Annotation of rpl/lapack/lapack/dla_gbrfsx_extended.f, revision 1.10
1.8 bertrand 1: *> \brief \b DLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
1.5 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLA_GBRFSX_EXTENDED + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dla_gbrfsx_extended.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dla_gbrfsx_extended.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dla_gbrfsx_extended.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
22: * NRHS, AB, LDAB, AFB, LDAFB, IPIV,
23: * COLEQU, C, B, LDB, Y, LDY,
24: * BERR_OUT, N_NORMS, ERR_BNDS_NORM,
25: * ERR_BNDS_COMP, RES, AYB, DY,
26: * Y_TAIL, RCOND, ITHRESH, RTHRESH,
27: * DZ_UB, IGNORE_CWISE, INFO )
28: *
29: * .. Scalar Arguments ..
30: * INTEGER INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
31: * $ PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
32: * LOGICAL COLEQU, IGNORE_CWISE
33: * DOUBLE PRECISION RTHRESH, DZ_UB
34: * ..
35: * .. Array Arguments ..
36: * INTEGER IPIV( * )
37: * DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
38: * $ Y( LDY, * ), RES(*), DY(*), Y_TAIL(*)
39: * DOUBLE PRECISION C( * ), AYB(*), RCOND, BERR_OUT(*),
40: * $ ERR_BNDS_NORM( NRHS, * ),
41: * $ ERR_BNDS_COMP( NRHS, * )
42: * ..
43: *
44: *
45: *> \par Purpose:
46: * =============
47: *>
48: *> \verbatim
49: *>
50: *>
51: *> DLA_GBRFSX_EXTENDED improves the computed solution to a system of
52: *> linear equations by performing extra-precise iterative refinement
53: *> and provides error bounds and backward error estimates for the solution.
54: *> This subroutine is called by DGBRFSX to perform iterative refinement.
55: *> In addition to normwise error bound, the code provides maximum
56: *> componentwise error bound if possible. See comments for ERR_BNDS_NORM
57: *> and ERR_BNDS_COMP for details of the error bounds. Note that this
58: *> subroutine is only resonsible for setting the second fields of
59: *> ERR_BNDS_NORM and ERR_BNDS_COMP.
60: *> \endverbatim
61: *
62: * Arguments:
63: * ==========
64: *
65: *> \param[in] PREC_TYPE
66: *> \verbatim
67: *> PREC_TYPE is INTEGER
68: *> Specifies the intermediate precision to be used in refinement.
69: *> The value is defined by ILAPREC(P) where P is a CHARACTER and
70: *> P = 'S': Single
71: *> = 'D': Double
72: *> = 'I': Indigenous
73: *> = 'X', 'E': Extra
74: *> \endverbatim
75: *>
76: *> \param[in] TRANS_TYPE
77: *> \verbatim
78: *> TRANS_TYPE is INTEGER
79: *> Specifies the transposition operation on A.
80: *> The value is defined by ILATRANS(T) where T is a CHARACTER and
81: *> T = 'N': No transpose
82: *> = 'T': Transpose
83: *> = 'C': Conjugate transpose
84: *> \endverbatim
85: *>
86: *> \param[in] N
87: *> \verbatim
88: *> N is INTEGER
89: *> The number of linear equations, i.e., the order of the
90: *> matrix A. N >= 0.
91: *> \endverbatim
92: *>
93: *> \param[in] KL
94: *> \verbatim
95: *> KL is INTEGER
96: *> The number of subdiagonals within the band of A. KL >= 0.
97: *> \endverbatim
98: *>
99: *> \param[in] KU
100: *> \verbatim
101: *> KU is INTEGER
102: *> The number of superdiagonals within the band of A. KU >= 0
103: *> \endverbatim
104: *>
105: *> \param[in] NRHS
106: *> \verbatim
107: *> NRHS is INTEGER
108: *> The number of right-hand-sides, i.e., the number of columns of the
109: *> matrix B.
110: *> \endverbatim
111: *>
112: *> \param[in] AB
113: *> \verbatim
114: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
115: *> On entry, the N-by-N matrix AB.
