1: *> \brief \b ZGERFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGERFS + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgerfs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
22: * X, LDX, FERR, BERR, WORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER TRANS
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: *> ZGERFS improves the computed solution to a system of linear
42: *> equations and provides error bounds and backward error estimates for
43: *> the solution.
44: *> \endverbatim
45: *
46: * Arguments:
47: * ==========
48: *
49: *> \param[in] TRANS
50: *> \verbatim
51: *> TRANS is CHARACTER*1
52: *> Specifies the form of the system of equations:
53: *> = 'N': A * X = B (No transpose)
54: *> = 'T': A**T * X = B (Transpose)
55: *> = 'C': A**H * X = B (Conjugate transpose)
56: *> \endverbatim
57: *>
58: *> \param[in] N
59: *> \verbatim
60: *> N is INTEGER
61: *> The order of the matrix A. N >= 0.
62: *> \endverbatim
63: *>
64: *> \param[in] NRHS
65: *> \verbatim
66: *> NRHS is INTEGER
67: *> The number of right hand sides, i.e., the number of columns
68: *> of the matrices B and X. NRHS >= 0.
69: *> \endverbatim
70: *>
71: *> \param[in] A
72: *> \verbatim
73: *> A is COMPLEX*16 array, dimension (LDA,N)
74: *> The original N-by-N matrix A.
75: *> \endverbatim
76: *>
77: *> \param[in] LDA
78: *> \verbatim
79: *> LDA is INTEGER
80: *> The leading dimension of the array A. LDA >= max(1,N).
81: *> \endverbatim
82: *>
83: *> \param[in] AF
84: *> \verbatim
85: *> AF is COMPLEX*16 array, dimension (LDAF,N)
86: *> The factors L and U from the factorization A = P*L*U
87: *> as computed by ZGETRF.
88: *> \endverbatim
89: *>
90: *> \param[in] LDAF
91: *> \verbatim
92: *> LDAF is INTEGER
93: *> The leading dimension of the array AF. LDAF >= max(1,N).
94: *> \endverbatim
95: *>
96: *> \param[in] IPIV
97: *> \verbatim
98: *> IPIV is INTEGER array, dimension (N)
99: *> The pivot indices from ZGETRF; for 1<=i<=N, row i of the
100: *> matrix was interchanged with row IPIV(i).
101: *> \endverbatim
102: *>
103: *> \param[in] B
104: *> \verbatim
105: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
106: *> The right hand side matrix B.
107: *> \endverbatim
108: *>
109: *> \param[in] LDB
110: *> \verbatim
111: *> LDB is INTEGER
112: *> The leading dimension of the array B. LDB >= max(1,N).
113: *> \endverbatim
114: *>
115: *> \param[in,out] X
116: *> \verbatim
117: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
118: *> On entry, the solution matrix X, as computed by ZGETRS.
119: *> On exit, the improved solution matrix X.
120: *> \endverbatim
121: *>
122: *> \param[in] LDX
123: *> \verbatim
124: *> LDX is INTEGER
125: *> The leading dimension of the array X. LDX >= max(1,N).
126: *> \endverbatim
127: *>
128: *> \param[out] FERR
129: *> \verbatim
130: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
131: *> The estimated forward error bound for each solution vector
132: *> X(j) (the j-th column of the solution matrix X).
133: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
134: *> is an estimated upper bound for the magnitude of the largest
135: *> element in (X(j) - XTRUE) divided by the magnitude of the
136: *> largest element in X(j). The estimate is as reliable as
137: *> the estimate for RCOND, and is almost always a slight
138: *> overestimate of the true error.
139: *> \endverbatim
140: *>
141: *> \param[out] BERR
142: *> \verbatim
143: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
144: *> The componentwise relative backward error of each solution
145: *> vector X(j) (i.e., the smallest relative change in
146: *> any element of A or B that makes X(j) an exact solution).
147: *> \endverbatim
148: *>
149: *> \param[out] WORK
150: *> \verbatim
151: *> WORK is COMPLEX*16 array, dimension (2*N)
152: *> \endverbatim
153: *>
154: *> \param[out] RWORK
155: *> \verbatim
156: *> RWORK is DOUBLE PRECISION array, dimension (N)
157: *> \endverbatim
158: *>
159: *> \param[out] INFO
160: *> \verbatim
161: *> INFO is INTEGER
162: *> = 0: successful exit
163: *> < 0: if INFO = -i, the i-th argument had an illegal value
164: *> \endverbatim
165: *
166: *> \par Internal Parameters:
167: * =========================
168: *>
169: *> \verbatim
170: *> ITMAX is the maximum number of steps of iterative refinement.
