1: *> \brief \b ZPBRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
22: * LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
30: * COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
31: * $ WORK( * ), X( LDX, * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> ZPBRFS improves the computed solution to a system of linear
41: *> equations when the coefficient matrix is Hermitian positive definite
42: *> and banded, and provides error bounds and backward error estimates
43: *> for the solution.
44: *> \endverbatim
45: *
46: * Arguments:
47: * ==========
48: *
49: *> \param[in] UPLO
50: *> \verbatim
51: *> UPLO is CHARACTER*1
52: *> = 'U': Upper triangle of A is stored;
53: *> = 'L': Lower triangle of A is stored.
54: *> \endverbatim
55: *>
56: *> \param[in] N
57: *> \verbatim
58: *> N is INTEGER
59: *> The order of the matrix A. N >= 0.
60: *> \endverbatim
61: *>
62: *> \param[in] KD
63: *> \verbatim
64: *> KD is INTEGER
65: *> The number of superdiagonals of the matrix A if UPLO = 'U',
66: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
67: *> \endverbatim
68: *>
69: *> \param[in] NRHS
70: *> \verbatim
71: *> NRHS is INTEGER
72: *> The number of right hand sides, i.e., the number of columns
73: *> of the matrices B and X. NRHS >= 0.
74: *> \endverbatim
75: *>
76: *> \param[in] AB
77: *> \verbatim
78: *> AB is COMPLEX*16 array, dimension (LDAB,N)
79: *> The upper or lower triangle of the Hermitian band matrix A,
80: *> stored in the first KD+1 rows of the array. The j-th column
81: *> of A is stored in the j-th column of the array AB as follows:
82: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
83: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
84: *> \endverbatim
85: *>
86: *> \param[in] LDAB
87: *> \verbatim
88: *> LDAB is INTEGER
89: *> The leading dimension of the array AB. LDAB >= KD+1.
90: *> \endverbatim
91: *>
92: *> \param[in] AFB
93: *> \verbatim
94: *> AFB is COMPLEX*16 array, dimension (LDAFB,N)
95: *> The triangular factor U or L from the Cholesky factorization
96: *> A = U**H*U or A = L*L**H of the band matrix A as computed by
97: *> ZPBTRF, in the same storage format as A (see AB).
98: *> \endverbatim
99: *>
100: *> \param[in] LDAFB
101: *> \verbatim
102: *> LDAFB is INTEGER
103: *> The leading dimension of the array AFB. LDAFB >= KD+1.
104: *> \endverbatim
105: *>
106: *> \param[in] B
107: *> \verbatim
108: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
109: *> The right hand side matrix B.
110: *> \endverbatim
111: *>
112: *> \param[in] LDB
113: *> \verbatim
114: *> LDB is INTEGER
115: *> The leading dimension of the array B. LDB >= max(1,N).
116: *> \endverbatim
117: *>
118: *> \param[in,out] X
119: *> \verbatim
120: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
121: *> On entry, the solution matrix X, as computed by ZPBTRS.
122: *> On exit, the improved solution matrix X.
123: *> \endverbatim
124: *>
125: *> \param[in] LDX
126: *> \verbatim
127: *> LDX is INTEGER
128: *> The leading dimension of the array X. LDX >= max(1,N).
129: *> \endverbatim
130: *>
131: *> \param[out] FERR
132: *> \verbatim
133: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
134: *> The estimated forward error bound for each solution vector
135: *> X(j) (the j-th column of the solution matrix X).
136: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
137: *> is an estimated upper bound for the magnitude of the largest
138: *> element in (X(j) - XTRUE) divided by the magnitude of the
139: *> largest element in X(j). The estimate is as reliable as
140: *> the estimate for RCOND, and is almost always a slight
141: *> overestimate of the true error.
142: *> \endverbatim
143: *>
144: *> \param[out] BERR
145: *> \verbatim
146: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
147: *> The componentwise relative backward error of each solution
148: *> vector X(j) (i.e., the smallest relative change in
149: *> any element of A or B that makes X(j) an exact solution).
150: *> \endverbatim
151: *>
152: *> \param[out] WORK
153: *> \verbatim
154: *> WORK is COMPLEX*16 array, dimension (2*N)
155: *> \endverbatim
156: *>
157: *> \param[out] RWORK
158: *> \verbatim
159: *> RWORK is DOUBLE PRECISION array, dimension (N)
160: *> \endverbatim
161: *>
162: *> \param[out] INFO
163: *> \verbatim
164: *> INFO is INTEGER
165: *> = 0: successful exit
166: *> < 0: if INFO = -i, the i-th argument had an illegal value
167: *> \endverbatim
168: *
169: *> \par Internal Parameters:
170: * =========================
171: *>
172: *> \verbatim
173: *> ITMAX is the maximum number of steps of iterative refinement.
