1: *> \brief \b ZTBRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZTBRFS + 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/ztbrfs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
22: * LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER DIAG, TRANS, UPLO
26: * INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
30: * COMPLEX*16 AB( LDAB, * ), B( LDB, * ), WORK( * ),
31: * $ X( LDX, * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> ZTBRFS provides error bounds and backward error estimates for the
41: *> solution to a system of linear equations with a triangular band
42: *> coefficient matrix.
43: *>
44: *> The solution matrix X must be computed by ZTBTRS or some other
45: *> means before entering this routine. ZTBRFS 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] KD
82: *> \verbatim
83: *> KD is INTEGER
84: *> The number of superdiagonals or subdiagonals of the
85: *> triangular band matrix A. KD >= 0.
86: *> \endverbatim
87: *>
88: *> \param[in] NRHS
89: *> \verbatim
90: *> NRHS is INTEGER
91: *> The number of right hand sides, i.e., the number of columns
92: *> of the matrices B and X. NRHS >= 0.
93: *> \endverbatim
94: *>
95: *> \param[in] AB
96: *> \verbatim
97: *> AB is COMPLEX*16 array, dimension (LDAB,N)
98: *> The upper or lower triangular band matrix A, stored in the
99: *> first kd+1 rows of the array. The j-th column of A is stored
100: *> in the j-th column of the array AB as follows:
101: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
102: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
103: *> If DIAG = 'U', the diagonal elements of A are not referenced
104: *> and are assumed to be 1.
105: *> \endverbatim
106: *>
107: *> \param[in] LDAB
108: *> \verbatim
109: *> LDAB is INTEGER
110: *> The leading dimension of the array AB. LDAB >= KD+1.
111: *> \endverbatim
112: *>
113: *> \param[in] B
114: *> \verbatim
115: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
116: *> The right hand side matrix B.
117: *> \endverbatim
118: *>
119: *> \param[in] LDB
120: *> \verbatim
121: *> LDB is INTEGER
122: *> The leading dimension of the array B. LDB >= max(1,N).
123: *> \endverbatim
124: *>
125: *> \param[in] X
126: *> \verbatim
127: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
128: *> The solution matrix X.
129: *> \endverbatim
130: *>
131: *> \param[in] LDX
132: *> \verbatim
133: *> LDX is INTEGER
134: *> The leading dimension of the array X. LDX >= max(1,N).
135: *> \endverbatim
136: *>
137: *> \param[out] FERR
138: *> \verbatim
139: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
140: *> The estimated forward error bound for each solution vector
141: *> X(j) (the j-th column of the solution matrix X).
142: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
143: *> is an estimated upper bound for the magnitude of the largest
144: *> element in (X(j) - XTRUE) divided by the magnitude of the
145: *> largest element in X(j). The estimate is as reliable as
146: *> the estimate for RCOND, and is almost always a slight
147: *> overestimate of the true error.
148: *> \endverbatim
149: *>
150: *> \param[out] BERR
151: *> \verbatim
152: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
153: *> The componentwise relative backward error of each solution
154: *> vector X(j) (i.e., the smallest relative change in
155: *> any element of A or B that makes X(j) an exact solution).
156: *> \endverbatim
157: *>
158: *> \param[out] WORK
159: *> \verbatim
160: *> WORK is COMPLEX*16 array, dimension (2*N)
161: *> \endverbatim
162: *>
163: *> \param[out] RWORK
164: *> \verbatim
165: *> RWORK is DOUBLE PRECISION array, dimension (N)
166: *> \endverbatim
167: *>
168: *> \param[out] INFO
169: *> \verbatim
170: *> INFO is INTEGER
171: *> = 0: successful exit
172: *> < 0: if INFO = -i, the i-th argument had an illegal value
173: *> \endverbatim
174: *
175: * Authors:
176: * ========
177: *
178: *> \author Univ. of Tennessee
179: *> \author Univ. of California Berkeley
180: *> \author Univ. of Colorado Denver
181: *> \author NAG Ltd.
182: *
183: *> \date November 2011
184: *
185: *> \ingroup complex16OTHERcomputational
186: *
187: * =====================================================================
188: SUBROUTINE ZTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
189: $ LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
190: *
191: * -- LAPACK computational routine (version 3.4.0) --
192: * -- LAPACK is a software package provided by Univ. of Tennessee, --
193: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
194: * November 2011
195: *
196: * .. Scalar Arguments ..
197: CHARACTER DIAG, TRANS, UPLO
198: INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS
199: * ..
200: * .. Array Arguments ..
201: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
202: COMPLEX*16 AB( LDAB, * ), B( LDB, * ), WORK( * ),
203: $ X( LDX, * )
204: * ..
205: *
206: * =====================================================================
207: *
208: * .. Parameters ..
