Annotation of rpl/lapack/lapack/zhprfs.f, revision 1.10
1.9 bertrand 1: *> \brief \b ZHPRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZHPRFS + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhprfs.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhprfs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
22: * FERR, BERR, WORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IPIV( * )
30: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
31: * COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
32: * $ X( LDX, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> ZHPRFS improves the computed solution to a system of linear
42: *> equations when the coefficient matrix is Hermitian indefinite
43: *> and packed, and provides error bounds and backward error estimates
44: *> for the solution.
45: *> \endverbatim
46: *
47: * Arguments:
48: * ==========
49: *
50: *> \param[in] UPLO
51: *> \verbatim
52: *> UPLO is CHARACTER*1
53: *> = 'U': Upper triangle of A is stored;
54: *> = 'L': Lower triangle of A is stored.
55: *> \endverbatim
56: *>
57: *> \param[in] N
58: *> \verbatim
59: *> N is INTEGER
60: *> The order of the matrix A. N >= 0.
61: *> \endverbatim
62: *>
63: *> \param[in] NRHS
64: *> \verbatim
65: *> NRHS is INTEGER
66: *> The number of right hand sides, i.e., the number of columns
67: *> of the matrices B and X. NRHS >= 0.
68: *> \endverbatim
69: *>
70: *> \param[in] AP
71: *> \verbatim
72: *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
73: *> The upper or lower triangle of the Hermitian matrix A, packed
74: *> columnwise in a linear array. The j-th column of A is stored
75: *> in the array AP as follows:
76: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
77: *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
78: *> \endverbatim
79: *>
80: *> \param[in] AFP
81: *> \verbatim
82: *> AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
83: *> The factored form of the matrix A. AFP contains the block
84: *> diagonal matrix D and the multipliers used to obtain the
85: *> factor U or L from the factorization A = U*D*U**H or
86: *> A = L*D*L**H as computed by ZHPTRF, stored as a packed
87: *> triangular matrix.
88: *> \endverbatim
89: *>
90: *> \param[in] IPIV
91: *> \verbatim
92: *> IPIV is INTEGER array, dimension (N)
93: *> Details of the interchanges and the block structure of D
94: *> as determined by ZHPTRF.
95: *> \endverbatim
96: *>
97: *> \param[in] B
98: *> \verbatim
99: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
100: *> The right hand side matrix B.
101: *> \endverbatim
102: *>
103: *> \param[in] LDB
104: *> \verbatim
105: *> LDB is INTEGER
106: *> The leading dimension of the array B. LDB >= max(1,N).
107: *> \endverbatim
108: *>
109: *> \param[in,out] X
110: *> \verbatim
111: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
112: *> On entry, the solution matrix X, as computed by ZHPTRS.
113: *> On exit, the improved solution matrix X.
114: *> \endverbatim
115: *>
116: *> \param[in] LDX
117: *> \verbatim
118: *> LDX is INTEGER
119: *> The leading dimension of the array X. LDX >= max(1,N).
120: *> \endverbatim
121: *>
122: *> \param[out] FERR
123: *> \verbatim
124: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
125: *> The estimated forward error bound for each solution vector
126: *> X(j) (the j-th column of the solution matrix X).
127: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
128: *> is an estimated upper bound for the magnitude of the largest
129: *> element in (X(j) - XTRUE) divided by the magnitude of the
130: *> largest element in X(j). The estimate is as reliable as
131: *> the estimate for RCOND, and is almost always a slight
132: *> overestimate of the true error.
133: *> \endverbatim
134: *>
135: *> \param[out] BERR
136: *> \verbatim
137: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
138: *> The componentwise relative backward error of each solution
139: *> vector X(j) (i.e., the smallest relative change in
140: *> any element of A or B that makes X(j) an exact solution).
141: *> \endverbatim
142: *>
143: *> \param[out] WORK
144: *> \verbatim
145: *> WORK is COMPLEX*16 array, dimension (2*N)
146: *> \endverbatim
147: *>
148: *> \param[out] RWORK
149: *> \verbatim
150: *> RWORK is DOUBLE PRECISION array, dimension (N)
151: *> \endverbatim
152: *>
153: *> \param[out] INFO
154: *> \verbatim
155: *> INFO is INTEGER
156: *> = 0: successful exit
157: *> < 0: if INFO = -i, the i-th argument had an illegal value
158: *> \endverbatim
159: *
160: *> \par Internal Parameters:
161: * =========================
162: *>
163: *> \verbatim
164: *> ITMAX is the maximum number of steps of iterative refinement.
