Annotation of rpl/lapack/lapack/ztrrfs.f, revision 1.16
1.8 bertrand 1: *> \brief \b ZTRRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.14 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.8 bertrand 7: *
8: *> \htmlonly
1.14 bertrand 9: *> Download ZTRRFS + dependencies
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11: *> [TGZ]</a>
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1.8 bertrand 15: *> [TXT]</a>
1.14 bertrand 16: *> \endhtmlonly
1.8 bertrand 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 )
1.14 bertrand 23: *
1.8 bertrand 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: * ..
1.14 bertrand 33: *
1.8 bertrand 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: *
1.14 bertrand 172: *> \author Univ. of Tennessee
173: *> \author Univ. of California Berkeley
174: *> \author Univ. of Colorado Denver
175: *> \author NAG Ltd.
1.8 bertrand 176: *
1.14 bertrand 177: *> \date December 2016
1.8 bertrand 178: *
179: *> \ingroup complex16OTHERcomputational
180: *
181: * =====================================================================
1.1 bertrand 182: SUBROUTINE ZTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
183: $ LDX, FERR, BERR, WORK, RWORK, INFO )
184: *
1.14 bertrand 185: * -- LAPACK computational routine (version 3.7.0) --
1.1 bertrand 186: * -- LAPACK is a software package provided by Univ. of Tennessee, --
187: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.14 bertrand 188: * December 2016
1.1 bertrand 189: *
190: * .. Scalar Arguments ..
191: CHARACTER DIAG, TRANS, UPLO
192: INTEGER INFO, LDA, LDB, LDX, N, NRHS
193: * ..
194: * .. Array Arguments ..
195: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
196: COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ),
197: $ X( LDX, * )
198: * ..
199: *
200: * =====================================================================
201: *
202: * .. Parameters ..
203: DOUBLE PRECISION ZERO
204: PARAMETER ( ZERO = 0.0D+0 )
205: COMPLEX*16 ONE
206: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
207: * ..
208: * .. Local Scalars ..
209: LOGICAL NOTRAN, NOUNIT, UPPER
210: CHARACTER TRANSN, TRANST
211: INTEGER I, J, K, KASE, NZ
212: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
213: COMPLEX*16 ZDUM
214: * ..
215: * .. Local Arrays ..
216: INTEGER ISAVE( 3 )
217: * ..
218: * .. External Subroutines ..
219: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZTRMV, ZTRSV
220: * ..
221: * .. Intrinsic Functions ..
222: INTRINSIC ABS, DBLE, DIMAG, MAX
223: * ..
224: * .. External Functions ..
225: LOGICAL LSAME
226: DOUBLE PRECISION DLAMCH
227: EXTERNAL LSAME, DLAMCH
228: * ..
229: * .. Statement Functions ..
230: DOUBLE PRECISION CABS1
231: * ..
232: * .. Statement Function definitions ..
233: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
234: * ..
235: * .. Executable Statements ..
236: *
237: * Test the input parameters.
238: *
239: INFO = 0
240: UPPER = LSAME( UPLO, 'U' )
241: NOTRAN = LSAME( TRANS, 'N' )
242: NOUNIT = LSAME( DIAG, 'N' )
243: *
244: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
245: INFO = -1
246: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
247: $ LSAME( TRANS, 'C' ) ) THEN
248: INFO = -2
249: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
250: INFO = -3
251: ELSE IF( N.LT.0 ) THEN
252: INFO = -4
253: ELSE IF( NRHS.LT.0 ) THEN
254: INFO = -5
255: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
256: INFO = -7
257: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
258: INFO = -9
259: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
260: INFO = -11
261: END IF
262: IF( INFO.NE.0 ) THEN
263: CALL XERBLA( 'ZTRRFS', -INFO )
264: RETURN
265: END IF
266: *
267: * Quick return if possible
268: *
269: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
270: DO 10 J = 1, NRHS
271: FERR( J ) = ZERO
272: BERR( J ) = ZERO
273: 10 CONTINUE
274: RETURN
275: END IF
276: *
277: IF( NOTRAN ) THEN
278: TRANSN = 'N'
279: TRANST = 'C'
280: ELSE
281: TRANSN = 'C'
282: TRANST = 'N'
283: END IF
284: *
285: * NZ = maximum number of nonzero elements in each row of A, plus 1
286: *
287: NZ = N + 1
288: EPS = DLAMCH( 'Epsilon' )
289: SAFMIN = DLAMCH( 'Safe minimum' )
290: SAFE1 = NZ*SAFMIN
291: SAFE2 = SAFE1 / EPS
292: *
293: * Do for each right hand side
294: *
295: DO 250 J = 1, NRHS
296: *
297: * Compute residual R = B - op(A) * X,
298: * where op(A) = A, A**T, or A**H, depending on TRANS.
299: *
300: CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
301: CALL ZTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK, 1 )
302: CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
303: *
304: * Compute componentwise relative backward error from formula
305: *
306: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
307: *
308: * where abs(Z) is the componentwise absolute value of the matrix
309: * or vector Z. If the i-th component of the denominator is less
310: * than SAFE2, then SAFE1 is added to the i-th components of the
311: * numerator and denominator before dividing.
