Annotation of rpl/lapack/lapack/zporfs.f, revision 1.18
1.9 bertrand 1: *> \brief \b ZPORFS
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 ZPORFS + dependencies
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11: *> [TGZ]</a>
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14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zporfs.f">
1.9 bertrand 15: *> [TXT]</a>
1.15 bertrand 16: *> \endhtmlonly
1.9 bertrand 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 )
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: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
30: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
31: * $ WORK( * ), X( LDX, * )
32: * ..
1.15 bertrand 33: *
1.9 bertrand 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: *
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: *
178: *> \ingroup complex16POcomputational
179: *
180: * =====================================================================
1.1 bertrand 181: SUBROUTINE ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
182: $ LDX, FERR, BERR, WORK, RWORK, INFO )
183: *
1.18 ! bertrand 184: * -- LAPACK computational routine --
1.1 bertrand 185: * -- LAPACK is a software package provided by Univ. of Tennessee, --
186: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
187: *
188: * .. Scalar Arguments ..
189: CHARACTER UPLO
190: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
191: * ..
192: * .. Array Arguments ..
193: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
194: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
195: $ WORK( * ), X( LDX, * )
196: * ..
197: *
198: * ====================================================================
199: *
200: * .. Parameters ..
201: INTEGER ITMAX
202: PARAMETER ( ITMAX = 5 )
203: DOUBLE PRECISION ZERO
204: PARAMETER ( ZERO = 0.0D+0 )
205: COMPLEX*16 ONE
206: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
207: DOUBLE PRECISION TWO
208: PARAMETER ( TWO = 2.0D+0 )
209: DOUBLE PRECISION THREE
210: PARAMETER ( THREE = 3.0D+0 )
211: * ..
212: * .. Local Scalars ..
213: LOGICAL UPPER
214: INTEGER COUNT, I, J, K, KASE, NZ
215: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
216: COMPLEX*16 ZDUM
217: * ..
218: * .. Local Arrays ..
219: INTEGER ISAVE( 3 )
220: * ..
221: * .. External Subroutines ..
222: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZLACN2, ZPOTRS
223: * ..
224: * .. Intrinsic Functions ..
225: INTRINSIC ABS, DBLE, DIMAG, MAX
226: * ..
227: * .. External Functions ..
228: LOGICAL LSAME
229: DOUBLE PRECISION DLAMCH
230: EXTERNAL LSAME, DLAMCH
231: * ..
232: * .. Statement Functions ..
233: DOUBLE PRECISION CABS1
234: * ..
235: * .. Statement Function definitions ..
236: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
237: * ..
238: * .. Executable Statements ..
239: *
240: * Test the input parameters.
241: *
242: INFO = 0
243: UPPER = LSAME( UPLO, 'U' )
244: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
245: INFO = -1
246: ELSE IF( N.LT.0 ) THEN
247: INFO = -2
248: ELSE IF( NRHS.LT.0 ) THEN
249: INFO = -3
250: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
251: INFO = -5
252: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
253: INFO = -7
254: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
255: INFO = -9
256: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
257: INFO = -11
258: END IF
259: IF( INFO.NE.0 ) THEN
260: CALL XERBLA( 'ZPORFS', -INFO )
261: RETURN
262: END IF
263: *
264: * Quick return if possible
265: *
266: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
267: DO 10 J = 1, NRHS
268: FERR( J ) = ZERO
269: BERR( J ) = ZERO
270: 10 CONTINUE
271: RETURN
272: END IF
273: *
274: * NZ = maximum number of nonzero elements in each row of A, plus 1
275: *
276: NZ = N + 1
277: EPS = DLAMCH( 'Epsilon' )
278: SAFMIN = DLAMCH( 'Safe minimum' )
279: SAFE1 = NZ*SAFMIN
280: SAFE2 = SAFE1 / EPS
281: *
282: * Do for each right hand side
283: *
284: DO 140 J = 1, NRHS
285: *
286: COUNT = 1
287: LSTRES = THREE
288: 20 CONTINUE
289: *
290: * Loop until stopping criterion is satisfied.
