001:       SUBROUTINE ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
002:      $                   LDB, X, LDX, FERR, BERR, WORK, RWORK, 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 ZLACN2 in place of ZLACON, 10 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:       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
017:       COMPLEX*16         AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
018:      $                   WORK( * ), X( LDX, * )
019: *     ..
020: *
021: *  Purpose
022: *  =======
023: *
024: *  ZPBRFS improves the computed solution to a system of linear
025: *  equations when the coefficient matrix is Hermitian 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 Hermitian 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) COMPLEX*16 array, dimension (LDAFB,N)
058: *          The triangular factor U or L from the Cholesky factorization
059: *          A = U**H*U or A = L*L**H of the band matrix A as computed by
060: *          ZPBTRF, 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) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
072: *          On entry, the solution matrix X, as computed by ZPBTRS.
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) COMPLEX*16 array, dimension (2*N)
094: *
095: *  RWORK   (workspace) DOUBLE PRECISION 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:       COMPLEX*16         ONE
114:       PARAMETER          ( ONE = ( 1.0D+0, 0.0D+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:       COMPLEX*16         ZDUM
125: *     ..
126: *     .. Local Arrays ..
127:       INTEGER            ISAVE( 3 )
128: *     ..
129: *     .. External Subroutines ..
130:       EXTERNAL           XERBLA, ZAXPY, ZCOPY, ZHBMV, ZLACN2, ZPBTRS
131: *     ..
132: *     .. Intrinsic Functions ..
133:       INTRINSIC          ABS, DBLE, DIMAG, MAX, MIN
134: *     ..
135: *     .. External Functions ..
136:       LOGICAL            LSAME
137:       DOUBLE PRECISION   DLAMCH
138:       EXTERNAL           LSAME, DLAMCH
139: *     ..
140: *     .. Statement Functions ..
141:       DOUBLE PRECISION   CABS1
142: *     ..
143: *     .. Statement Function definitions ..
144:       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
145: *     ..
146: *     .. Executable Statements ..
147: *
148: *     Test the input parameters.
149: *
150:       INFO = 0
151:       UPPER = LSAME( UPLO, 'U' )
152:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
153:          INFO = -1
154:       ELSE IF( N.LT.0 ) THEN
155:          INFO = -2
156:       ELSE IF( KD.LT.0 ) THEN
157:          INFO = -3
158:       ELSE IF( NRHS.LT.0 ) THEN
159:          INFO = -4
160:       ELSE IF( LDAB.LT.KD+1 ) THEN
161:          INFO = -6
162:       ELSE IF( LDAFB.LT.KD+1 ) THEN
163:          INFO = -8
164:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
165:          INFO = -10
166:       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
167:          INFO = -12
168:       END IF
169:       IF( INFO.NE.0 ) THEN
170:          CALL XERBLA( 'ZPBRFS', -INFO )
171:          RETURN
172:       END IF
173: *
174: *     Quick return if possible
175: *
176:       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
177:          DO 10 J = 1, NRHS
178:             FERR( J ) = ZERO
179:             BERR( J ) = ZERO
180:    10    CONTINUE
181:          RETURN
182:       END IF
183: *
184: *     NZ = maximum number of nonzero elements in each row of A, plus 1
185: *
186:       NZ = MIN( N+1, 2*KD+2 )
187:       EPS = DLAMCH( 'Epsilon' )
188:       SAFMIN = DLAMCH( 'Safe minimum' )
189:       SAFE1 = NZ*SAFMIN
190:       SAFE2 = SAFE1 / EPS
191: *
192: *     Do for each right hand side
193: *
194:       DO 140 J = 1, NRHS
195: *
196:          COUNT = 1
197:          LSTRES = THREE
198:    20    CONTINUE
199: *
200: *        Loop until stopping criterion is satisfied.
201: *
202: *        Compute residual R = B - A * X
203: *
204:          CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
205:          CALL ZHBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
206:      $               WORK, 1 )
207: *
208: *        Compute componentwise relative backward error from formula
209: *
210: *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
211: *
212: *        where abs(Z) is the componentwise absolute value of the matrix
213: *        or vector Z.  If the i-th component of the denominator is less
214: *        than SAFE2, then SAFE1 is added to the i-th components of the
215: *        numerator and denominator before dividing.
216: *
217:          DO 30 I = 1, N
218:             RWORK( I ) = CABS1( B( I, J ) )
219:    30    CONTINUE
220: *
221: *        Compute abs(A)*abs(X) + abs(B).