165: *> \endverbatim
166: *
167: * Authors:
168: * ========
169: *
170: *> \author Univ. of Tennessee
171: *> \author Univ. of California Berkeley
172: *> \author Univ. of Colorado Denver
173: *> \author NAG Ltd.
174: *
175: *> \date November 2011
176: *
177: *> \ingroup complex16OTHERcomputational
178: *
179: * =====================================================================
1.1 bertrand 180: SUBROUTINE ZHPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
181: $ FERR, BERR, WORK, RWORK, INFO )
182: *
1.9 bertrand 183: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 184: * -- LAPACK is a software package provided by Univ. of Tennessee, --
185: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 bertrand 186: * November 2011
1.1 bertrand 187: *
188: * .. Scalar Arguments ..
189: CHARACTER UPLO
190: INTEGER INFO, LDB, LDX, N, NRHS
191: * ..
192: * .. Array Arguments ..
193: INTEGER IPIV( * )
194: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
195: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
196: $ X( LDX, * )
197: * ..
198: *
199: * =====================================================================
200: *
201: * .. Parameters ..
202: INTEGER ITMAX
203: PARAMETER ( ITMAX = 5 )
204: DOUBLE PRECISION ZERO
205: PARAMETER ( ZERO = 0.0D+0 )
206: COMPLEX*16 ONE
207: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
208: DOUBLE PRECISION TWO
209: PARAMETER ( TWO = 2.0D+0 )
210: DOUBLE PRECISION THREE
211: PARAMETER ( THREE = 3.0D+0 )
212: * ..
213: * .. Local Scalars ..
214: LOGICAL UPPER
215: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
216: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
217: COMPLEX*16 ZDUM
218: * ..
219: * .. Local Arrays ..
220: INTEGER ISAVE( 3 )
221: * ..
222: * .. External Subroutines ..
223: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHPMV, ZHPTRS, ZLACN2
224: * ..
225: * .. Intrinsic Functions ..
226: INTRINSIC ABS, DBLE, DIMAG, MAX
227: * ..
228: * .. External Functions ..
229: LOGICAL LSAME
230: DOUBLE PRECISION DLAMCH
231: EXTERNAL LSAME, DLAMCH
232: * ..
233: * .. Statement Functions ..
234: DOUBLE PRECISION CABS1
235: * ..
236: * .. Statement Function definitions ..
237: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
238: * ..
239: * .. Executable Statements ..
240: *
241: * Test the input parameters.
242: *
243: INFO = 0
244: UPPER = LSAME( UPLO, 'U' )
245: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
246: INFO = -1
247: ELSE IF( N.LT.0 ) THEN
248: INFO = -2
249: ELSE IF( NRHS.LT.0 ) THEN
250: INFO = -3
251: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
252: INFO = -8
253: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
254: INFO = -10
255: END IF
256: IF( INFO.NE.0 ) THEN
257: CALL XERBLA( 'ZHPRFS', -INFO )
258: RETURN
259: END IF
260: *
261: * Quick return if possible
262: *
263: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
264: DO 10 J = 1, NRHS
265: FERR( J ) = ZERO
266: BERR( J ) = ZERO
267: 10 CONTINUE
268: RETURN
269: END IF
270: *
271: * NZ = maximum number of nonzero elements in each row of A, plus 1
272: *
273: NZ = N + 1
274: EPS = DLAMCH( 'Epsilon' )
275: SAFMIN = DLAMCH( 'Safe minimum' )
276: SAFE1 = NZ*SAFMIN
277: SAFE2 = SAFE1 / EPS
278: *
279: * Do for each right hand side
280: *
281: DO 140 J = 1, NRHS
282: *
283: COUNT = 1
284: LSTRES = THREE
285: 20 CONTINUE
286: *
287: * Loop until stopping criterion is satisfied.