291: *
292: * Compute residual R = B - A * X
293: *
294: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
295: CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
296: *
297: * Compute componentwise relative backward error from formula
298: *
299: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
300: *
301: * where abs(Z) is the componentwise absolute value of the matrix
302: * or vector Z. If the i-th component of the denominator is less
303: * than SAFE2, then SAFE1 is added to the i-th components of the
304: * numerator and denominator before dividing.
305: *
306: DO 30 I = 1, N
307: RWORK( I ) = CABS1( B( I, J ) )
308: 30 CONTINUE
309: *
310: * Compute abs(A)*abs(X) + abs(B).
311: *
312: IF( UPPER ) THEN
313: DO 50 K = 1, N
314: S = ZERO
315: XK = CABS1( X( K, J ) )
316: DO 40 I = 1, K - 1
317: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
318: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
319: 40 CONTINUE
320: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S
321: 50 CONTINUE
322: ELSE
323: DO 70 K = 1, N
324: S = ZERO
325: XK = CABS1( X( K, J ) )
326: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK
327: DO 60 I = K + 1, N
328: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
329: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
330: 60 CONTINUE
331: RWORK( K ) = RWORK( K ) + S
332: 70 CONTINUE
333: END IF
334: S = ZERO
335: DO 80 I = 1, N
336: IF( RWORK( I ).GT.SAFE2 ) THEN
337: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
338: ELSE
339: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
340: $ ( RWORK( I )+SAFE1 ) )
341: END IF
342: 80 CONTINUE
343: BERR( J ) = S
344: *
345: * Test stopping criterion. Continue iterating if
346: * 1) The residual BERR(J) is larger than machine epsilon, and
347: * 2) BERR(J) decreased by at least a factor of 2 during the
348: * last iteration, and
349: * 3) At most ITMAX iterations tried.
350: *
351: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
352: $ COUNT.LE.ITMAX ) THEN
353: *
354: * Update solution and try again.
355: *
356: CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
357: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
358: LSTRES = BERR( J )
359: COUNT = COUNT + 1
360: GO TO 20
361: END IF
362: *
363: * Bound error from formula
364: *
365: * norm(X - XTRUE) / norm(X) .le. FERR =
366: * norm( abs(inv(A))*
367: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
368: *
369: * where
370: * norm(Z) is the magnitude of the largest component of Z
371: * inv(A) is the inverse of A
372: * abs(Z) is the componentwise absolute value of the matrix or
373: * vector Z
374: * NZ is the maximum number of nonzeros in any row of A, plus 1
375: * EPS is machine epsilon
376: *
377: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
378: * is incremented by SAFE1 if the i-th component of
379: * abs(A)*abs(X) + abs(B) is less than SAFE2.
380: *
381: * Use ZLACN2 to estimate the infinity-norm of the matrix
382: * inv(A) * diag(W),
383: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
384: *
385: DO 90 I = 1, N
386: IF( RWORK( I ).GT.SAFE2 ) THEN
387: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
388: ELSE
389: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
390: $ SAFE1
391: END IF
392: 90 CONTINUE
393: *
394: KASE = 0
395: 100 CONTINUE
396: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
397: IF( KASE.NE.0 ) THEN
398: IF( KASE.EQ.1 ) THEN
399: *
1.8 bertrand 400: * Multiply by diag(W)*inv(A**H).
1.1 bertrand 401: *
402: CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
403: DO 110 I = 1, N
404: WORK( I ) = RWORK( I )*WORK( I )
405: 110 CONTINUE
406: ELSE IF( KASE.EQ.2 ) THEN
407: *
408: * Multiply by inv(A)*diag(W).
409: *
410: DO 120 I = 1, N
411: WORK( I ) = RWORK( I )*WORK( I )
412: 120 CONTINUE
413: CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
414: END IF
415: GO TO 100
416: END IF
417: *
418: * Normalize error.
419: *
420: LSTRES = ZERO
421: DO 130 I = 1, N
422: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
423: 130 CONTINUE
424: IF( LSTRES.NE.ZERO )
425: $ FERR( J ) = FERR( J ) / LSTRES
426: *
427: 140 CONTINUE
428: *
429: RETURN
430: *
431: * End of ZPORFS
432: *
433: END
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