Annotation of rpl/lapack/lapack/zporfs.f, revision 1.14
1.9 bertrand 1: *> \brief \b ZPORFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZPORFS + 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/zporfs.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zporfs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
22: * LDX, FERR, BERR, WORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
30: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
31: * $ WORK( * ), X( LDX, * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> ZPORFS improves the computed solution to a system of linear
41: *> equations when the coefficient matrix is Hermitian positive definite,
42: *> and provides error bounds and backward error estimates for the
43: *> 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] NRHS
63: *> \verbatim
64: *> NRHS is INTEGER
65: *> The number of right hand sides, i.e., the number of columns
66: *> of the matrices B and X. NRHS >= 0.
67: *> \endverbatim
68: *>
69: *> \param[in] A
70: *> \verbatim
71: *> A is COMPLEX*16 array, dimension (LDA,N)
72: *> The Hermitian matrix A. If UPLO = 'U', the leading N-by-N
73: *> upper triangular part of A contains the upper triangular part
74: *> of the matrix A, and the strictly lower triangular part of A
75: *> is not referenced. If UPLO = 'L', the leading N-by-N lower
76: *> triangular part of A contains the lower triangular part of
77: *> the matrix A, and the strictly upper triangular part of A is
78: *> not referenced.
79: *> \endverbatim
80: *>
81: *> \param[in] LDA
82: *> \verbatim
83: *> LDA is INTEGER
84: *> The leading dimension of the array A. LDA >= max(1,N).
85: *> \endverbatim
86: *>
87: *> \param[in] AF
88: *> \verbatim
89: *> AF is COMPLEX*16 array, dimension (LDAF,N)
90: *> The triangular factor U or L from the Cholesky factorization
91: *> A = U**H*U or A = L*L**H, as computed by ZPOTRF.
92: *> \endverbatim
93: *>
94: *> \param[in] LDAF
95: *> \verbatim
96: *> LDAF is INTEGER
97: *> The leading dimension of the array AF. LDAF >= max(1,N).
98: *> \endverbatim
99: *>
100: *> \param[in] B
101: *> \verbatim
102: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
103: *> The right hand side matrix B.
104: *> \endverbatim
105: *>
106: *> \param[in] LDB
107: *> \verbatim
108: *> LDB is INTEGER
109: *> The leading dimension of the array B. LDB >= max(1,N).
110: *> \endverbatim
111: *>
112: *> \param[in,out] X
113: *> \verbatim
114: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
115: *> On entry, the solution matrix X, as computed by ZPOTRS.
116: *> On exit, the improved solution matrix X.
117: *> \endverbatim
118: *>
119: *> \param[in] LDX
120: *> \verbatim
121: *> LDX is INTEGER
122: *> The leading dimension of the array X. LDX >= max(1,N).
123: *> \endverbatim
124: *>
125: *> \param[out] FERR
126: *> \verbatim
127: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
128: *> The estimated forward error bound for each solution vector
129: *> X(j) (the j-th column of the solution matrix X).
130: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
131: *> is an estimated upper bound for the magnitude of the largest
132: *> element in (X(j) - XTRUE) divided by the magnitude of the
133: *> largest element in X(j). The estimate is as reliable as
134: *> the estimate for RCOND, and is almost always a slight
135: *> overestimate of the true error.
136: *> \endverbatim
137: *>
138: *> \param[out] BERR
139: *> \verbatim
140: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
141: *> The componentwise relative backward error of each solution
142: *> vector X(j) (i.e., the smallest relative change in
143: *> any element of A or B that makes X(j) an exact solution).
144: *> \endverbatim
145: *>
146: *> \param[out] WORK
147: *> \verbatim
148: *> WORK is COMPLEX*16 array, dimension (2*N)
149: *> \endverbatim
150: *>
151: *> \param[out] RWORK
152: *> \verbatim
153: *> RWORK is DOUBLE PRECISION array, dimension (N)
154: *> \endverbatim
155: *>
156: *> \param[out] INFO
157: *> \verbatim
158: *> INFO is INTEGER
159: *> = 0: successful exit
160: *> < 0: if INFO = -i, the i-th argument had an illegal value
161: *> \endverbatim
162: *
163: *> \par Internal Parameters:
164: * =========================
165: *>
166: *> \verbatim
167: *> ITMAX is the maximum number of steps of iterative refinement.
