: *          is an estimated upper bound for the magnitude of the largest
083: *          element in (X(j) - XTRUE) divided by the magnitude of the
084: *          largest element in X(j).  The estimate is as reliable as
085: *          the estimate for RCOND, and is almost always a slight
086: *          overestimate of the true error.
087: *
088: *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
089: *          The componentwise relative backward error of each solution
090: *          vector X(j) (i.e., the smallest relative change in
091: *          any element of A or B that makes X(j) an exact solution).
092: *
093: *  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)
094: *
095: *  IWORK   (workspace) INTEGER array, dimension (N)
096: *
097: *  INFO    (output) INTEGER
098: *          = 0:  successful exit
099: *          < 0:  if INFO = -i, the i-th argument had an illegal value
100: *
101: *  Internal Parameters
102: *  ===================
103: *
104: *  ITMAX is the maximum number of steps of iterative refinement.
105: *
106: *  =====================================================================
107: *
108: *     .. Parameters ..
109:       INTEGER            ITMAX
110:       PARAMETER          ( ITMAX = 5 )
111:       DOUBLE PRECISION   ZERO
112:       PARAMETER          ( ZERO = 0.0D+0 )
113:       DOUBLE PRECISION   ONE
114:       PARAMETER          ( ONE = 1.0D+0 )
115:       DOUBLE PRECISION   TWO
116:       PARAMETER          ( TWO = 2.0D+0 )
117:       DOUBLE PRECISION   THREE
118:       PARAMETER          ( THREE = 3.0D+0 )
119: *     ..
120: *     .. Local Scalars ..
121:       LOGICAL            UPPER
122:       INTEGER            COUNT, I, J, K, KASE, L, NZ
123:       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
124: *     ..
125: *     .. Local Arrays ..
126:       INTEGER            ISAVE( 3 )
127: *     ..
128: *     .. External Subroutines ..
129:       EXTERNAL           DAXPY, DCOPY, DLACN2, DPBTRS, DSBMV, XERBLA
130: *     ..
131: *     .. Intrinsic Functions ..
132:       INTRINSIC          ABS, MAX, MIN
133: *     ..
134: *     .. External Functions ..
135:       LOGICAL            LSAME
136:       DOUBLE PRECISION   DLAMCH
137:       EXTERNAL           LSAME, DLAMCH
138: *     ..
139: *     .. Executable Statements ..
140: *
141: *     Test the input parameters.
142: *
143:       INFO = 0
144:       UPPER = LSAME( UPLO, 'U' )
145:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
146:          INFO = -1
147:       ELSE IF( N.LT.0 ) THEN
148:          INFO = -2
149:       ELSE IF( KD.LT.0 ) THEN
150:          INFO = -3
151:       ELSE IF( NRHS.LT.0 ) THEN
152:          INFO = -4
153:       ELSE IF( LDAB.LT.KD+1 ) THEN
154:          INFO = -6
155:       ELSE IF( LDAFB.LT.KD+1 ) THEN
156:          INFO = -8
157:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
158:          INFO = -10
159:       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
160:          INFO = -12
161:       END IF
162:       IF( INFO.NE.0 ) THEN
163:          CALL XERBLA( 'DPBRFS', -INFO )
164:          RETURN
165:       END IF
166: *
167: *     Quick return if possible
168: *
169:       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
170:          DO 10 J = 1, NRHS
171:             FERR( J ) = ZERO
172:             BERR( J ) = ZERO
173:    10    CONTINUE
174:          RETURN
175:       END IF
176: *
177: *     NZ = maximum number of nonzero elements in each row of A, plus 1
178: *
179:       NZ = MIN( N+1, 2*KD+2 )
180:       EPS = DLAMCH( 'Epsilon' )
181:       SAFMIN = DLAMCH( 'Safe minimum' )
182:       SAFE1 = NZ*SAFMIN
183:       SAFE2 = SAFE1 / EPS
184: *
185: *     Do for each right hand side
186: *
187:       DO 140 J = 1, NRHS
188: *
189:          COUNT = 1
190:          LSTRES = THREE
191:    20    CONTINUE
192: *
193: *        Loop until stopping criterion is satisfied.
194: *
195: *        Compute residual R = B - A * X
196: *
197:          CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
198:          CALL DSBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
199:      $               WORK( N+1 ), 1 )
200: *
201: *        Compute componentwise relative backward error from formula
202: *
203: *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
204: *
205: *        where abs(Z) is the componentwise absolute value of the matrix
206: *        or vector Z.  If the i-th component of the denominator is less
207: *        than SAFE2, then SAFE1 is added to the i-th components of the
208: *        numerator and denominator before dividing.
