Annotation of rpl/lapack/lapack/dporfs.f, revision 1.16
1.9 bertrand 1: *> \brief \b DPORFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.15 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.9 bertrand 7: *
8: *> \htmlonly
1.15 bertrand 9: *> Download DPORFS + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dporfs.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dporfs.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dporfs.f">
1.9 bertrand 15: *> [TXT]</a>
1.15 bertrand 16: *> \endhtmlonly
1.9 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
22: * LDX, FERR, BERR, WORK, IWORK, INFO )
1.15 bertrand 23: *
1.9 bertrand 24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IWORK( * )
30: * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
31: * $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
32: * ..
1.15 bertrand 33: *
1.9 bertrand 34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> DPORFS improves the computed solution to a system of linear
41: *> equations when the coefficient matrix is symmetric 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 DOUBLE PRECISION array, dimension (LDA,N)
72: *> The symmetric 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 DOUBLE PRECISION array, dimension (LDAF,N)
90: *> The triangular factor U or L from the Cholesky factorization
91: *> A = U**T*U or A = L*L**T, as computed by DPOTRF.
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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)
115: *> On entry, the solution matrix X, as computed by DPOTRS.
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 DOUBLE PRECISION array, dimension (3*N)
149: *> \endverbatim
150: *>
151: *> \param[out] IWORK
152: *> \verbatim
153: *> IWORK is INTEGER 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: *
1.15 bertrand 173: *> \author Univ. of Tennessee
174: *> \author Univ. of California Berkeley
175: *> \author Univ. of Colorado Denver
176: *> \author NAG Ltd.
1.9 bertrand 177: *
1.15 bertrand 178: *> \date December 2016
1.9 bertrand 179: *
180: *> \ingroup doublePOcomputational
181: *
182: * =====================================================================
1.1 bertrand 183: SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
184: $ LDX, FERR, BERR, WORK, IWORK, INFO )
185: *
1.15 bertrand 186: * -- LAPACK computational routine (version 3.7.0) --
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..--
1.15 bertrand 189: * December 2016
1.1 bertrand 190: *
191: * .. Scalar Arguments ..
192: CHARACTER UPLO
193: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
194: * ..
195: * .. Array Arguments ..
196: INTEGER IWORK( * )
197: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
198: $ BERR( * ), FERR( * ), 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: DOUBLE PRECISION ONE
209: PARAMETER ( ONE = 1.0D+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: * ..
220: * .. Local Arrays ..
221: INTEGER ISAVE( 3 )
222: * ..
223: * .. External Subroutines ..
224: EXTERNAL DAXPY, DCOPY, DLACN2, DPOTRS, DSYMV, XERBLA
225: * ..
226: * .. Intrinsic Functions ..
227: INTRINSIC ABS, MAX
228: * ..
229: * .. External Functions ..
230: LOGICAL LSAME
231: DOUBLE PRECISION DLAMCH
232: EXTERNAL LSAME, DLAMCH
233: * ..
234: * .. Executable Statements ..
235: *
236: * Test the input parameters.
237: *
238: INFO = 0
239: UPPER = LSAME( UPLO, 'U' )
240: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
241: INFO = -1
242: ELSE IF( N.LT.0 ) THEN
243: INFO = -2
244: ELSE IF( NRHS.LT.0 ) THEN
245: INFO = -3
246: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
247: INFO = -5
248: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
249: INFO = -7
250: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
251: INFO = -9
252: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
253: INFO = -11
254: END IF
255: IF( INFO.NE.0 ) THEN
256: CALL XERBLA( 'DPORFS', -INFO )
257: RETURN
258: END IF
259: *
260: * Quick return if possible
261: *
262: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
263: DO 10 J = 1, NRHS
264: FERR( J ) = ZERO
265: BERR( J ) = ZERO
266: 10 CONTINUE
267: RETURN
268: END IF
269: *
270: * NZ = maximum number of nonzero elements in each row of A, plus 1
271: *
272: NZ = N + 1
273: EPS = DLAMCH( 'Epsilon' )
274: SAFMIN = DLAMCH( 'Safe minimum' )
275: SAFE1 = NZ*SAFMIN
276: SAFE2 = SAFE1 / EPS
277: *
278: * Do for each right hand side
279: *
280: DO 140 J = 1, NRHS
281: *
282: COUNT = 1
283: LSTRES = THREE
284: 20 CONTINUE
285: *
286: * Loop until stopping criterion is satisfied.
