1: *> \brief \b ZTRRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
22: * LDX, FERR, BERR, WORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER DIAG, TRANS, UPLO
26: * INTEGER INFO, LDA, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
30: * COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ),
31: * $ X( LDX, * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> ZTRRFS provides error bounds and backward error estimates for the
41: *> solution to a system of linear equations with a triangular
42: *> coefficient matrix.
43: *>
44: *> The solution matrix X must be computed by ZTRTRS or some other
45: *> means before entering this routine. ZTRRFS does not do iterative
46: *> refinement because doing so cannot improve the backward error.
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] UPLO
53: *> \verbatim
54: *> UPLO is CHARACTER*1
55: *> = 'U': A is upper triangular;
56: *> = 'L': A is lower triangular.
57: *> \endverbatim
58: *>
59: *> \param[in] TRANS
60: *> \verbatim
61: *> TRANS is CHARACTER*1
62: *> Specifies the form of the system of equations:
63: *> = 'N': A * X = B (No transpose)
64: *> = 'T': A**T * X = B (Transpose)
65: *> = 'C': A**H * X = B (Conjugate transpose)
66: *> \endverbatim
67: *>
68: *> \param[in] DIAG
69: *> \verbatim
70: *> DIAG is CHARACTER*1
71: *> = 'N': A is non-unit triangular;
72: *> = 'U': A is unit triangular.
73: *> \endverbatim
74: *>
75: *> \param[in] N
76: *> \verbatim
77: *> N is INTEGER
78: *> The order of the matrix A. N >= 0.
79: *> \endverbatim
80: *>
81: *> \param[in] NRHS
82: *> \verbatim
83: *> NRHS is INTEGER
84: *> The number of right hand sides, i.e., the number of columns
85: *> of the matrices B and X. NRHS >= 0.
86: *> \endverbatim
87: *>
88: *> \param[in] A
89: *> \verbatim
90: *> A is COMPLEX*16 array, dimension (LDA,N)
91: *> The triangular matrix A. If UPLO = 'U', the leading N-by-N
92: *> upper triangular part of the array A contains the upper
93: *> triangular matrix, and the strictly lower triangular part of
94: *> A is not referenced. If UPLO = 'L', the leading N-by-N lower
95: *> triangular part of the array A contains the lower triangular
96: *> matrix, and the strictly upper triangular part of A is not
97: *> referenced. If DIAG = 'U', the diagonal elements of A are
98: *> also not referenced and are assumed to be 1.
99: *> \endverbatim
100: *>
101: *> \param[in] LDA
102: *> \verbatim
103: *> LDA is INTEGER
104: *> The leading dimension of the array A. LDA >= max(1,N).
105: *> \endverbatim
106: *>
107: *> \param[in] B
108: *> \verbatim
109: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
110: *> The right hand side matrix B.
111: *> \endverbatim
112: *>
113: *> \param[in] LDB
114: *> \verbatim
115: *> LDB is INTEGER
116: *> The leading dimension of the array B. LDB >= max(1,N).
117: *> \endverbatim
118: *>
119: *> \param[in] X
120: *> \verbatim
121: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
122: *> The solution matrix X.
123: *> \endverbatim
124: *>
125: *> \param[in] LDX
126: *> \verbatim
127: *> LDX is INTEGER
128: *> The leading dimension of the array X. LDX >= max(1,N).
129: *> \endverbatim
130: *>
131: *> \param[out] FERR
132: *> \verbatim
133: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
134: *> The estimated forward error bound for each solution vector
135: *> X(j) (the j-th column of the solution matrix X).
136: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
137: *> is an estimated upper bound for the magnitude of the largest
138: *> element in (X(j) - XTRUE) divided by the magnitude of the
139: *> largest element in X(j). The estimate is as reliable as
140: *> the estimate for RCOND, and is almost always a slight
141: *> overestimate of the true error.
142: *> \endverbatim
143: *>
144: *> \param[out] BERR
145: *> \verbatim
146: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
147: *> The componentwise relative backward error of each solution
148: *> vector X(j) (i.e., the smallest relative change in
149: *> any element of A or B that makes X(j) an exact solution).
150: *> \endverbatim
151: *>
152: *> \param[out] WORK
153: *> \verbatim
154: *> WORK is COMPLEX*16 array, dimension (2*N)
155: *> \endverbatim
156: *>
157: *> \param[out] RWORK
158: *> \verbatim
159: *> RWORK is DOUBLE PRECISION array, dimension (N)
160: *> \endverbatim
161: *>
162: *> \param[out] INFO
163: *> \verbatim
164: *> INFO is INTEGER
165: *> = 0: successful exit
166: *> < 0: if INFO = -i, the i-th argument had an illegal value
167: *> \endverbatim
168: *
169: * Authors:
170: * ========
171: *
172: *> \author Univ. of Tennessee
173: *> \author Univ. of California Berkeley
174: *> \author Univ. of Colorado Denver
175: *> \author NAG Ltd.