168: *> \endverbatim
169: *
170: * Authors:
171: * ========
172: *
173: *> \author Univ. of Tennessee
174: *> \author Univ. of California Berkeley
175: *> \author Univ. of Colorado Denver
176: *> \author NAG Ltd.
177: *
178: *> \date November 2011
179: *
180: *> \ingroup complex16POcomputational
181: *
182: * =====================================================================
1.1 bertrand 183: SUBROUTINE ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
184: $ LDX, FERR, BERR, WORK, RWORK, INFO )
185: *
1.9 bertrand 186: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 187: * -- LAPACK is a software package provided by Univ. of Tennessee, --
188: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 bertrand 189: * November 2011
1.1 bertrand 190: *
191: * .. Scalar Arguments ..
192: CHARACTER UPLO
193: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
194: * ..
195: * .. Array Arguments ..
196: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
197: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
198: $ WORK( * ), X( LDX, * )
199: * ..
200: *
201: * ====================================================================
202: *
203: * .. Parameters ..
204: INTEGER ITMAX
205: PARAMETER ( ITMAX = 5 )
206: DOUBLE PRECISION ZERO
207: PARAMETER ( ZERO = 0.0D+0 )
208: COMPLEX*16 ONE
209: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
210: DOUBLE PRECISION TWO
211: PARAMETER ( TWO = 2.0D+0 )
212: DOUBLE PRECISION THREE
213: PARAMETER ( THREE = 3.0D+0 )
214: * ..
215: * .. Local Scalars ..
216: LOGICAL UPPER
217: INTEGER COUNT, I, J, K, KASE, NZ
218: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
219: COMPLEX*16 ZDUM
220: * ..
221: * .. Local Arrays ..
222: INTEGER ISAVE( 3 )
223: * ..
224: * .. External Subroutines ..
225: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZLACN2, ZPOTRS
226: * ..
227: * .. Intrinsic Functions ..
228: INTRINSIC ABS, DBLE, DIMAG, MAX
229: * ..
230: * .. External Functions ..
231: LOGICAL LSAME
232: DOUBLE PRECISION DLAMCH
233: EXTERNAL LSAME, DLAMCH
234: * ..
235: * .. Statement Functions ..
236: DOUBLE PRECISION CABS1
237: * ..
238: * .. Statement Function definitions ..
239: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
240: * ..
241: * .. Executable Statements ..
242: *
243: * Test the input parameters.
244: *
245: INFO = 0
246: UPPER = LSAME( UPLO, 'U' )
247: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
248: INFO = -1
249: ELSE IF( N.LT.0 ) THEN
250: INFO = -2
251: ELSE IF( NRHS.LT.0 ) THEN
252: INFO = -3
253: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
254: INFO = -5
255: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
256: INFO = -7
257: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
258: INFO = -9
259: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
260: INFO = -11
261: END IF
262: IF( INFO.NE.0 ) THEN
263: CALL XERBLA( 'ZPORFS', -INFO )
264: RETURN
265: END IF
266: *
267: * Quick return if possible
268: *
269: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
270: DO 10 J = 1, NRHS
271: FERR( J ) = ZERO
272: BERR( J ) = ZERO
273: 10 CONTINUE
274: RETURN
275: END IF
276: *
277: * NZ = maximum number of nonzero elements in each row of A, plus 1
278: *
279: NZ = N + 1
280: EPS = DLAMCH( 'Epsilon' )
281: SAFMIN = DLAMCH( 'Safe minimum' )
282: SAFE1 = NZ*SAFMIN
283: SAFE2 = SAFE1 / EPS
284: *
285: * Do for each right hand side
286: *
287: DO 140 J = 1, NRHS
288: *
289: COUNT = 1
290: LSTRES = THREE
291: 20 CONTINUE
292: *
293: * Loop until stopping criterion is satisfied.