209: DOUBLE PRECISION ZERO
210: PARAMETER ( ZERO = 0.0D+0 )
211: COMPLEX*16 ONE
212: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
213: * ..
214: * .. Local Scalars ..
215: LOGICAL NOTRAN, NOUNIT, UPPER
216: CHARACTER TRANSN, TRANST
217: INTEGER I, J, K, KASE, NZ
218: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
219: COMPLEX*16 ZDUM
220: * ..
221: * .. Local Arrays ..
222: INTEGER ISAVE( 3 )
223: * ..
224: * .. External Subroutines ..
225: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZTBMV, ZTBSV
226: * ..
227: * .. Intrinsic Functions ..
228: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
229: * ..
230: * .. External Functions ..
231: LOGICAL LSAME
232: DOUBLE PRECISION DLAMCH
233: EXTERNAL LSAME, DLAMCH
234: * ..
235: * .. Statement Functions ..
236: DOUBLE PRECISION CABS1
237: * ..
238: * .. Statement Function definitions ..
239: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
240: * ..
241: * .. Executable Statements ..
242: *
243: * Test the input parameters.
244: *
245: INFO = 0
246: UPPER = LSAME( UPLO, 'U' )
247: NOTRAN = LSAME( TRANS, 'N' )
248: NOUNIT = LSAME( DIAG, 'N' )
249: *
250: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
251: INFO = -1
252: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
253: $ LSAME( TRANS, 'C' ) ) THEN
254: INFO = -2
255: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
256: INFO = -3
257: ELSE IF( N.LT.0 ) THEN
258: INFO = -4
259: ELSE IF( KD.LT.0 ) THEN
260: INFO = -5
261: ELSE IF( NRHS.LT.0 ) THEN
262: INFO = -6
263: ELSE IF( LDAB.LT.KD+1 ) THEN
264: INFO = -8
265: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
266: INFO = -10
267: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
268: INFO = -12
269: END IF
270: IF( INFO.NE.0 ) THEN
271: CALL XERBLA( 'ZTBRFS', -INFO )
272: RETURN
273: END IF
274: *
275: * Quick return if possible
276: *
277: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
278: DO 10 J = 1, NRHS
279: FERR( J ) = ZERO
280: BERR( J ) = ZERO
281: 10 CONTINUE
282: RETURN
283: END IF
284: *
285: IF( NOTRAN ) THEN
286: TRANSN = 'N'
287: TRANST = 'C'
288: ELSE
289: TRANSN = 'C'
290: TRANST = 'N'
291: END IF
292: *
293: * NZ = maximum number of nonzero elements in each row of A, plus 1
294: *
295: NZ = KD + 2
296: EPS = DLAMCH( 'Epsilon' )
297: SAFMIN = DLAMCH( 'Safe minimum' )
298: SAFE1 = NZ*SAFMIN
299: SAFE2 = SAFE1 / EPS
300: *
301: * Do for each right hand side
302: *
303: DO 250 J = 1, NRHS
304: *
305: * Compute residual R = B - op(A) * X,
306: * where op(A) = A, A**T, or A**H, depending on TRANS.
307: *
308: CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
309: CALL ZTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 )
310: CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
311: *
312: * Compute componentwise relative backward error from formula
313: *
314: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
315: *
316: * where abs(Z) is the componentwise absolute value of the matrix
317: * or vector Z. If the i-th component of the denominator is less
318: * than SAFE2, then SAFE1 is added to the i-th components of the
319: * numerator and denominator before dividing.
320: *
321: DO 20 I = 1, N
322: RWORK( I ) = CABS1( B( I, J ) )
323: 20 CONTINUE
324: *
325: IF( NOTRAN ) THEN
326: *
327: * Compute abs(A)*abs(X) + abs(B).
328: *
329: IF( UPPER ) THEN
330: IF( NOUNIT ) THEN
331: DO 40 K = 1, N
332: XK = CABS1( X( K, J ) )
333: DO 30 I = MAX( 1, K-KD ), K
334: RWORK( I ) = RWORK( I ) +
335: $ CABS1( AB( KD+1+I-K, K ) )*XK
336: 30 CONTINUE
337: 40 CONTINUE
338: ELSE
339: DO 60 K = 1, N
340: XK = CABS1( X( K, J ) )
341: DO 50 I = MAX( 1, K-KD ), K - 1
342: RWORK( I ) = RWORK( I ) +
343: $ CABS1( AB( KD+1+I-K, K ) )*XK
344: 50 CONTINUE
345: RWORK( K ) = RWORK( K ) + XK
346: 60 CONTINUE
347: END IF
348: ELSE
349: IF( NOUNIT ) THEN
350: DO 80 K = 1, N
351: XK = CABS1( X( K, J ) )
352: DO 70 I = K, MIN( N, K+KD )
353: RWORK( I ) = RWORK( I ) +
354: $ CABS1( AB( 1+I-K, K ) )*XK
355: 70 CONTINUE
356: 80 CONTINUE
357: ELSE
358: DO 100 K = 1, N
359: XK = CABS1( X( K, J ) )
360: DO 90 I = K + 1, MIN( N, K+KD )
361: RWORK( I ) = RWORK( I ) +
362: $ CABS1( AB( 1+I-K, K ) )*XK
363: 90 CONTINUE
364: RWORK( K ) = RWORK( K ) + XK
365: 100 CONTINUE
366: END IF
367: END IF
368: ELSE
369: *
370: * Compute abs(A**H)*abs(X) + abs(B).
371: *
372: IF( UPPER ) THEN
373: IF( NOUNIT ) THEN
374: DO 120 K = 1, N
375: S = ZERO
376: DO 110 I = MAX( 1, K-KD ), K
377: S = S + CABS1( AB( KD+1+I-K, K ) )*
378: $ CABS1( X( I, J ) )
379: 110 CONTINUE
380: RWORK( K ) = RWORK( K ) + S
381: 120 CONTINUE
382: ELSE
383: DO 140 K = 1, N
384: S = CABS1( X( K, J ) )
385: DO 130 I = MAX( 1, K-KD ), K - 1
386: S = S + CABS1( AB( KD+1+I-K, K ) )*
387: $ CABS1( X( I, J ) )
388: 130 CONTINUE
389: RWORK( K ) = RWORK( K ) + S
390: 140 CONTINUE
391: END IF
392: ELSE
393: IF( NOUNIT ) THEN
394: DO 160 K = 1, N
395: S = ZERO
396: DO 150 I = K, MIN( N, K+KD )
397: S = S + CABS1( AB( 1+I-K, K ) )*
398: $ CABS1( X( I, J ) )
399: 150 CONTINUE
400: RWORK( K ) = RWORK( K ) + S
401: 160 CONTINUE
402: ELSE
403: DO 180 K = 1, N
404: S = CABS1( X( K, J ) )
405: DO 170 I = K + 1, MIN( N, K+KD )
406: S = S + CABS1( AB( 1+I-K, K ) )*
407: $ CABS1( X( I, J ) )
408: 170 CONTINUE
409: RWORK( K ) = RWORK( K ) + S
410: 180 CONTINUE
411: END IF
412: END IF
413: END IF
414: S = ZERO
415: DO 190 I = 1, N
416: IF( RWORK( I ).GT.SAFE2 ) THEN
417: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
418: ELSE
419: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
420: $ ( RWORK( I )+SAFE1 ) )
421: END IF
422: 190 CONTINUE
423: BERR( J ) = S
424: *
425: * Bound error from formula
426: *
427: * norm(X - XTRUE) / norm(X) .le. FERR =
428: * norm( abs(inv(op(A)))*
429: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
430: *
431: * where
432: * norm(Z) is the magnitude of the largest component of Z
433: * inv(op(A)) is the inverse of op(A)
434: * abs(Z) is the componentwise absolute value of the matrix or
435: * vector Z
436: * NZ is the maximum number of nonzeros in any row of A, plus 1
437: * EPS is machine epsilon
438: *
439: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
440: * is incremented by SAFE1 if the i-th component of
441: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
442: *
443: * Use ZLACN2 to estimate the infinity-norm of the matrix
444: * inv(op(A)) * diag(W),
445: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
446: *
447: DO 200 I = 1, N
448: IF( RWORK( I ).GT.SAFE2 ) THEN
449: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
450: ELSE
451: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
452: $ SAFE1
453: END IF
454: 200 CONTINUE
455: *
456: KASE = 0
457: 210 CONTINUE
458: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
459: IF( KASE.NE.0 ) THEN
460: IF( KASE.EQ.1 ) THEN
461: *
462: * Multiply by diag(W)*inv(op(A)**H).
463: *
464: CALL ZTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB, WORK,
465: $ 1 )
466: DO 220 I = 1, N
467: WORK( I ) = RWORK( I )*WORK( I )
468: 220 CONTINUE
469: ELSE
470: *
471: * Multiply by inv(op(A))*diag(W).
472: *
473: DO 230 I = 1, N
474: WORK( I ) = RWORK( I )*WORK( I )
475: 230 CONTINUE
476: CALL ZTBSV( UPLO, TRANSN, DIAG, N, KD, AB, LDAB, WORK,
477: $ 1 )
478: END IF
479: GO TO 210
480: END IF
481: *
482: * Normalize error.
483: *
484: LSTRES = ZERO
485: DO 240 I = 1, N
486: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
487: 240 CONTINUE
488: IF( LSTRES.NE.ZERO )
489: $ FERR( J ) = FERR( J ) / LSTRES
490: *
491: 250 CONTINUE
492: *
493: RETURN
494: *
495: * End of ZTBRFS
496: *
497: END
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