174: *> \endverbatim
175: *
176: * Authors:
177: * ========
178: *
179: *> \author Univ. of Tennessee
180: *> \author Univ. of California Berkeley
181: *> \author Univ. of Colorado Denver
182: *> \author NAG Ltd.
183: *
184: *> \ingroup complex16OTHERcomputational
185: *
186: * =====================================================================
187: SUBROUTINE ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
188: $ LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
189: *
190: * -- LAPACK computational routine --
191: * -- LAPACK is a software package provided by Univ. of Tennessee, --
192: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
193: *
194: * .. Scalar Arguments ..
195: CHARACTER UPLO
196: INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
197: * ..
198: * .. Array Arguments ..
199: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
200: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
201: $ WORK( * ), X( LDX, * )
202: * ..
203: *
204: * =====================================================================
205: *
206: * .. Parameters ..
207: INTEGER ITMAX
208: PARAMETER ( ITMAX = 5 )
209: DOUBLE PRECISION ZERO
210: PARAMETER ( ZERO = 0.0D+0 )
211: COMPLEX*16 ONE
212: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
213: DOUBLE PRECISION TWO
214: PARAMETER ( TWO = 2.0D+0 )
215: DOUBLE PRECISION THREE
216: PARAMETER ( THREE = 3.0D+0 )
217: * ..
218: * .. Local Scalars ..
219: LOGICAL UPPER
220: INTEGER COUNT, I, J, K, KASE, L, NZ
221: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
222: COMPLEX*16 ZDUM
223: * ..
224: * .. Local Arrays ..
225: INTEGER ISAVE( 3 )
226: * ..
227: * .. External Subroutines ..
228: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHBMV, ZLACN2, ZPBTRS
229: * ..
230: * .. Intrinsic Functions ..
231: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
232: * ..
233: * .. External Functions ..
234: LOGICAL LSAME
235: DOUBLE PRECISION DLAMCH
236: EXTERNAL LSAME, DLAMCH
237: * ..
238: * .. Statement Functions ..
239: DOUBLE PRECISION CABS1
240: * ..
241: * .. Statement Function definitions ..
242: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
243: * ..
244: * .. Executable Statements ..
245: *
246: * Test the input parameters.
247: *
248: INFO = 0
249: UPPER = LSAME( UPLO, 'U' )
250: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
251: INFO = -1
252: ELSE IF( N.LT.0 ) THEN
253: INFO = -2
254: ELSE IF( KD.LT.0 ) THEN
255: INFO = -3
256: ELSE IF( NRHS.LT.0 ) THEN
257: INFO = -4
258: ELSE IF( LDAB.LT.KD+1 ) THEN
259: INFO = -6
260: ELSE IF( LDAFB.LT.KD+1 ) THEN
261: INFO = -8
262: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
263: INFO = -10
264: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
265: INFO = -12
266: END IF
267: IF( INFO.NE.0 ) THEN
268: CALL XERBLA( 'ZPBRFS', -INFO )
269: RETURN
270: END IF
271: *
272: * Quick return if possible
273: *
274: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
275: DO 10 J = 1, NRHS
276: FERR( J ) = ZERO
277: BERR( J ) = ZERO
278: 10 CONTINUE
279: RETURN
280: END IF
281: *
282: * NZ = maximum number of nonzero elements in each row of A, plus 1
283: *
284: NZ = MIN( N+1, 2*KD+2 )
285: EPS = DLAMCH( 'Epsilon' )
286: SAFMIN = DLAMCH( 'Safe minimum' )
287: SAFE1 = NZ*SAFMIN
288: SAFE2 = SAFE1 / EPS
289: *
290: * Do for each right hand side
291: *
292: DO 140 J = 1, NRHS
293: *
294: COUNT = 1
295: LSTRES = THREE
296: 20 CONTINUE
297: *
298: * Loop until stopping criterion is satisfied.
299: *
300: * Compute residual R = B - A * X
301: *
302: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
303: CALL ZHBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
304: $ WORK, 1 )
305: *
306: * Compute componentwise relative backward error from formula
307: *
308: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
309: *
310: * where abs(Z) is the componentwise absolute value of the matrix
311: * or vector Z. If the i-th component of the denominator is less
312: * than SAFE2, then SAFE1 is added to the i-th components of the
313: * numerator and denominator before dividing.