287: *
288: * Compute residual R = B - A * X
289: *
290: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
291: CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
292: $ WORK( N+1 ), 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: WORK( I ) = ABS( B( I, J ) )
305: 30 CONTINUE
306: *
307: * Compute abs(A)*abs(X) + abs(B).
308: *
309: IF( UPPER ) THEN
310: DO 50 K = 1, N
311: S = ZERO
312: XK = ABS( X( K, J ) )
313: DO 40 I = 1, K - 1
314: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
315: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
316: 40 CONTINUE
317: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
318: 50 CONTINUE
319: ELSE
320: DO 70 K = 1, N
321: S = ZERO
322: XK = ABS( X( K, J ) )
323: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
324: DO 60 I = K + 1, N
325: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
326: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
327: 60 CONTINUE
328: WORK( K ) = WORK( K ) + S
329: 70 CONTINUE
330: END IF
331: S = ZERO
332: DO 80 I = 1, N
333: IF( WORK( I ).GT.SAFE2 ) THEN
334: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
335: ELSE
336: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
337: $ ( WORK( I )+SAFE1 ) )
338: END IF
339: 80 CONTINUE
340: BERR( J ) = S
341: *
342: * Test stopping criterion. Continue iterating if
343: * 1) The residual BERR(J) is larger than machine epsilon, and
344: * 2) BERR(J) decreased by at least a factor of 2 during the
345: * last iteration, and
346: * 3) At most ITMAX iterations tried.
347: *
348: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
349: $ COUNT.LE.ITMAX ) THEN
350: *
351: * Update solution and try again.
352: *
353: CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
354: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
355: LSTRES = BERR( J )
356: COUNT = COUNT + 1
357: GO TO 20
358: END IF
359: *
360: * Bound error from formula
361: *
362: * norm(X - XTRUE) / norm(X) .le. FERR =
363: * norm( abs(inv(A))*
364: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
365: *
366: * where
367: * norm(Z) is the magnitude of the largest component of Z
368: * inv(A) is the inverse of A
369: * abs(Z) is the componentwise absolute value of the matrix or
370: * vector Z
371: * NZ is the maximum number of nonzeros in any row of A, plus 1
372: * EPS is machine epsilon
373: *
374: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
375: * is incremented by SAFE1 if the i-th component of
376: * abs(A)*abs(X) + abs(B) is less than SAFE2.
377: *
378: * Use DLACN2 to estimate the infinity-norm of the matrix
379: * inv(A) * diag(W),
380: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
381: *
382: DO 90 I = 1, N
383: IF( WORK( I ).GT.SAFE2 ) THEN
384: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
385: ELSE
386: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
387: END IF
388: 90 CONTINUE
389: *
390: KASE = 0
391: 100 CONTINUE
392: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
393: $ KASE, ISAVE )
394: IF( KASE.NE.0 ) THEN
395: IF( KASE.EQ.1 ) THEN
396: *
1.8 bertrand 397: * Multiply by diag(W)*inv(A**T).
1.1 bertrand 398: *
399: CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
400: DO 110 I = 1, N
401: WORK( N+I ) = WORK( I )*WORK( N+I )
402: 110 CONTINUE
403: ELSE IF( KASE.EQ.2 ) THEN
404: *
405: * Multiply by inv(A)*diag(W).
406: *
407: DO 120 I = 1, N
408: WORK( N+I ) = WORK( I )*WORK( N+I )
409: 120 CONTINUE
410: CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
411: END IF
412: GO TO 100
413: END IF
414: *
415: * Normalize error.
416: *
417: LSTRES = ZERO
418: DO 130 I = 1, N
419: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
420: 130 CONTINUE
421: IF( LSTRES.NE.ZERO )
422: $ FERR( J ) = FERR( J ) / LSTRES
423: *
424: 140 CONTINUE
425: *
426: RETURN
427: *
428: * End of DPORFS
429: *
430: END
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