1: *> \brief \b DGBRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
22: * IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
23: * INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER TRANS
27: * INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
28: * ..
29: * .. Array Arguments ..
30: * INTEGER IPIV( * ), IWORK( * )
31: * DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
32: * $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> DGBRFS improves the computed solution to a system of linear
42: *> equations when the coefficient matrix is banded, and provides
43: *> error bounds and backward error estimates for the solution.
44: *> \endverbatim
45: *
46: * Arguments:
47: * ==========
48: *
49: *> \param[in] TRANS
50: *> \verbatim
51: *> TRANS is CHARACTER*1
52: *> Specifies the form of the system of equations:
53: *> = 'N': A * X = B (No transpose)
54: *> = 'T': A**T * X = B (Transpose)
55: *> = 'C': A**H * X = B (Conjugate transpose = Transpose)
56: *> \endverbatim
57: *>
58: *> \param[in] N
59: *> \verbatim
60: *> N is INTEGER
61: *> The order of the matrix A. N >= 0.
62: *> \endverbatim
63: *>
64: *> \param[in] KL
65: *> \verbatim
66: *> KL is INTEGER
67: *> The number of subdiagonals within the band of A. KL >= 0.
68: *> \endverbatim
69: *>
70: *> \param[in] KU
71: *> \verbatim
72: *> KU is INTEGER
73: *> The number of superdiagonals within the band of A. KU >= 0.
74: *> \endverbatim
75: *>
76: *> \param[in] NRHS
77: *> \verbatim
78: *> NRHS is INTEGER
79: *> The number of right hand sides, i.e., the number of columns
80: *> of the matrices B and X. NRHS >= 0.
81: *> \endverbatim
82: *>
83: *> \param[in] AB
84: *> \verbatim
85: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
86: *> The original band matrix A, stored in rows 1 to KL+KU+1.
87: *> The j-th column of A is stored in the j-th column of the
88: *> array AB as follows:
89: *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
90: *> \endverbatim
91: *>
92: *> \param[in] LDAB
93: *> \verbatim
94: *> LDAB is INTEGER
95: *> The leading dimension of the array AB. LDAB >= KL+KU+1.
96: *> \endverbatim
97: *>
98: *> \param[in] AFB
99: *> \verbatim
100: *> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
101: *> Details of the LU factorization of the band matrix A, as
102: *> computed by DGBTRF. U is stored as an upper triangular band
103: *> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
104: *> the multipliers used during the factorization are stored in
105: *> rows KL+KU+2 to 2*KL+KU+1.
106: *> \endverbatim
107: *>
108: *> \param[in] LDAFB
109: *> \verbatim
110: *> LDAFB is INTEGER
111: *> The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
112: *> \endverbatim
113: *>
114: *> \param[in] IPIV
115: *> \verbatim
116: *> IPIV is INTEGER array, dimension (N)
117: *> The pivot indices from DGBTRF; for 1<=i<=N, row i of the
118: *> matrix was interchanged with row IPIV(i).
119: *> \endverbatim
120: *>
121: *> \param[in] B
122: *> \verbatim
123: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
124: *> The right hand side matrix B.
125: *> \endverbatim
126: *>
127: *> \param[in] LDB
128: *> \verbatim
129: *> LDB is INTEGER
130: *> The leading dimension of the array B. LDB >= max(1,N).
131: *> \endverbatim
132: *>
133: *> \param[in,out] X
134: *> \verbatim
135: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
136: *> On entry, the solution matrix X, as computed by DGBTRS.
137: *> On exit, the improved solution matrix X.
138: *> \endverbatim
139: *>
140: *> \param[in] LDX
141: *> \verbatim
142: *> LDX is INTEGER
143: *> The leading dimension of the array X. LDX >= max(1,N).
144: *> \endverbatim
145: *>
146: *> \param[out] FERR
147: *> \verbatim
148: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
149: *> The estimated forward error bound for each solution vector
150: *> X(j) (the j-th column of the solution matrix X).
151: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
152: *> is an estimated upper bound for the magnitude of the largest
153: *> element in (X(j) - XTRUE) divided by the magnitude of the
154: *> largest element in X(j). The estimate is as reliable as
155: *> the estimate for RCOND, and is almost always a slight
156: *> overestimate of the true error.
157: *> \endverbatim
158: *>
159: *> \param[out] BERR
160: *> \verbatim
161: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
162: *> The componentwise relative backward error of each solution
163: *> vector X(j) (i.e., the smallest relative change in
164: *> any element of A or B that makes X(j) an exact solution).