314: *
315: DO 30 I = 1, N
316: RWORK( I ) = CABS1( B( I, J ) )
317: 30 CONTINUE
318: *
319: * Compute abs(A)*abs(X) + abs(B).
320: *
321: IF( UPPER ) THEN
322: DO 50 K = 1, N
323: S = ZERO
324: XK = CABS1( X( K, J ) )
325: L = KD + 1 - K
326: DO 40 I = MAX( 1, K-KD ), K - 1
327: RWORK( I ) = RWORK( I ) + CABS1( AB( L+I, K ) )*XK
328: S = S + CABS1( AB( L+I, K ) )*CABS1( X( I, J ) )
329: 40 CONTINUE
330: RWORK( K ) = RWORK( K ) + ABS( DBLE( AB( KD+1, K ) ) )*
331: $ XK + S
332: 50 CONTINUE
333: ELSE
334: DO 70 K = 1, N
335: S = ZERO
336: XK = CABS1( X( K, J ) )
337: RWORK( K ) = RWORK( K ) + ABS( DBLE( AB( 1, K ) ) )*XK
338: L = 1 - K
339: DO 60 I = K + 1, MIN( N, K+KD )
340: RWORK( I ) = RWORK( I ) + CABS1( AB( L+I, K ) )*XK
341: S = S + CABS1( AB( L+I, K ) )*CABS1( X( I, J ) )
342: 60 CONTINUE
343: RWORK( K ) = RWORK( K ) + S
344: 70 CONTINUE
345: END IF
346: S = ZERO
347: DO 80 I = 1, N
348: IF( RWORK( I ).GT.SAFE2 ) THEN
349: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
350: ELSE
351: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
352: $ ( RWORK( I )+SAFE1 ) )
353: END IF
354: 80 CONTINUE
355: BERR( J ) = S
356: *
357: * Test stopping criterion. Continue iterating if
358: * 1) The residual BERR(J) is larger than machine epsilon, and
359: * 2) BERR(J) decreased by at least a factor of 2 during the
360: * last iteration, and
361: * 3) At most ITMAX iterations tried.
362: *
363: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
364: $ COUNT.LE.ITMAX ) THEN
365: *
366: * Update solution and try again.
367: *
368: CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
369: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
370: LSTRES = BERR( J )
371: COUNT = COUNT + 1
372: GO TO 20
373: END IF
374: *
375: * Bound error from formula
376: *
377: * norm(X - XTRUE) / norm(X) .le. FERR =
378: * norm( abs(inv(A))*
379: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
380: *
381: * where
382: * norm(Z) is the magnitude of the largest component of Z
383: * inv(A) is the inverse of A
384: * abs(Z) is the componentwise absolute value of the matrix or
385: * vector Z
386: * NZ is the maximum number of nonzeros in any row of A, plus 1
387: * EPS is machine epsilon
388: *
389: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
390: * is incremented by SAFE1 if the i-th component of
391: * abs(A)*abs(X) + abs(B) is less than SAFE2.
392: *
393: * Use ZLACN2 to estimate the infinity-norm of the matrix
394: * inv(A) * diag(W),
395: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
396: *
397: DO 90 I = 1, N
398: IF( RWORK( I ).GT.SAFE2 ) THEN
399: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
400: ELSE
401: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
402: $ SAFE1
403: END IF
404: 90 CONTINUE
405: *
406: KASE = 0
407: 100 CONTINUE
408: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
409: IF( KASE.NE.0 ) THEN
410: IF( KASE.EQ.1 ) THEN
411: *
412: * Multiply by diag(W)*inv(A**H).
413: *
414: CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
415: DO 110 I = 1, N
416: WORK( I ) = RWORK( I )*WORK( I )
417: 110 CONTINUE
418: ELSE IF( KASE.EQ.2 ) THEN
419: *
420: * Multiply by inv(A)*diag(W).
421: *
422: DO 120 I = 1, N
423: WORK( I ) = RWORK( I )*WORK( I )
424: 120 CONTINUE
425: CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
426: END IF
427: GO TO 100
428: END IF
429: *
430: * Normalize error.
431: *
432: LSTRES = ZERO
433: DO 130 I = 1, N
434: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
435: 130 CONTINUE
436: IF( LSTRES.NE.ZERO )
437: $ FERR( J ) = FERR( J ) / LSTRES
438: *
439: 140 CONTINUE
440: *
441: RETURN
442: *
443: * End of ZPBRFS
444: *
445: END
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