176: *
177: *> \ingroup complex16OTHERcomputational
178: *
179: * =====================================================================
180: SUBROUTINE ZTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
181: $ LDX, FERR, BERR, WORK, RWORK, INFO )
182: *
183: * -- LAPACK computational routine --
184: * -- LAPACK is a software package provided by Univ. of Tennessee, --
185: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
186: *
187: * .. Scalar Arguments ..
188: CHARACTER DIAG, TRANS, UPLO
189: INTEGER INFO, LDA, LDB, LDX, N, NRHS
190: * ..
191: * .. Array Arguments ..
192: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
193: COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ),
194: $ X( LDX, * )
195: * ..
196: *
197: * =====================================================================
198: *
199: * .. Parameters ..
200: DOUBLE PRECISION ZERO
201: PARAMETER ( ZERO = 0.0D+0 )
202: COMPLEX*16 ONE
203: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
204: * ..
205: * .. Local Scalars ..
206: LOGICAL NOTRAN, NOUNIT, UPPER
207: CHARACTER TRANSN, TRANST
208: INTEGER I, J, K, KASE, NZ
209: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
210: COMPLEX*16 ZDUM
211: * ..
212: * .. Local Arrays ..
213: INTEGER ISAVE( 3 )
214: * ..
215: * .. External Subroutines ..
216: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZTRMV, ZTRSV
217: * ..
218: * .. Intrinsic Functions ..
219: INTRINSIC ABS, DBLE, DIMAG, MAX
220: * ..
221: * .. External Functions ..
222: LOGICAL LSAME
223: DOUBLE PRECISION DLAMCH
224: EXTERNAL LSAME, DLAMCH
225: * ..
226: * .. Statement Functions ..
227: DOUBLE PRECISION CABS1
228: * ..
229: * .. Statement Function definitions ..
230: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
231: * ..
232: * .. Executable Statements ..
233: *
234: * Test the input parameters.
235: *
236: INFO = 0
237: UPPER = LSAME( UPLO, 'U' )
238: NOTRAN = LSAME( TRANS, 'N' )
239: NOUNIT = LSAME( DIAG, 'N' )
240: *
241: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
242: INFO = -1
243: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
244: $ LSAME( TRANS, 'C' ) ) THEN
245: INFO = -2
246: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
247: INFO = -3
248: ELSE IF( N.LT.0 ) THEN
249: INFO = -4
250: ELSE IF( NRHS.LT.0 ) THEN
251: INFO = -5
252: ELSE IF( LDA.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( 'ZTRRFS', -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: IF( NOTRAN ) THEN
275: TRANSN = 'N'
276: TRANST = 'C'
277: ELSE
278: TRANSN = 'C'
279: TRANST = 'N'
280: END IF
281: *
282: * NZ = maximum number of nonzero elements in each row of A, plus 1
283: *
284: NZ = N + 1
285: EPS = DLAMCH( 'Epsilon' )
286: SAFMIN = DLAMCH( 'Safe minimum' )
287: SAFE1 = NZ*SAFMIN
288: SAFE2 = SAFE1 / EPS
289: *
290: * Do for each right hand side
291: *
292: DO 250 J = 1, NRHS
293: *
294: * Compute residual R = B - op(A) * X,
295: * where op(A) = A, A**T, or A**H, depending on TRANS.
296: *
297: CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
298: CALL ZTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK, 1 )
299: CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
300: *
301: * Compute componentwise relative backward error from formula
302: *
303: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
304: *
305: * where abs(Z) is the componentwise absolute value of the matrix
306: * or vector Z. If the i-th component of the denominator is less
307: * than SAFE2, then SAFE1 is added to the i-th components of the
308: * numerator and denominator before dividing.
309: *
310: DO 20 I = 1, N
311: RWORK( I ) = CABS1( B( I, J ) )
312: 20 CONTINUE
313: *
314: IF( NOTRAN ) THEN
315: *
316: * Compute abs(A)*abs(X) + abs(B).