288: *
289: * Compute residual R = B - A * X
290: *
291: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
292: CALL ZHPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK, 1 )
293: *
294: * Compute componentwise relative backward error from formula
295: *
296: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
297: *
298: * where abs(Z) is the componentwise absolute value of the matrix
299: * or vector Z. If the i-th component of the denominator is less
300: * than SAFE2, then SAFE1 is added to the i-th components of the
301: * numerator and denominator before dividing.
302: *
303: DO 30 I = 1, N
304: RWORK( I ) = CABS1( B( I, J ) )
305: 30 CONTINUE
306: *
307: * Compute abs(A)*abs(X) + abs(B).
308: *
309: KK = 1
310: IF( UPPER ) THEN
311: DO 50 K = 1, N
312: S = ZERO
313: XK = CABS1( X( K, J ) )
314: IK = KK
315: DO 40 I = 1, K - 1
316: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
317: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
318: IK = IK + 1
319: 40 CONTINUE
320: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )*
321: $ XK + S
322: KK = KK + K
323: 50 CONTINUE
324: ELSE
325: DO 70 K = 1, N
326: S = ZERO
327: XK = CABS1( X( K, J ) )
328: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK
329: IK = KK + 1
330: DO 60 I = K + 1, N
331: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
332: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
333: IK = IK + 1
334: 60 CONTINUE
335: RWORK( K ) = RWORK( K ) + S
336: KK = KK + ( N-K+1 )
337: 70 CONTINUE
338: END IF
339: S = ZERO
340: DO 80 I = 1, N
341: IF( RWORK( I ).GT.SAFE2 ) THEN
342: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
343: ELSE
344: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
345: $ ( RWORK( I )+SAFE1 ) )
346: END IF
347: 80 CONTINUE
348: BERR( J ) = S
349: *
350: * Test stopping criterion. Continue iterating if
351: * 1) The residual BERR(J) is larger than machine epsilon, and
352: * 2) BERR(J) decreased by at least a factor of 2 during the
353: * last iteration, and
354: * 3) At most ITMAX iterations tried.
355: *
356: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
357: $ COUNT.LE.ITMAX ) THEN
358: *
359: * Update solution and try again.
360: *
361: CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
362: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
363: LSTRES = BERR( J )
364: COUNT = COUNT + 1
365: GO TO 20
366: END IF
367: *
368: * Bound error from formula
369: *
370: * norm(X - XTRUE) / norm(X) .le. FERR =
371: * norm( abs(inv(A))*
372: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
373: *
374: * where
375: * norm(Z) is the magnitude of the largest component of Z
376: * inv(A) is the inverse of A
377: * abs(Z) is the componentwise absolute value of the matrix or
378: * vector Z
379: * NZ is the maximum number of nonzeros in any row of A, plus 1
380: * EPS is machine epsilon
381: *
382: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
383: * is incremented by SAFE1 if the i-th component of
384: * abs(A)*abs(X) + abs(B) is less than SAFE2.
385: *
386: * Use ZLACN2 to estimate the infinity-norm of the matrix
387: * inv(A) * diag(W),
388: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
389: *
390: DO 90 I = 1, N
391: IF( RWORK( I ).GT.SAFE2 ) THEN
392: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
393: ELSE
394: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
395: $ SAFE1
396: END IF
397: 90 CONTINUE
398: *
399: KASE = 0
400: 100 CONTINUE
401: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
402: IF( KASE.NE.0 ) THEN
403: IF( KASE.EQ.1 ) THEN
404: *
1.8 bertrand 405: * Multiply by diag(W)*inv(A**H).
1.1 bertrand 406: *
407: CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
408: DO 110 I = 1, N
409: WORK( I ) = RWORK( I )*WORK( I )
410: 110 CONTINUE
411: ELSE IF( KASE.EQ.2 ) THEN
412: *
413: * Multiply by inv(A)*diag(W).
414: *
415: DO 120 I = 1, N
416: WORK( I ) = RWORK( I )*WORK( I )
417: 120 CONTINUE
418: CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
419: END IF
420: GO TO 100
421: END IF
422: *
423: * Normalize error.
424: *
425: LSTRES = ZERO
426: DO 130 I = 1, N
427: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
428: 130 CONTINUE
429: IF( LSTRES.NE.ZERO )
430: $ FERR( J ) = FERR( J ) / LSTRES
431: *
432: 140 CONTINUE
433: *
434: RETURN
435: *
436: * End of ZHPRFS
437: *
438: END
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