171: *> \endverbatim
172: *
173: * Authors:
174: * ========
175: *
176: *> \author Univ. of Tennessee
177: *> \author Univ. of California Berkeley
178: *> \author Univ. of Colorado Denver
179: *> \author NAG Ltd.
180: *
181: *> \date November 2011
182: *
183: *> \ingroup complex16GEcomputational
184: *
185: * =====================================================================
186: SUBROUTINE ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
187: $ X, LDX, FERR, BERR, WORK, RWORK, INFO )
188: *
189: * -- LAPACK computational routine (version 3.4.0) --
190: * -- LAPACK is a software package provided by Univ. of Tennessee, --
191: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
192: * November 2011
193: *
194: * .. Scalar Arguments ..
195: CHARACTER TRANS
196: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
197: * ..
198: * .. Array Arguments ..
199: INTEGER IPIV( * )
200: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
201: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
202: $ WORK( * ), X( LDX, * )
203: * ..
204: *
205: * =====================================================================
206: *
207: * .. Parameters ..
208: INTEGER ITMAX
209: PARAMETER ( ITMAX = 5 )
210: DOUBLE PRECISION ZERO
211: PARAMETER ( ZERO = 0.0D+0 )
212: COMPLEX*16 ONE
213: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
214: DOUBLE PRECISION TWO
215: PARAMETER ( TWO = 2.0D+0 )
216: DOUBLE PRECISION THREE
217: PARAMETER ( THREE = 3.0D+0 )
218: * ..
219: * .. Local Scalars ..
220: LOGICAL NOTRAN
221: CHARACTER TRANSN, TRANST
222: INTEGER COUNT, I, J, K, KASE, NZ
223: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
224: COMPLEX*16 ZDUM
225: * ..
226: * .. Local Arrays ..
227: INTEGER ISAVE( 3 )
228: * ..
229: * .. External Functions ..
230: LOGICAL LSAME
231: DOUBLE PRECISION DLAMCH
232: EXTERNAL LSAME, DLAMCH
233: * ..
234: * .. External Subroutines ..
235: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGETRS, ZLACN2
236: * ..
237: * .. Intrinsic Functions ..
238: INTRINSIC ABS, DBLE, DIMAG, MAX
239: * ..
240: * .. Statement Functions ..
241: DOUBLE PRECISION CABS1
242: * ..
243: * .. Statement Function definitions ..
244: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
245: * ..
246: * .. Executable Statements ..
247: *
248: * Test the input parameters.
249: *
250: INFO = 0
251: NOTRAN = LSAME( TRANS, 'N' )
252: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
253: $ LSAME( TRANS, 'C' ) ) THEN
254: INFO = -1
255: ELSE IF( N.LT.0 ) THEN
256: INFO = -2
257: ELSE IF( NRHS.LT.0 ) THEN
258: INFO = -3
259: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
260: INFO = -5
261: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
262: INFO = -7
263: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
264: INFO = -10
265: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
266: INFO = -12
267: END IF
268: IF( INFO.NE.0 ) THEN
269: CALL XERBLA( 'ZGERFS', -INFO )
270: RETURN
271: END IF
272: *
273: * Quick return if possible
274: *
275: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
276: DO 10 J = 1, NRHS
277: FERR( J ) = ZERO
278: BERR( J ) = ZERO
279: 10 CONTINUE
280: RETURN
281: END IF
282: *
283: IF( NOTRAN ) THEN
284: TRANSN = 'N'
285: TRANST = 'C'
286: ELSE
287: TRANSN = 'C'
288: TRANST = 'N'
289: END IF
290: *
291: * NZ = maximum number of nonzero elements in each row of A, plus 1
292: *
293: NZ = N + 1
294: EPS = DLAMCH( 'Epsilon' )
295: SAFMIN = DLAMCH( 'Safe minimum' )
296: SAFE1 = NZ*SAFMIN
297: SAFE2 = SAFE1 / EPS
298: *
299: * Do for each right hand side
300: *
301: DO 140 J = 1, NRHS
302: *
303: COUNT = 1
304: LSTRES = THREE
305: 20 CONTINUE
306: *
307: * Loop until stopping criterion is satisfied.