222: *
223:          IF( UPPER ) THEN
224:             DO 50 K = 1, N
225:                S = ZERO
226:                XK = CABS1( X( K, J ) )
227:                L = KD + 1 - K
228:                DO 40 I = MAX( 1, K-KD ), K - 1
229:                   RWORK( I ) = RWORK( I ) + CABS1( AB( L+I, K ) )*XK
230:                   S = S + CABS1( AB( L+I, K ) )*CABS1( X( I, J ) )
231:    40          CONTINUE
232:                RWORK( K ) = RWORK( K ) + ABS( DBLE( AB( KD+1, K ) ) )*
233:      $                      XK + S
234:    50       CONTINUE
235:          ELSE
236:             DO 70 K = 1, N
237:                S = ZERO
238:                XK = CABS1( X( K, J ) )
239:                RWORK( K ) = RWORK( K ) + ABS( DBLE( AB( 1, K ) ) )*XK
240:                L = 1 - K
241:                DO 60 I = K + 1, MIN( N, K+KD )
242:                   RWORK( I ) = RWORK( I ) + CABS1( AB( L+I, K ) )*XK
243:                   S = S + CABS1( AB( L+I, K ) )*CABS1( X( I, J ) )
244:    60          CONTINUE
245:                RWORK( K ) = RWORK( K ) + S
246:    70       CONTINUE
247:          END IF
248:          S = ZERO
249:          DO 80 I = 1, N
250:             IF( RWORK( I ).GT.SAFE2 ) THEN
251:                S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
252:             ELSE
253:                S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
254:      $             ( RWORK( I )+SAFE1 ) )
255:             END IF
256:    80    CONTINUE
257:          BERR( J ) = S
258: *
259: *        Test stopping criterion. Continue iterating if
260: *           1) The residual BERR(J) is larger than machine epsilon, and
261: *           2) BERR(J) decreased by at least a factor of 2 during the
262: *              last iteration, and
263: *           3) At most ITMAX iterations tried.
264: *
265:          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
266:      $       COUNT.LE.ITMAX ) THEN
267: *
268: *           Update solution and try again.
269: *
270:             CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
271:             CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
272:             LSTRES = BERR( J )
273:             COUNT = COUNT + 1
274:             GO TO 20
275:          END IF
276: *
277: *        Bound error from formula
278: *
279: *        norm(X - XTRUE) / norm(X) .le. FERR =
280: *        norm( abs(inv(A))*
281: *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
282: *
283: *        where
284: *          norm(Z) is the magnitude of the largest component of Z
285: *          inv(A) is the inverse of A
286: *          abs(Z) is the componentwise absolute value of the matrix or
287: *             vector Z
288: *          NZ is the maximum number of nonzeros in any row of A, plus 1
289: *          EPS is machine epsilon
290: *
291: *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
292: *        is incremented by SAFE1 if the i-th component of
293: *        abs(A)*abs(X) + abs(B) is less than SAFE2.
294: *
295: *        Use ZLACN2 to estimate the infinity-norm of the matrix
296: *           inv(A) * diag(W),
297: *        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
298: *
299:          DO 90 I = 1, N
300:             IF( RWORK( I ).GT.SAFE2 ) THEN
301:                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
302:             ELSE
303:                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
304:      $                      SAFE1
305:             END IF
306:    90    CONTINUE
307: *
308:          KASE = 0
309:   100    CONTINUE
310:          CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
311:          IF( KASE.NE.0 ) THEN
312:             IF( KASE.EQ.1 ) THEN
313: *
314: *              Multiply by diag(W)*inv(A').
315: *
316:                CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
317:                DO 110 I = 1, N
318:                   WORK( I ) = RWORK( I )*WORK( I )
319:   110          CONTINUE
320:             ELSE IF( KASE.EQ.2 ) THEN
321: *
322: *              Multiply by inv(A)*diag(W).
323: *
324:                DO 120 I = 1, N
325:                   WORK( I ) = RWORK( I )*WORK( I )
326:   120          CONTINUE
327:                CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
328:             END IF
329:             GO TO 100
330:          END IF
331: *
332: *        Normalize error.
333: *
334:          LSTRES = ZERO
335:          DO 130 I = 1, N
336:             LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
337:   130    CONTINUE
338:          IF( LSTRES.NE.ZERO )
339:      $      FERR( J ) = FERR( J ) / LSTRES
340: *
341:   140 CONTINUE
342: *
343:       RETURN
344: *
345: *     End of ZPBRFS
346: *
347:       END
348: