1: SUBROUTINE ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
2: $ LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
3: *
4: * -- LAPACK routine (version 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
8: *
9: * Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
10: *
11: * .. Scalar Arguments ..
12: CHARACTER UPLO
13: INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
14: * ..
15: * .. Array Arguments ..
16: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
18: $ WORK( * ), X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * ZPBRFS improves the computed solution to a system of linear
25: * equations when the coefficient matrix is Hermitian positive definite
26: * and banded, and provides error bounds and backward error estimates
27: * for the solution.
28: *
29: * Arguments
30: * =========
31: *
32: * UPLO (input) CHARACTER*1
33: * = 'U': Upper triangle of A is stored;
34: * = 'L': Lower triangle of A is stored.
35: *
36: * N (input) INTEGER
37: * The order of the matrix A. N >= 0.
38: *
39: * KD (input) INTEGER
40: * The number of superdiagonals of the matrix A if UPLO = 'U',
41: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
42: *
43: * NRHS (input) INTEGER
44: * The number of right hand sides, i.e., the number of columns
45: * of the matrices B and X. NRHS >= 0.
46: *
47: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
48: * The upper or lower triangle of the Hermitian band matrix A,
49: * stored in the first KD+1 rows of the array. The j-th column
50: * of A is stored in the j-th column of the array AB as follows:
51: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
52: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
53: *
54: * LDAB (input) INTEGER
55: * The leading dimension of the array AB. LDAB >= KD+1.
56: *
57: * AFB (input) COMPLEX*16 array, dimension (LDAFB,N)
58: * The triangular factor U or L from the Cholesky factorization
59: * A = U**H*U or A = L*L**H of the band matrix A as computed by
60: * ZPBTRF, in the same storage format as A (see AB).
61: *
62: * LDAFB (input) INTEGER
63: * The leading dimension of the array AFB. LDAFB >= KD+1.
64: *
65: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
66: * The right hand side matrix B.
67: *
68: * LDB (input) INTEGER
69: * The leading dimension of the array B. LDB >= max(1,N).
70: *
71: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
72: * On entry, the solution matrix X, as computed by ZPBTRS.
73: * On exit, the improved solution matrix X.
74: *
75: * LDX (input) INTEGER
76: * The leading dimension of the array X. LDX >= max(1,N).
77: *
78: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
79: * The estimated forward error bound for each solution vector
80: * X(j) (the j-th column of the solution matrix X).
81: * If XTRUE is the true solution corresponding to X(j), FERR(j)
82: * is an estimated upper bound for the magnitude of the largest
83: * element in (X(j) - XTRUE) divided by the magnitude of the
84: * largest element in X(j). The estimate is as reliable as
85: * the estimate for RCOND, and is almost always a slight
86: * overestimate of the true error.
87: *
88: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
89: * The componentwise relative backward error of each solution
90: * vector X(j) (i.e., the smallest relative change in
91: * any element of A or B that makes X(j) an exact solution).
92: *
93: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
94: *
95: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
96: *
97: * INFO (output) INTEGER
98: * = 0: successful exit
99: * < 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**H).
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
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