209: *
210:          DO 30 I = 1, N
211:             WORK( I ) = ABS( B( I, J ) )
212:    30    CONTINUE
213: *
214: *        Compute abs(A)*abs(X) + abs(B).
215: *
216:          IF( UPPER ) THEN
217:             DO 50 K = 1, N
218:                S = ZERO
219:                XK = ABS( X( K, J ) )
220:                L = KD + 1 - K
221:                DO 40 I = MAX( 1, K-KD ), K - 1
222:                   WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
223:                   S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
224:    40          CONTINUE
225:                WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S
226:    50       CONTINUE
227:          ELSE
228:             DO 70 K = 1, N
229:                S = ZERO
230:                XK = ABS( X( K, J ) )
231:                WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK
232:                L = 1 - K
233:                DO 60 I = K + 1, MIN( N, K+KD )
234:                   WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
235:                   S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
236:    60          CONTINUE
237:                WORK( K ) = WORK( K ) + S
238:    70       CONTINUE
239:          END IF
240:          S = ZERO
241:          DO 80 I = 1, N
242:             IF( WORK( I ).GT.SAFE2 ) THEN
243:                S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
244:             ELSE
245:                S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
246:      $             ( WORK( I )+SAFE1 ) )
247:             END IF
248:    80    CONTINUE
249:          BERR( J ) = S
250: *
251: *        Test stopping criterion. Continue iterating if
252: *           1) The residual BERR(J) is larger than machine epsilon, and
253: *           2) BERR(J) decreased by at least a factor of 2 during the
254: *              last iteration, and
255: *           3) At most ITMAX iterations tried.
256: *
257:          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
258:      $       COUNT.LE.ITMAX ) THEN
259: *
260: *           Update solution and try again.
261: *
262:             CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
263:      $                   INFO )
264:             CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
265:             LSTRES = BERR( J )
266:             COUNT = COUNT + 1
267:             GO TO 20
268:          END IF
269: *
270: *        Bound error from formula
271: *
272: *        norm(X - XTRUE) / norm(X) .le. FERR =
273: *        norm( abs(inv(A))*
274: *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
275: *
276: *        where
277: *          norm(Z) is the magnitude of the largest component of Z
278: *          inv(A) is the inverse of A
279: *          abs(Z) is the componentwise absolute value of the matrix or
280: *             vector Z
281: *          NZ is the maximum number of nonzeros in any row of A, plus 1
282: *          EPS is machine epsilon
283: *
284: *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
285: *        is incremented by SAFE1 if the i-th component of
286: *        abs(A)*abs(X) + abs(B) is less than SAFE2.
287: *
288: *        Use DLACN2 to estimate the infinity-norm of the matrix
289: *           inv(A) * diag(W),
290: *        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
291: *
292:          DO 90 I = 1, N
293:             IF( WORK( I ).GT.SAFE2 ) THEN
294:                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
295:             ELSE
296:                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
297:             END IF
298:    90    CONTINUE
299: *
300:          KASE = 0
301:   100    CONTINUE
302:          CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
303:      $                KASE, ISAVE )
304:          IF( KASE.NE.0 ) THEN
305:             IF( KASE.EQ.1 ) THEN
306: *
307: *              Multiply by diag(W)*inv(A').
308: *
309:                CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
310:      $                      INFO )
311:                DO 110 I = 1, N
312:                   WORK( N+I ) = WORK( N+I )*WORK( I )
313:   110          CONTINUE
314:             ELSE IF( KASE.EQ.2 ) THEN
315: *
316: *              Multiply by inv(A)*diag(W).
317: *
318:                DO 120 I = 1, N
319:                   WORK( N+I ) = WORK( N+I )*WORK( I )
320:   120          CONTINUE
321:                CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
322:      $                      INFO )
323:             END IF
324:             GO TO 100
325:          END IF
326: *
327: *        Normalize error.
328: *
329:          LSTRES = ZERO
330:          DO 130 I = 1, N
331:             LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
332:   130    CONTINUE
333:          IF( LSTRES.NE.ZERO )
334:      $      FERR( J ) = FERR( J ) / LSTRES
335: *
336:   140 CONTINUE
337: *
338:       RETURN
339: *
340: *     End of DPBRFS
341: *
342:       END
343: