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: *> \ingroup complex16OTHERcomputational
184: *
185: * =====================================================================
186: SUBROUTINE ZTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
187: $ LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
188: *
189: * -- LAPACK computational routine --
190: * -- LAPACK is a software package provided by Univ. of Tennessee, --
191: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
192: *
193: * .. Scalar Arguments ..
194: CHARACTER DIAG, TRANS, UPLO
195: INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS
196: * ..
197: * .. Array Arguments ..
198: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
199: COMPLEX*16 AB( LDAB, * ), B( LDB, * ), WORK( * ),
200: $ X( LDX, * )
201: * ..
202: *
203: * =====================================================================
204: *
205: * .. Parameters ..
206: DOUBLE PRECISION ZERO
207: PARAMETER ( ZERO = 0.0D+0 )
208: COMPLEX*16 ONE
209: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
210: * ..
211: * .. Local Scalars ..
212: LOGICAL NOTRAN, NOUNIT, UPPER
213: CHARACTER TRANSN, TRANST
214: INTEGER I, J, K, KASE, NZ
215: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
216: COMPLEX*16 ZDUM
217: * ..
218: * .. Local Arrays ..
219: INTEGER ISAVE( 3 )
220: * ..
221: * .. External Subroutines ..
222: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZTBMV, ZTBSV
223: * ..
224: * .. Intrinsic Functions ..
225: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
226: * ..
227: * .. External Functions ..
228: LOGICAL LSAME
229: DOUBLE PRECISION DLAMCH
230: EXTERNAL LSAME, DLAMCH
231: * ..
232: * .. Statement Functions ..
233: DOUBLE PRECISION CABS1
234: * ..
235: * .. Statement Function definitions ..
236: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
237: * ..
238: * .. Executable Statements ..
239: *
240: * Test the input parameters.
241: *
242: INFO = 0
243: UPPER = LSAME( UPLO, 'U' )
244: NOTRAN = LSAME( TRANS, 'N' )
245: NOUNIT = LSAME( DIAG, 'N' )
246: *
247: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
248: INFO = -1
249: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
250: $ LSAME( TRANS, 'C' ) ) THEN
251: INFO = -2
252: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
253: INFO = -3
254: ELSE IF( N.LT.0 ) THEN
255: INFO = -4
256: ELSE IF( KD.LT.0 ) THEN
257: INFO = -5
258: ELSE IF( NRHS.LT.0 ) THEN
259: INFO = -6
260: ELSE IF( LDAB.LT.KD+1 ) THEN
261: INFO = -8
262: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
263: INFO = -10
264: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
265: INFO = -12
266: END IF
267: IF( INFO.NE.0 ) THEN
268: CALL XERBLA( 'ZTBRFS', -INFO )
269: RETURN
270: END IF
271: *
272: * Quick return if possible
273: *
274: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
275: DO 10 J = 1, NRHS
276: FERR( J ) = ZERO
277: BERR( J ) = ZERO
278: 10 CONTINUE
279: RETURN
280: END IF
281: *
282: IF( NOTRAN ) THEN
283: TRANSN = 'N'
284: TRANST = 'C'
285: ELSE
286: TRANSN = 'C'
287: TRANST = 'N'
288: END IF
289: *
290: * NZ = maximum number of nonzero elements in each row of A, plus 1
291: *
292: NZ = KD + 2
293: EPS = DLAMCH( 'Epsilon' )
294: SAFMIN = DLAMCH( 'Safe minimum' )
295: SAFE1 = NZ*SAFMIN
296: SAFE2 = SAFE1 / EPS
297: *
298: * Do for each right hand side
299: *
300: DO 250 J = 1, NRHS
301: *
302: * Compute residual R = B - op(A) * X,
303: * where op(A) = A, A**T, or A**H, depending on TRANS.
304: *
305: CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
306: CALL ZTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 )
307: CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
308: *
309: * Compute componentwise relative backward error from formula
310: *
311: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
312: *
313: * where abs(Z) is the componentwise absolute value of the matrix
314: * or vector Z. If the i-th component of the denominator is less
315: * than SAFE2, then SAFE1 is added to the i-th components of the
316: * numerator and denominator before dividing.
317: *
318: DO 20 I = 1, N
319: RWORK( I ) = CABS1( B( I, J ) )
320: 20 CONTINUE
321: *
322: IF( NOTRAN ) THEN
323: *
324: * Compute abs(A)*abs(X) + abs(B).
325: *
326: IF( UPPER ) THEN
327: IF( NOUNIT ) THEN
328: DO 40 K = 1, N
329: XK = CABS1( X( K, J ) )
330: DO 30 I = MAX( 1, K-KD ), K
331: RWORK( I ) = RWORK( I ) +
332: $ CABS1( AB( KD+1+I-K, K ) )*XK
333: 30 CONTINUE
334: 40 CONTINUE
335: ELSE
336: DO 60 K = 1, N
337: XK = CABS1( X( K, J ) )
338: DO 50 I = MAX( 1, K-KD ), K - 1
339: RWORK( I ) = RWORK( I ) +
340: $ CABS1( AB( KD+1+I-K, K ) )*XK
341: 50 CONTINUE
342: RWORK( K ) = RWORK( K ) + XK
343: 60 CONTINUE
344: END IF
345: ELSE
346: IF( NOUNIT ) THEN
347: DO 80 K = 1, N
348: XK = CABS1( X( K, J ) )
349: DO 70 I = K, MIN( N, K+KD )
350: RWORK( I ) = RWORK( I ) +
351: $ CABS1( AB( 1+I-K, K ) )*XK
352: 70 CONTINUE
353: 80 CONTINUE
354: ELSE
355: DO 100 K = 1, N
356: XK = CABS1( X( K, J ) )
357: DO 90 I = K + 1, MIN( N, K+KD )
358: RWORK( I ) = RWORK( I ) +
359: $ CABS1( AB( 1+I-K, K ) )*XK
360: 90 CONTINUE
361: RWORK( K ) = RWORK( K ) + XK
362: 100 CONTINUE
363: END IF
364: END IF
365: ELSE
366: *
367: * Compute abs(A**H)*abs(X) + abs(B).
368: *
369: IF( UPPER ) THEN
370: IF( NOUNIT ) THEN
371: DO 120 K = 1, N
372: S = ZERO
373: DO 110 I = MAX( 1, K-KD ), K
374: S = S + CABS1( AB( KD+1+I-K, K ) )*
375: $ CABS1( X( I, J ) )
376: 110 CONTINUE
377: RWORK( K ) = RWORK( K ) + S
378: 120 CONTINUE
379: ELSE
380: DO 140 K = 1, N
381: S = CABS1( X( K, J ) )
382: DO 130 I = MAX( 1, K-KD ), K - 1
383: S = S + CABS1( AB( KD+1+I-K, K ) )*
384: $ CABS1( X( I, J ) )
385: 130 CONTINUE
386: RWORK( K ) = RWORK( K ) + S
387: 140 CONTINUE
388: END IF
389: ELSE
390: IF( NOUNIT ) THEN
391: DO 160 K = 1, N
392: S = ZERO
393: DO 150 I = K, MIN( N, K+KD )
394: S = S + CABS1( AB( 1+I-K, K ) )*
395: $ CABS1( X( I, J ) )
396: 150 CONTINUE
397: RWORK( K ) = RWORK( K ) + S
398: 160 CONTINUE
399: ELSE
400: DO 180 K = 1, N
401: S = CABS1( X( K, J ) )
402: DO 170 I = K + 1, MIN( N, K+KD )
403: S = S + CABS1( AB( 1+I-K, K ) )*
404: $ CABS1( X( I, J ) )
405: 170 CONTINUE
406: RWORK( K ) = RWORK( K ) + S
407: 180 CONTINUE
408: END IF
409: END IF
410: END IF
411: S = ZERO
412: DO 190 I = 1, N
413: IF( RWORK( I ).GT.SAFE2 ) THEN
414: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
415: ELSE
416: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
417: $ ( RWORK( I )+SAFE1 ) )
418: END IF
419: 190 CONTINUE
420: BERR( J ) = S
421: *
422: * Bound error from formula
423: *
424: * norm(X - XTRUE) / norm(X) .le. FERR =
425: * norm( abs(inv(op(A)))*
426: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
427: *
428: * where
429: * norm(Z) is the magnitude of the largest component of Z
430: * inv(op(A)) is the inverse of op(A)
431: * abs(Z) is the componentwise absolute value of the matrix or
432: * vector Z
433: * NZ is the maximum number of nonzeros in any row of A, plus 1
434: * EPS is machine epsilon
435: *
436: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
437: * is incremented by SAFE1 if the i-th component of
438: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
439: *
440: * Use ZLACN2 to estimate the infinity-norm of the matrix
441: * inv(op(A)) * diag(W),
442: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
443: *
444: DO 200 I = 1, N
445: IF( RWORK( I ).GT.SAFE2 ) THEN
446: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
447: ELSE
448: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
449: $ SAFE1
450: END IF
451: 200 CONTINUE
452: *
453: KASE = 0
454: 210 CONTINUE
455: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
456: IF( KASE.NE.0 ) THEN
457: IF( KASE.EQ.1 ) THEN
458: *
459: * Multiply by diag(W)*inv(op(A)**H).
460: *
461: CALL ZTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB, WORK,
462: $ 1 )
463: DO 220 I = 1, N
464: WORK( I ) = RWORK( I )*WORK( I )
465: 220 CONTINUE
466: ELSE
467: *
468: * Multiply by inv(op(A))*diag(W).
469: *
470: DO 230 I = 1, N
471: WORK( I ) = RWORK( I )*WORK( I )
472: 230 CONTINUE
473: CALL ZTBSV( UPLO, TRANSN, DIAG, N, KD, AB, LDAB, WORK,
474: $ 1 )
475: END IF
476: GO TO 210
477: END IF
478: *
479: * Normalize error.
480: *
481: LSTRES = ZERO
482: DO 240 I = 1, N
483: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
484: 240 CONTINUE
485: IF( LSTRES.NE.ZERO )
486: $ FERR( J ) = FERR( J ) / LSTRES
487: *
488: 250 CONTINUE
489: *
490: RETURN
491: *
492: * End of ZTBRFS
493: *
494: END
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