165: *> \endverbatim
166: *>
167: *> \param[out] WORK
168: *> \verbatim
169: *> WORK is DOUBLE PRECISION array, dimension (3*N)
170: *> \endverbatim
171: *>
172: *> \param[out] IWORK
173: *> \verbatim
174: *> IWORK is INTEGER array, dimension (N)
175: *> \endverbatim
176: *>
177: *> \param[out] INFO
178: *> \verbatim
179: *> INFO is INTEGER
180: *> = 0: successful exit
181: *> < 0: if INFO = -i, the i-th argument had an illegal value
182: *> \endverbatim
183: *
184: *> \par Internal Parameters:
185: * =========================
186: *>
187: *> \verbatim
188: *> ITMAX is the maximum number of steps of iterative refinement.
189: *> \endverbatim
190: *
191: * Authors:
192: * ========
193: *
194: *> \author Univ. of Tennessee
195: *> \author Univ. of California Berkeley
196: *> \author Univ. of Colorado Denver
197: *> \author NAG Ltd.
198: *
199: *> \ingroup doubleGBcomputational
200: *
201: * =====================================================================
202: SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
203: $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
204: $ INFO )
205: *
206: * -- LAPACK computational routine --
207: * -- LAPACK is a software package provided by Univ. of Tennessee, --
208: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
209: *
210: * .. Scalar Arguments ..
211: CHARACTER TRANS
212: INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
213: * ..
214: * .. Array Arguments ..
215: INTEGER IPIV( * ), IWORK( * )
216: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
217: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
218: * ..
219: *
220: * =====================================================================
221: *
222: * .. Parameters ..
223: INTEGER ITMAX
224: PARAMETER ( ITMAX = 5 )
225: DOUBLE PRECISION ZERO
226: PARAMETER ( ZERO = 0.0D+0 )
227: DOUBLE PRECISION ONE
228: PARAMETER ( ONE = 1.0D+0 )
229: DOUBLE PRECISION TWO
230: PARAMETER ( TWO = 2.0D+0 )
231: DOUBLE PRECISION THREE
232: PARAMETER ( THREE = 3.0D+0 )
233: * ..
234: * .. Local Scalars ..
235: LOGICAL NOTRAN
236: CHARACTER TRANST
237: INTEGER COUNT, I, J, K, KASE, KK, NZ
238: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
239: * ..
240: * .. Local Arrays ..
241: INTEGER ISAVE( 3 )
242: * ..
243: * .. External Subroutines ..
244: EXTERNAL DAXPY, DCOPY, DGBMV, DGBTRS, DLACN2, XERBLA
245: * ..
246: * .. Intrinsic Functions ..
247: INTRINSIC ABS, MAX, MIN
248: * ..
249: * .. External Functions ..
250: LOGICAL LSAME
251: DOUBLE PRECISION DLAMCH
252: EXTERNAL LSAME, DLAMCH
253: * ..
254: * .. Executable Statements ..
255: *
256: * Test the input parameters.
257: *
258: INFO = 0
259: NOTRAN = LSAME( TRANS, 'N' )
260: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
261: $ LSAME( TRANS, 'C' ) ) THEN
262: INFO = -1
263: ELSE IF( N.LT.0 ) THEN
264: INFO = -2
265: ELSE IF( KL.LT.0 ) THEN
266: INFO = -3
267: ELSE IF( KU.LT.0 ) THEN
268: INFO = -4
269: ELSE IF( NRHS.LT.0 ) THEN
270: INFO = -5
271: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
272: INFO = -7
273: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
274: INFO = -9
275: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
276: INFO = -12
277: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
278: INFO = -14
279: END IF
280: IF( INFO.NE.0 ) THEN
281: CALL XERBLA( 'DGBRFS', -INFO )
282: RETURN
283: END IF
284: *
285: * Quick return if possible
286: *
287: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
288: DO 10 J = 1, NRHS
289: FERR( J ) = ZERO
290: BERR( J ) = ZERO
291: 10 CONTINUE
292: RETURN
293: END IF
294: *
295: IF( NOTRAN ) THEN
296: TRANST = 'T'
297: ELSE
298: TRANST = 'N'
299: END IF
300: *
301: * NZ = maximum number of nonzero elements in each row of A, plus 1
302: *
303: NZ = MIN( KL+KU+2, N+1 )
304: EPS = DLAMCH( 'Epsilon' )
305: SAFMIN = DLAMCH( 'Safe minimum' )
306: SAFE1 = NZ*SAFMIN
307: SAFE2 = SAFE1 / EPS
308: *
309: * Do for each right hand side
310: *
311: DO 140 J = 1, NRHS
312: *
313: COUNT = 1
314: LSTRES = THREE
315: 20 CONTINUE
316: *
317: * Loop until stopping criterion is satisfied.
