1: *> \brief \b DTRRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrrfs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
22: * LDX, FERR, BERR, WORK, IWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER DIAG, TRANS, UPLO
26: * INTEGER INFO, LDA, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IWORK( * )
30: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
31: * $ WORK( * ), X( LDX, * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> DTRRFS 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 DTRTRS or some other
45: *> means before entering this routine. DTRRFS 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 = 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (3*N)
155: *> \endverbatim
156: *>
157: *> \param[out] IWORK
158: *> \verbatim
159: *> IWORK is INTEGER 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: *> \date December 2016
178: *
179: *> \ingroup doubleOTHERcomputational
180: *
181: * =====================================================================
182: SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
183: $ LDX, FERR, BERR, WORK, IWORK, INFO )
184: *
185: * -- LAPACK computational routine (version 3.7.0) --
186: * -- LAPACK is a software package provided by Univ. of Tennessee, --
187: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
188: * December 2016
189: *
190: * .. Scalar Arguments ..
191: CHARACTER DIAG, TRANS, UPLO
192: INTEGER INFO, LDA, LDB, LDX, N, NRHS
193: * ..
194: * .. Array Arguments ..
195: INTEGER IWORK( * )
196: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
197: $ WORK( * ), X( LDX, * )
198: * ..
199: *
200: * =====================================================================
201: *
202: * .. Parameters ..
203: DOUBLE PRECISION ZERO
204: PARAMETER ( ZERO = 0.0D+0 )
205: DOUBLE PRECISION ONE
206: PARAMETER ( ONE = 1.0D+0 )
207: * ..
208: * .. Local Scalars ..
209: LOGICAL NOTRAN, NOUNIT, UPPER
210: CHARACTER TRANST
211: INTEGER I, J, K, KASE, NZ
212: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
213: * ..
214: * .. Local Arrays ..
215: INTEGER ISAVE( 3 )
216: * ..
217: * .. External Subroutines ..
218: EXTERNAL DAXPY, DCOPY, DLACN2, DTRMV, DTRSV, XERBLA
219: * ..
220: * .. Intrinsic Functions ..
221: INTRINSIC ABS, MAX
222: * ..
223: * .. External Functions ..
224: LOGICAL LSAME
225: DOUBLE PRECISION DLAMCH
226: EXTERNAL LSAME, DLAMCH
227: * ..
228: * .. Executable Statements ..
229: *
230: * Test the input parameters.
231: *
232: INFO = 0
233: UPPER = LSAME( UPLO, 'U' )
234: NOTRAN = LSAME( TRANS, 'N' )
235: NOUNIT = LSAME( DIAG, 'N' )
236: *
237: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
238: INFO = -1
239: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
240: $ LSAME( TRANS, 'C' ) ) THEN
241: INFO = -2
242: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
243: INFO = -3
244: ELSE IF( N.LT.0 ) THEN
245: INFO = -4
246: ELSE IF( NRHS.LT.0 ) THEN
247: INFO = -5
248: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
249: INFO = -7
250: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
251: INFO = -9
252: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
253: INFO = -11
254: END IF
255: IF( INFO.NE.0 ) THEN
256: CALL XERBLA( 'DTRRFS', -INFO )
257: RETURN
258: END IF
259: *
260: * Quick return if possible
261: *
262: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
263: DO 10 J = 1, NRHS
264: FERR( J ) = ZERO
265: BERR( J ) = ZERO
266: 10 CONTINUE
267: RETURN
268: END IF
269: *
270: IF( NOTRAN ) THEN
271: TRANST = 'T'
272: ELSE
273: TRANST = 'N'
274: END IF
275: *
276: * NZ = maximum number of nonzero elements in each row of A, plus 1
277: *
278: NZ = N + 1
279: EPS = DLAMCH( 'Epsilon' )
280: SAFMIN = DLAMCH( 'Safe minimum' )
281: SAFE1 = NZ*SAFMIN
282: SAFE2 = SAFE1 / EPS
283: *
284: * Do for each right hand side
285: *
286: DO 250 J = 1, NRHS
287: *
288: * Compute residual R = B - op(A) * X,
289: * where op(A) = A or A**T, depending on TRANS.
290: *
291: CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
292: CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 )
293: CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
294: *
295: * Compute componentwise relative backward error from formula
296: *
297: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
298: *
299: * where abs(Z) is the componentwise absolute value of the matrix
300: * or vector Z. If the i-th component of the denominator is less
301: * than SAFE2, then SAFE1 is added to the i-th components of the
302: * numerator and denominator before dividing.
303: *
304: DO 20 I = 1, N
305: WORK( I ) = ABS( B( I, J ) )
306: 20 CONTINUE
307: *
308: IF( NOTRAN ) THEN
309: *
310: * Compute abs(A)*abs(X) + abs(B).