308: *
309: * Compute residual R = B - op(A) * X,
310: * where op(A) = A, A**T, or A**H, depending on TRANS.
311: *
312: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
313: CALL ZGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK,
314: $ 1 )
315: *
316: * Compute componentwise relative backward error from formula
317: *
318: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
319: *
320: * where abs(Z) is the componentwise absolute value of the matrix
321: * or vector Z. If the i-th component of the denominator is less
322: * than SAFE2, then SAFE1 is added to the i-th components of the
323: * numerator and denominator before dividing.
324: *
325: DO 30 I = 1, N
326: RWORK( I ) = CABS1( B( I, J ) )
327: 30 CONTINUE
328: *
329: * Compute abs(op(A))*abs(X) + abs(B).
330: *
331: IF( NOTRAN ) THEN
332: DO 50 K = 1, N
333: XK = CABS1( X( K, J ) )
334: DO 40 I = 1, N
335: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
336: 40 CONTINUE
337: 50 CONTINUE
338: ELSE
339: DO 70 K = 1, N
340: S = ZERO
341: DO 60 I = 1, N
342: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
343: 60 CONTINUE
344: RWORK( K ) = RWORK( K ) + S
345: 70 CONTINUE
346: END IF
347: S = ZERO
348: DO 80 I = 1, N
349: IF( RWORK( I ).GT.SAFE2 ) THEN
350: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
351: ELSE
352: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
353: $ ( RWORK( I )+SAFE1 ) )
354: END IF
355: 80 CONTINUE
356: BERR( J ) = S
357: *
358: * Test stopping criterion. Continue iterating if
359: * 1) The residual BERR(J) is larger than machine epsilon, and
360: * 2) BERR(J) decreased by at least a factor of 2 during the
361: * last iteration, and
362: * 3) At most ITMAX iterations tried.
363: *
364: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
365: $ COUNT.LE.ITMAX ) THEN
366: *
367: * Update solution and try again.
368: *
369: CALL ZGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
370: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
371: LSTRES = BERR( J )
372: COUNT = COUNT + 1
373: GO TO 20
374: END IF
375: *
376: * Bound error from formula
377: *
378: * norm(X - XTRUE) / norm(X) .le. FERR =
379: * norm( abs(inv(op(A)))*
380: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
381: *
382: * where
383: * norm(Z) is the magnitude of the largest component of Z
384: * inv(op(A)) is the inverse of op(A)
385: * abs(Z) is the componentwise absolute value of the matrix or
386: * vector Z
387: * NZ is the maximum number of nonzeros in any row of A, plus 1
388: * EPS is machine epsilon
389: *
390: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
391: * is incremented by SAFE1 if the i-th component of
392: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
393: *
394: * Use ZLACN2 to estimate the infinity-norm of the matrix
395: * inv(op(A)) * diag(W),
396: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
397: *
398: DO 90 I = 1, N
399: IF( RWORK( I ).GT.SAFE2 ) THEN
400: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
401: ELSE
402: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
403: $ SAFE1
404: END IF
405: 90 CONTINUE
406: *
407: KASE = 0
408: 100 CONTINUE
409: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
410: IF( KASE.NE.0 ) THEN
411: IF( KASE.EQ.1 ) THEN
412: *
413: * Multiply by diag(W)*inv(op(A)**H).
414: *
415: CALL ZGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK, N,
416: $ INFO )
417: DO 110 I = 1, N
418: WORK( I ) = RWORK( I )*WORK( I )
419: 110 CONTINUE
420: ELSE
421: *
422: * Multiply by inv(op(A))*diag(W).
423: *
424: DO 120 I = 1, N
425: WORK( I ) = RWORK( I )*WORK( I )
426: 120 CONTINUE
427: CALL ZGETRS( TRANSN, N, 1, AF, LDAF, IPIV, WORK, N,
428: $ INFO )
429: END IF
430: GO TO 100
431: END IF
432: *
433: * Normalize error.
434: *
435: LSTRES = ZERO
436: DO 130 I = 1, N
437: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
438: 130 CONTINUE
439: IF( LSTRES.NE.ZERO )
440: $ FERR( J ) = FERR( J ) / LSTRES
441: *
442: 140 CONTINUE
443: *
444: RETURN
445: *
446: * End of ZGERFS
447: *
448: END
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