318: *
319: * Compute residual R = B - op(A) * X,
320: * where op(A) = A, A**T, or A**H, depending on TRANS.
321: *
322: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
323: CALL DGBMV( TRANS, N, N, KL, KU, -ONE, AB, LDAB, X( 1, J ), 1,
324: $ ONE, WORK( N+1 ), 1 )
325: *
326: * Compute componentwise relative backward error from formula
327: *
328: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
329: *
330: * where abs(Z) is the componentwise absolute value of the matrix
331: * or vector Z. If the i-th component of the denominator is less
332: * than SAFE2, then SAFE1 is added to the i-th components of the
333: * numerator and denominator before dividing.
334: *
335: DO 30 I = 1, N
336: WORK( I ) = ABS( B( I, J ) )
337: 30 CONTINUE
338: *
339: * Compute abs(op(A))*abs(X) + abs(B).
340: *
341: IF( NOTRAN ) THEN
342: DO 50 K = 1, N
343: KK = KU + 1 - K
344: XK = ABS( X( K, J ) )
345: DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL )
346: WORK( I ) = WORK( I ) + ABS( AB( KK+I, K ) )*XK
347: 40 CONTINUE
348: 50 CONTINUE
349: ELSE
350: DO 70 K = 1, N
351: S = ZERO
352: KK = KU + 1 - K
353: DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL )
354: S = S + ABS( AB( KK+I, K ) )*ABS( X( I, J ) )
355: 60 CONTINUE
356: WORK( K ) = WORK( K ) + S
357: 70 CONTINUE
358: END IF
359: S = ZERO
360: DO 80 I = 1, N
361: IF( WORK( I ).GT.SAFE2 ) THEN
362: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
363: ELSE
364: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
365: $ ( WORK( I )+SAFE1 ) )
366: END IF
367: 80 CONTINUE
368: BERR( J ) = S
369: *
370: * Test stopping criterion. Continue iterating if
371: * 1) The residual BERR(J) is larger than machine epsilon, and
372: * 2) BERR(J) decreased by at least a factor of 2 during the
373: * last iteration, and
374: * 3) At most ITMAX iterations tried.
375: *
376: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
377: $ COUNT.LE.ITMAX ) THEN
378: *
379: * Update solution and try again.
380: *
381: CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
382: $ WORK( N+1 ), N, INFO )
383: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
384: LSTRES = BERR( J )
385: COUNT = COUNT + 1
386: GO TO 20
387: END IF
388: *
389: * Bound error from formula
390: *
391: * norm(X - XTRUE) / norm(X) .le. FERR =
392: * norm( abs(inv(op(A)))*
393: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
394: *
395: * where
396: * norm(Z) is the magnitude of the largest component of Z
397: * inv(op(A)) is the inverse of op(A)
398: * abs(Z) is the componentwise absolute value of the matrix or
399: * vector Z
400: * NZ is the maximum number of nonzeros in any row of A, plus 1
401: * EPS is machine epsilon
402: *
403: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
404: * is incremented by SAFE1 if the i-th component of
405: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
406: *
407: * Use DLACN2 to estimate the infinity-norm of the matrix
408: * inv(op(A)) * diag(W),
409: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
410: *
411: DO 90 I = 1, N
412: IF( WORK( I ).GT.SAFE2 ) THEN
413: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
414: ELSE
415: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
416: END IF
417: 90 CONTINUE
418: *
419: KASE = 0
420: 100 CONTINUE
421: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
422: $ KASE, ISAVE )
423: IF( KASE.NE.0 ) THEN
424: IF( KASE.EQ.1 ) THEN
425: *
426: * Multiply by diag(W)*inv(op(A)**T).
427: *
428: CALL DGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV,
429: $ WORK( N+1 ), N, INFO )
430: DO 110 I = 1, N
431: WORK( N+I ) = WORK( N+I )*WORK( I )
432: 110 CONTINUE
433: ELSE
434: *
435: * Multiply by inv(op(A))*diag(W).
436: *
437: DO 120 I = 1, N
438: WORK( N+I ) = WORK( N+I )*WORK( I )
439: 120 CONTINUE
440: CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
441: $ WORK( N+1 ), N, INFO )
442: END IF
443: GO TO 100
444: END IF
445: *
446: * Normalize error.
447: *
448: LSTRES = ZERO
449: DO 130 I = 1, N
450: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
451: 130 CONTINUE
452: IF( LSTRES.NE.ZERO )
453: $ FERR( J ) = FERR( J ) / LSTRES
454: *
455: 140 CONTINUE
456: *
457: RETURN
458: *
459: * End of DGBRFS
460: *
461: END
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