312: *
313: DO 20 I = 1, N
314: RWORK( I ) = CABS1( B( I, J ) )
315: 20 CONTINUE
316: *
317: IF( NOTRAN ) THEN
318: *
319: * Compute abs(A)*abs(X) + abs(B).
320: *
321: IF( UPPER ) THEN
322: IF( NOUNIT ) THEN
323: DO 40 K = 1, N
324: XK = CABS1( X( K, J ) )
325: DO 30 I = 1, K
326: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
327: 30 CONTINUE
328: 40 CONTINUE
329: ELSE
330: DO 60 K = 1, N
331: XK = CABS1( X( K, J ) )
332: DO 50 I = 1, K - 1
333: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
334: 50 CONTINUE
335: RWORK( K ) = RWORK( K ) + XK
336: 60 CONTINUE
337: END IF
338: ELSE
339: IF( NOUNIT ) THEN
340: DO 80 K = 1, N
341: XK = CABS1( X( K, J ) )
342: DO 70 I = K, N
343: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
344: 70 CONTINUE
345: 80 CONTINUE
346: ELSE
347: DO 100 K = 1, N
348: XK = CABS1( X( K, J ) )
349: DO 90 I = K + 1, N
350: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
351: 90 CONTINUE
352: RWORK( K ) = RWORK( K ) + XK
353: 100 CONTINUE
354: END IF
355: END IF
356: ELSE
357: *
358: * Compute abs(A**H)*abs(X) + abs(B).
359: *
360: IF( UPPER ) THEN
361: IF( NOUNIT ) THEN
362: DO 120 K = 1, N
363: S = ZERO
364: DO 110 I = 1, K
365: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
366: 110 CONTINUE
367: RWORK( K ) = RWORK( K ) + S
368: 120 CONTINUE
369: ELSE
370: DO 140 K = 1, N
371: S = CABS1( X( K, J ) )
372: DO 130 I = 1, K - 1
373: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
374: 130 CONTINUE
375: RWORK( K ) = RWORK( K ) + S
376: 140 CONTINUE
377: END IF
378: ELSE
379: IF( NOUNIT ) THEN
380: DO 160 K = 1, N
381: S = ZERO
382: DO 150 I = K, N
383: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
384: 150 CONTINUE
385: RWORK( K ) = RWORK( K ) + S
386: 160 CONTINUE
387: ELSE
388: DO 180 K = 1, N
389: S = CABS1( X( K, J ) )
390: DO 170 I = K + 1, N
391: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
392: 170 CONTINUE
393: RWORK( K ) = RWORK( K ) + S
394: 180 CONTINUE
395: END IF
396: END IF
397: END IF
398: S = ZERO
399: DO 190 I = 1, N
400: IF( RWORK( I ).GT.SAFE2 ) THEN
401: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
402: ELSE
403: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
404: $ ( RWORK( I )+SAFE1 ) )
405: END IF
406: 190 CONTINUE
407: BERR( J ) = S
408: *
409: * Bound error from formula
410: *
411: * norm(X - XTRUE) / norm(X) .le. FERR =
412: * norm( abs(inv(op(A)))*
413: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
414: *
415: * where
416: * norm(Z) is the magnitude of the largest component of Z
417: * inv(op(A)) is the inverse of op(A)
418: * abs(Z) is the componentwise absolute value of the matrix or
419: * vector Z
420: * NZ is the maximum number of nonzeros in any row of A, plus 1
421: * EPS is machine epsilon
422: *
423: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
424: * is incremented by SAFE1 if the i-th component of
425: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
426: *
427: * Use ZLACN2 to estimate the infinity-norm of the matrix
428: * inv(op(A)) * diag(W),
429: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
430: *
431: DO 200 I = 1, N
432: IF( RWORK( I ).GT.SAFE2 ) THEN
433: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
434: ELSE
435: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
436: $ SAFE1
437: END IF
438: 200 CONTINUE
439: *
440: KASE = 0
441: 210 CONTINUE
442: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
443: IF( KASE.NE.0 ) THEN
444: IF( KASE.EQ.1 ) THEN
445: *
446: * Multiply by diag(W)*inv(op(A)**H).
447: *
448: CALL ZTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK, 1 )
449: DO 220 I = 1, N
450: WORK( I ) = RWORK( I )*WORK( I )
451: 220 CONTINUE
452: ELSE
453: *
454: * Multiply by inv(op(A))*diag(W).
455: *
456: DO 230 I = 1, N
457: WORK( I ) = RWORK( I )*WORK( I )
458: 230 CONTINUE
459: CALL ZTRSV( UPLO, TRANSN, DIAG, N, A, LDA, WORK, 1 )
460: END IF
461: GO TO 210
462: END IF
463: *
464: * Normalize error.
465: *
466: LSTRES = ZERO
467: DO 240 I = 1, N
468: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
469: 240 CONTINUE
470: IF( LSTRES.NE.ZERO )
471: $ FERR( J ) = FERR( J ) / LSTRES
472: *
473: 250 CONTINUE
474: *
475: RETURN
476: *
477: * End of ZTRRFS
478: *
479: END
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