311: *
312: IF( UPPER ) THEN
313: IF( NOUNIT ) THEN
314: DO 40 K = 1, N
315: XK = ABS( X( K, J ) )
316: DO 30 I = 1, K
317: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
318: 30 CONTINUE
319: 40 CONTINUE
320: ELSE
321: DO 60 K = 1, N
322: XK = ABS( X( K, J ) )
323: DO 50 I = 1, K - 1
324: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
325: 50 CONTINUE
326: WORK( K ) = WORK( K ) + XK
327: 60 CONTINUE
328: END IF
329: ELSE
330: IF( NOUNIT ) THEN
331: DO 80 K = 1, N
332: XK = ABS( X( K, J ) )
333: DO 70 I = K, N
334: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
335: 70 CONTINUE
336: 80 CONTINUE
337: ELSE
338: DO 100 K = 1, N
339: XK = ABS( X( K, J ) )
340: DO 90 I = K + 1, N
341: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
342: 90 CONTINUE
343: WORK( K ) = WORK( K ) + XK
344: 100 CONTINUE
345: END IF
346: END IF
347: ELSE
348: *
349: * Compute abs(A**T)*abs(X) + abs(B).
350: *
351: IF( UPPER ) THEN
352: IF( NOUNIT ) THEN
353: DO 120 K = 1, N
354: S = ZERO
355: DO 110 I = 1, K
356: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
357: 110 CONTINUE
358: WORK( K ) = WORK( K ) + S
359: 120 CONTINUE
360: ELSE
361: DO 140 K = 1, N
362: S = ABS( X( K, J ) )
363: DO 130 I = 1, K - 1
364: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
365: 130 CONTINUE
366: WORK( K ) = WORK( K ) + S
367: 140 CONTINUE
368: END IF
369: ELSE
370: IF( NOUNIT ) THEN
371: DO 160 K = 1, N
372: S = ZERO
373: DO 150 I = K, N
374: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
375: 150 CONTINUE
376: WORK( K ) = WORK( K ) + S
377: 160 CONTINUE
378: ELSE
379: DO 180 K = 1, N
380: S = ABS( X( K, J ) )
381: DO 170 I = K + 1, N
382: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
383: 170 CONTINUE
384: WORK( K ) = WORK( K ) + S
385: 180 CONTINUE
386: END IF
387: END IF
388: END IF
389: S = ZERO
390: DO 190 I = 1, N
391: IF( WORK( I ).GT.SAFE2 ) THEN
392: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
393: ELSE
394: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
395: $ ( WORK( I )+SAFE1 ) )
396: END IF
397: 190 CONTINUE
398: BERR( J ) = S
399: *
400: * Bound error from formula
401: *
402: * norm(X - XTRUE) / norm(X) .le. FERR =
403: * norm( abs(inv(op(A)))*
404: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
405: *
406: * where
407: * norm(Z) is the magnitude of the largest component of Z
408: * inv(op(A)) is the inverse of op(A)
409: * abs(Z) is the componentwise absolute value of the matrix or
410: * vector Z
411: * NZ is the maximum number of nonzeros in any row of A, plus 1
412: * EPS is machine epsilon
413: *
414: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
415: * is incremented by SAFE1 if the i-th component of
416: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
417: *
418: * Use DLACN2 to estimate the infinity-norm of the matrix
419: * inv(op(A)) * diag(W),
420: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
421: *
422: DO 200 I = 1, N
423: IF( WORK( I ).GT.SAFE2 ) THEN
424: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
425: ELSE
426: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
427: END IF
428: 200 CONTINUE
429: *
430: KASE = 0
431: 210 CONTINUE
432: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
433: $ KASE, ISAVE )
434: IF( KASE.NE.0 ) THEN
435: IF( KASE.EQ.1 ) THEN
436: *
437: * Multiply by diag(W)*inv(op(A)**T).
438: *
439: CALL DTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK( N+1 ),
440: $ 1 )
441: DO 220 I = 1, N
442: WORK( N+I ) = WORK( I )*WORK( N+I )
443: 220 CONTINUE
444: ELSE
445: *
446: * Multiply by inv(op(A))*diag(W).
447: *
448: DO 230 I = 1, N
449: WORK( N+I ) = WORK( I )*WORK( N+I )
450: 230 CONTINUE
451: CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ),
452: $ 1 )
453: END IF
454: GO TO 210
455: END IF
456: *
457: * Normalize error.
458: *
459: LSTRES = ZERO
460: DO 240 I = 1, N
461: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
462: 240 CONTINUE
463: IF( LSTRES.NE.ZERO )
464: $ FERR( J ) = FERR( J ) / LSTRES
465: *
466: 250 CONTINUE
467: *
468: RETURN
469: *
470: * End of DTRRFS
471: *
472: END
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