317: *
318: IF( UPPER ) THEN
319: IF( NOUNIT ) THEN
320: DO 40 K = 1, N
321: XK = CABS1( X( K, J ) )
322: DO 30 I = 1, K
323: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
324: 30 CONTINUE
325: 40 CONTINUE
326: ELSE
327: DO 60 K = 1, N
328: XK = CABS1( X( K, J ) )
329: DO 50 I = 1, K - 1
330: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
331: 50 CONTINUE
332: RWORK( K ) = RWORK( K ) + XK
333: 60 CONTINUE
334: END IF
335: ELSE
336: IF( NOUNIT ) THEN
337: DO 80 K = 1, N
338: XK = CABS1( X( K, J ) )
339: DO 70 I = K, N
340: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
341: 70 CONTINUE
342: 80 CONTINUE
343: ELSE
344: DO 100 K = 1, N
345: XK = CABS1( X( K, J ) )
346: DO 90 I = K + 1, N
347: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
348: 90 CONTINUE
349: RWORK( K ) = RWORK( K ) + XK
350: 100 CONTINUE
351: END IF
352: END IF
353: ELSE
354: *
355: * Compute abs(A**H)*abs(X) + abs(B).
356: *
357: IF( UPPER ) THEN
358: IF( NOUNIT ) THEN
359: DO 120 K = 1, N
360: S = ZERO
361: DO 110 I = 1, K
362: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
363: 110 CONTINUE
364: RWORK( K ) = RWORK( K ) + S
365: 120 CONTINUE
366: ELSE
367: DO 140 K = 1, N
368: S = CABS1( X( K, J ) )
369: DO 130 I = 1, K - 1
370: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
371: 130 CONTINUE
372: RWORK( K ) = RWORK( K ) + S
373: 140 CONTINUE
374: END IF
375: ELSE
376: IF( NOUNIT ) THEN
377: DO 160 K = 1, N
378: S = ZERO
379: DO 150 I = K, N
380: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
381: 150 CONTINUE
382: RWORK( K ) = RWORK( K ) + S
383: 160 CONTINUE
384: ELSE
385: DO 180 K = 1, N
386: S = CABS1( X( K, J ) )
387: DO 170 I = K + 1, N
388: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
389: 170 CONTINUE
390: RWORK( K ) = RWORK( K ) + S
391: 180 CONTINUE
392: END IF
393: END IF
394: END IF
395: S = ZERO
396: DO 190 I = 1, N
397: IF( RWORK( I ).GT.SAFE2 ) THEN
398: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
399: ELSE
400: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
401: $ ( RWORK( I )+SAFE1 ) )
402: END IF
403: 190 CONTINUE
404: BERR( J ) = S
405: *
406: * Bound error from formula
407: *
408: * norm(X - XTRUE) / norm(X) .le. FERR =
409: * norm( abs(inv(op(A)))*
410: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
411: *
412: * where
413: * norm(Z) is the magnitude of the largest component of Z
414: * inv(op(A)) is the inverse of op(A)
415: * abs(Z) is the componentwise absolute value of the matrix or
416: * vector Z
417: * NZ is the maximum number of nonzeros in any row of A, plus 1
418: * EPS is machine epsilon
419: *
420: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
421: * is incremented by SAFE1 if the i-th component of
422: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
423: *
424: * Use ZLACN2 to estimate the infinity-norm of the matrix
425: * inv(op(A)) * diag(W),
426: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
427: *
428: DO 200 I = 1, N
429: IF( RWORK( I ).GT.SAFE2 ) THEN
430: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
431: ELSE
432: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
433: $ SAFE1
434: END IF
435: 200 CONTINUE
436: *
437: KASE = 0
438: 210 CONTINUE
439: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
440: IF( KASE.NE.0 ) THEN
441: IF( KASE.EQ.1 ) THEN
442: *
443: * Multiply by diag(W)*inv(op(A)**H).
444: *
445: CALL ZTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK, 1 )
446: DO 220 I = 1, N
447: WORK( I ) = RWORK( I )*WORK( I )
448: 220 CONTINUE
449: ELSE
450: *
451: * Multiply by inv(op(A))*diag(W).
452: *
453: DO 230 I = 1, N
454: WORK( I ) = RWORK( I )*WORK( I )
455: 230 CONTINUE
456: CALL ZTRSV( UPLO, TRANSN, DIAG, N, A, LDA, WORK, 1 )
457: END IF
458: GO TO 210
459: END IF
460: *
461: * Normalize error.
462: *
463: LSTRES = ZERO
464: DO 240 I = 1, N
465: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
466: 240 CONTINUE
467: IF( LSTRES.NE.ZERO )
468: $ FERR( J ) = FERR( J ) / LSTRES
469: *
470: 250 CONTINUE
471: *
472: RETURN
473: *
474: * End of ZTRRFS
475: *
476: END
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