294: *
295: * Compute residual R = B - A * X
296: *
297: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
298: CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
299: *
300: * Compute componentwise relative backward error from formula
301: *
302: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
303: *
304: * where abs(Z) is the componentwise absolute value of the matrix
305: * or vector Z. If the i-th component of the denominator is less
306: * than SAFE2, then SAFE1 is added to the i-th components of the
307: * numerator and denominator before dividing.
308: *
309: DO 30 I = 1, N
310: RWORK( I ) = CABS1( B( I, J ) )
311: 30 CONTINUE
312: *
313: * Compute abs(A)*abs(X) + abs(B).
314: *
315: IF( UPPER ) THEN
316: DO 50 K = 1, N
317: S = ZERO
318: XK = CABS1( X( K, J ) )
319: DO 40 I = 1, K - 1
320: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
321: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
322: 40 CONTINUE
323: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S
324: 50 CONTINUE
325: ELSE
326: DO 70 K = 1, N
327: S = ZERO
328: XK = CABS1( X( K, J ) )
329: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK
330: DO 60 I = K + 1, N
331: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
332: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
333: 60 CONTINUE
334: RWORK( K ) = RWORK( K ) + S
335: 70 CONTINUE
336: END IF
337: S = ZERO
338: DO 80 I = 1, N
339: IF( RWORK( I ).GT.SAFE2 ) THEN
340: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
341: ELSE
342: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
343: $ ( RWORK( I )+SAFE1 ) )
344: END IF
345: 80 CONTINUE
346: BERR( J ) = S
347: *
348: * Test stopping criterion. Continue iterating if
349: * 1) The residual BERR(J) is larger than machine epsilon, and
350: * 2) BERR(J) decreased by at least a factor of 2 during the
351: * last iteration, and
352: * 3) At most ITMAX iterations tried.
353: *
354: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
355: $ COUNT.LE.ITMAX ) THEN
356: *
357: * Update solution and try again.
358: *
359: CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
360: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
361: LSTRES = BERR( J )
362: COUNT = COUNT + 1
363: GO TO 20
364: END IF
365: *
366: * Bound error from formula
367: *
368: * norm(X - XTRUE) / norm(X) .le. FERR =
369: * norm( abs(inv(A))*
370: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
371: *
372: * where
373: * norm(Z) is the magnitude of the largest component of Z
374: * inv(A) is the inverse of A
375: * abs(Z) is the componentwise absolute value of the matrix or
376: * vector Z
377: * NZ is the maximum number of nonzeros in any row of A, plus 1
378: * EPS is machine epsilon
379: *
380: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
381: * is incremented by SAFE1 if the i-th component of
382: * abs(A)*abs(X) + abs(B) is less than SAFE2.
383: *
384: * Use ZLACN2 to estimate the infinity-norm of the matrix
385: * inv(A) * diag(W),
386: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
387: *
388: DO 90 I = 1, N
389: IF( RWORK( I ).GT.SAFE2 ) THEN
390: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
391: ELSE
392: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
393: $ SAFE1
394: END IF
395: 90 CONTINUE
396: *
397: KASE = 0
398: 100 CONTINUE
399: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
400: IF( KASE.NE.0 ) THEN
401: IF( KASE.EQ.1 ) THEN
402: *
1.8 bertrand 403: * Multiply by diag(W)*inv(A**H).
1.1 bertrand 404: *
405: CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
406: DO 110 I = 1, N
407: WORK( I ) = RWORK( I )*WORK( I )
408: 110 CONTINUE
409: ELSE IF( KASE.EQ.2 ) THEN
410: *
411: * Multiply by inv(A)*diag(W).
412: *
413: DO 120 I = 1, N
414: WORK( I ) = RWORK( I )*WORK( I )
415: 120 CONTINUE
416: CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
417: END IF
418: GO TO 100
419: END IF
420: *
421: * Normalize error.
422: *
423: LSTRES = ZERO
424: DO 130 I = 1, N
425: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
426: 130 CONTINUE
427: IF( LSTRES.NE.ZERO )
428: $ FERR( J ) = FERR( J ) / LSTRES
429: *
430: 140 CONTINUE
431: *
432: RETURN
433: *
434: * End of ZPORFS
435: *
436: END
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