1: *> \brief \b ZSYRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
22: * X, 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: * INTEGER IPIV( * )
30: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
31: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
32: * $ WORK( * ), X( LDX, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> ZSYRFS improves the computed solution to a system of linear
42: *> equations when the coefficient matrix is symmetric indefinite, and
43: *> provides error bounds and backward error estimates 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] 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 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 COMPLEX*16 array, dimension (LDAF,N)
90: *> The factored form of the matrix A. AF contains the block
91: *> diagonal matrix D and the multipliers used to obtain the
92: *> factor U or L from the factorization A = U*D*U**T or
93: *> A = L*D*L**T as computed by ZSYTRF.
94: *> \endverbatim
95: *>
96: *> \param[in] LDAF
97: *> \verbatim
98: *> LDAF is INTEGER
99: *> The leading dimension of the array AF. LDAF >= max(1,N).
100: *> \endverbatim
101: *>
102: *> \param[in] IPIV
103: *> \verbatim
104: *> IPIV is INTEGER array, dimension (N)
105: *> Details of the interchanges and the block structure of D
106: *> as determined by ZSYTRF.
107: *> \endverbatim
108: *>
109: *> \param[in] B
110: *> \verbatim
111: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
112: *> The right hand side matrix B.
113: *> \endverbatim
114: *>
115: *> \param[in] LDB
116: *> \verbatim
117: *> LDB is INTEGER
118: *> The leading dimension of the array B. LDB >= max(1,N).
119: *> \endverbatim
120: *>
121: *> \param[in,out] X
122: *> \verbatim
123: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
124: *> On entry, the solution matrix X, as computed by ZSYTRS.
125: *> On exit, the improved solution matrix X.
126: *> \endverbatim
127: *>
128: *> \param[in] LDX
129: *> \verbatim
130: *> LDX is INTEGER
131: *> The leading dimension of the array X. LDX >= max(1,N).
132: *> \endverbatim
133: *>
134: *> \param[out] FERR
135: *> \verbatim
136: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
137: *> The estimated forward error bound for each solution vector
138: *> X(j) (the j-th column of the solution matrix X).
139: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
140: *> is an estimated upper bound for the magnitude of the largest
141: *> element in (X(j) - XTRUE) divided by the magnitude of the
142: *> largest element in X(j). The estimate is as reliable as
143: *> the estimate for RCOND, and is almost always a slight
144: *> overestimate of the true error.
145: *> \endverbatim
146: *>
147: *> \param[out] BERR
148: *> \verbatim
149: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
150: *> The componentwise relative backward error of each solution
151: *> vector X(j) (i.e., the smallest relative change in
152: *> any element of A or B that makes X(j) an exact solution).
153: *> \endverbatim
154: *>
155: *> \param[out] WORK
156: *> \verbatim
157: *> WORK is COMPLEX*16 array, dimension (2*N)
158: *> \endverbatim
159: *>
160: *> \param[out] RWORK
161: *> \verbatim
162: *> RWORK is DOUBLE PRECISION array, dimension (N)
163: *> \endverbatim
164: *>
165: *> \param[out] INFO
166: *> \verbatim
167: *> INFO is INTEGER
168: *> = 0: successful exit
169: *> < 0: if INFO = -i, the i-th argument had an illegal value
170: *> \endverbatim
171: *
172: *> \par Internal Parameters:
173: * =========================
174: *>
175: *> \verbatim
176: *> ITMAX is the maximum number of steps of iterative refinement.
177: *> \endverbatim
178: *
179: * Authors:
180: * ========
181: *
182: *> \author Univ. of Tennessee
183: *> \author Univ. of California Berkeley
184: *> \author Univ. of Colorado Denver
185: *> \author NAG Ltd.
186: *
187: *> \ingroup complex16SYcomputational
188: *
189: * =====================================================================
190: SUBROUTINE ZSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
191: $ X, LDX, FERR, BERR, WORK, RWORK, INFO )
192: *
193: * -- LAPACK computational routine --
194: * -- LAPACK is a software package provided by Univ. of Tennessee, --
195: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
196: *
197: * .. Scalar Arguments ..
198: CHARACTER UPLO
199: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
200: * ..
201: * .. Array Arguments ..
202: INTEGER IPIV( * )
203: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
204: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
205: $ WORK( * ), X( LDX, * )
206: * ..
207: *
208: * =====================================================================
209: *
210: * .. Parameters ..
211: INTEGER ITMAX
212: PARAMETER ( ITMAX = 5 )
213: DOUBLE PRECISION ZERO
214: PARAMETER ( ZERO = 0.0D+0 )
215: COMPLEX*16 ONE
216: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
217: DOUBLE PRECISION TWO
218: PARAMETER ( TWO = 2.0D+0 )
219: DOUBLE PRECISION THREE
220: PARAMETER ( THREE = 3.0D+0 )
221: * ..
222: * .. Local Scalars ..
223: LOGICAL UPPER
224: INTEGER COUNT, I, J, K, KASE, NZ
225: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
226: COMPLEX*16 ZDUM
227: * ..
228: * .. Local Arrays ..
229: INTEGER ISAVE( 3 )
230: * ..
231: * .. External Subroutines ..
232: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZSYMV, ZSYTRS
233: * ..
234: * .. Intrinsic Functions ..
235: INTRINSIC ABS, DBLE, DIMAG, MAX
236: * ..
237: * .. External Functions ..
238: LOGICAL LSAME
239: DOUBLE PRECISION DLAMCH
240: EXTERNAL LSAME, DLAMCH
241: * ..
242: * .. Statement Functions ..
243: DOUBLE PRECISION CABS1
244: * ..
245: * .. Statement Function definitions ..
246: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
247: * ..
248: * .. Executable Statements ..
249: *
250: * Test the input parameters.
251: *
252: INFO = 0
253: UPPER = LSAME( UPLO, 'U' )
254: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
255: INFO = -1
256: ELSE IF( N.LT.0 ) THEN
257: INFO = -2
258: ELSE IF( NRHS.LT.0 ) THEN
259: INFO = -3
260: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
261: INFO = -5
262: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
263: INFO = -7
264: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
265: INFO = -10
266: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
267: INFO = -12
268: END IF
269: IF( INFO.NE.0 ) THEN
270: CALL XERBLA( 'ZSYRFS', -INFO )
271: RETURN
272: END IF
273: *
274: * Quick return if possible
275: *
276: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
277: DO 10 J = 1, NRHS
278: FERR( J ) = ZERO
279: BERR( J ) = ZERO
280: 10 CONTINUE
281: RETURN
282: END IF
283: *
284: * NZ = maximum number of nonzero elements in each row of A, plus 1
285: *
286: NZ = N + 1
287: EPS = DLAMCH( 'Epsilon' )
288: SAFMIN = DLAMCH( 'Safe minimum' )
289: SAFE1 = NZ*SAFMIN
290: SAFE2 = SAFE1 / EPS
291: *
292: * Do for each right hand side
293: *
294: DO 140 J = 1, NRHS
295: *
296: COUNT = 1
297: LSTRES = THREE
298: 20 CONTINUE
299: *
300: * Loop until stopping criterion is satisfied.
301: *
302: * Compute residual R = B - A * X
303: *
304: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
305: CALL ZSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
306: *
307: * Compute componentwise relative backward error from formula
308: *
309: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
310: *
311: * where abs(Z) is the componentwise absolute value of the matrix
312: * or vector Z. If the i-th component of the denominator is less
313: * than SAFE2, then SAFE1 is added to the i-th components of the
314: * numerator and denominator before dividing.
315: *
316: DO 30 I = 1, N
317: RWORK( I ) = CABS1( B( I, J ) )
318: 30 CONTINUE
319: *
320: * Compute abs(A)*abs(X) + abs(B).
321: *
322: IF( UPPER ) THEN
323: DO 50 K = 1, N
324: S = ZERO
325: XK = CABS1( X( K, J ) )
326: DO 40 I = 1, K - 1
327: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
328: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
329: 40 CONTINUE
330: RWORK( K ) = RWORK( K ) + CABS1( A( K, K ) )*XK + S
331: 50 CONTINUE
332: ELSE
333: DO 70 K = 1, N
334: S = ZERO
335: XK = CABS1( X( K, J ) )
336: RWORK( K ) = RWORK( K ) + CABS1( A( K, K ) )*XK
337: DO 60 I = K + 1, N
338: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
339: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
340: 60 CONTINUE
341: RWORK( K ) = RWORK( K ) + S
342: 70 CONTINUE
343: END IF
344: S = ZERO
345: DO 80 I = 1, N
346: IF( RWORK( I ).GT.SAFE2 ) THEN
347: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
348: ELSE
349: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
350: $ ( RWORK( I )+SAFE1 ) )
351: END IF
352: 80 CONTINUE
353: BERR( J ) = S
354: *
355: * Test stopping criterion. Continue iterating if
356: * 1) The residual BERR(J) is larger than machine epsilon, and
357: * 2) BERR(J) decreased by at least a factor of 2 during the
358: * last iteration, and
359: * 3) At most ITMAX iterations tried.
360: *
361: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
362: $ COUNT.LE.ITMAX ) THEN
363: *
364: * Update solution and try again.
365: *
366: CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
367: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
368: LSTRES = BERR( J )
369: COUNT = COUNT + 1
370: GO TO 20
371: END IF
372: *
373: * Bound error from formula
374: *
375: * norm(X - XTRUE) / norm(X) .le. FERR =
376: * norm( abs(inv(A))*
377: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
378: *
379: * where
380: * norm(Z) is the magnitude of the largest component of Z
381: * inv(A) is the inverse of A
382: * abs(Z) is the componentwise absolute value of the matrix or
383: * vector Z
384: * NZ is the maximum number of nonzeros in any row of A, plus 1
385: * EPS is machine epsilon
386: *
387: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
388: * is incremented by SAFE1 if the i-th component of
389: * abs(A)*abs(X) + abs(B) is less than SAFE2.
390: *
391: * Use ZLACN2 to estimate the infinity-norm of the matrix
392: * inv(A) * diag(W),
393: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
394: *
395: DO 90 I = 1, N
396: IF( RWORK( I ).GT.SAFE2 ) THEN
397: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
398: ELSE
399: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
400: $ SAFE1
401: END IF
402: 90 CONTINUE
403: *
404: KASE = 0
405: 100 CONTINUE
406: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
407: IF( KASE.NE.0 ) THEN
408: IF( KASE.EQ.1 ) THEN
409: *
410: * Multiply by diag(W)*inv(A**T).
411: *
412: CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
413: DO 110 I = 1, N
414: WORK( I ) = RWORK( I )*WORK( I )
415: 110 CONTINUE
416: ELSE IF( KASE.EQ.2 ) THEN
417: *
418: * Multiply by inv(A)*diag(W).
419: *
420: DO 120 I = 1, N
421: WORK( I ) = RWORK( I )*WORK( I )
422: 120 CONTINUE
423: CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
424: END IF
425: GO TO 100
426: END IF
427: *
428: * Normalize error.
429: *
430: LSTRES = ZERO
431: DO 130 I = 1, N
432: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
433: 130 CONTINUE
434: IF( LSTRES.NE.ZERO )
435: $ FERR( J ) = FERR( J ) / LSTRES
436: *
437: 140 CONTINUE
438: *
439: RETURN
440: *
441: * End of ZSYRFS
442: *
443: END
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