116: *> \endverbatim
117: *>
118: *> \param[in] LDAB
119: *> \verbatim
120: *> LDAB is INTEGER
121: *> The leading dimension of the array AB. LDBA >= max(1,N).
122: *> \endverbatim
123: *>
124: *> \param[in] AFB
125: *> \verbatim
126: *> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
127: *> The factors L and U from the factorization
128: *> A = P*L*U as computed by DGBTRF.
129: *> \endverbatim
130: *>
131: *> \param[in] LDAFB
132: *> \verbatim
133: *> LDAFB is INTEGER
134: *> The leading dimension of the array AF. LDAFB >= max(1,N).
135: *> \endverbatim
136: *>
137: *> \param[in] IPIV
138: *> \verbatim
139: *> IPIV is INTEGER array, dimension (N)
140: *> The pivot indices from the factorization A = P*L*U
141: *> as computed by DGBTRF; row i of the matrix was interchanged
142: *> with row IPIV(i).
143: *> \endverbatim
144: *>
145: *> \param[in] COLEQU
146: *> \verbatim
147: *> COLEQU is LOGICAL
148: *> If .TRUE. then column equilibration was done to A before calling
149: *> this routine. This is needed to compute the solution and error
150: *> bounds correctly.
151: *> \endverbatim
152: *>
153: *> \param[in] C
154: *> \verbatim
155: *> C is DOUBLE PRECISION array, dimension (N)
156: *> The column scale factors for A. If COLEQU = .FALSE., C
157: *> is not accessed. If C is input, each element of C should be a power
158: *> of the radix to ensure a reliable solution and error estimates.
159: *> Scaling by powers of the radix does not cause rounding errors unless
160: *> the result underflows or overflows. Rounding errors during scaling
161: *> lead to refining with a matrix that is not equivalent to the
162: *> input matrix, producing error estimates that may not be
163: *> reliable.
164: *> \endverbatim
165: *>
166: *> \param[in] B
167: *> \verbatim
168: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
169: *> The right-hand-side matrix B.
170: *> \endverbatim
171: *>
172: *> \param[in] LDB
173: *> \verbatim
174: *> LDB is INTEGER
175: *> The leading dimension of the array B. LDB >= max(1,N).
176: *> \endverbatim
177: *>
178: *> \param[in,out] Y
179: *> \verbatim
180: *> Y is DOUBLE PRECISION array, dimension
181: *> (LDY,NRHS)
182: *> On entry, the solution matrix X, as computed by DGBTRS.
183: *> On exit, the improved solution matrix Y.
184: *> \endverbatim
185: *>
186: *> \param[in] LDY
187: *> \verbatim
188: *> LDY is INTEGER
189: *> The leading dimension of the array Y. LDY >= max(1,N).
190: *> \endverbatim
191: *>
192: *> \param[out] BERR_OUT
193: *> \verbatim
194: *> BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
195: *> On exit, BERR_OUT(j) contains the componentwise relative backward
196: *> error for right-hand-side j from the formula
197: *> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
198: *> where abs(Z) is the componentwise absolute value of the matrix
199: *> or vector Z. This is computed by DLA_LIN_BERR.
200: *> \endverbatim
201: *>
202: *> \param[in] N_NORMS
203: *> \verbatim
204: *> N_NORMS is INTEGER
205: *> Determines which error bounds to return (see ERR_BNDS_NORM
206: *> and ERR_BNDS_COMP).
207: *> If N_NORMS >= 1 return normwise error bounds.
208: *> If N_NORMS >= 2 return componentwise error bounds.
209: *> \endverbatim
210: *>
211: *> \param[in,out] ERR_BNDS_NORM
212: *> \verbatim
213: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension
214: *> (NRHS, N_ERR_BNDS)
215: *> For each right-hand side, this array contains information about
216: *> various error bounds and condition numbers corresponding to the
217: *> normwise relative error, which is defined as follows:
218: *>
219: *> Normwise relative error in the ith solution vector:
220: *> max_j (abs(XTRUE(j,i) - X(j,i)))
221: *> ------------------------------
222: *> max_j abs(X(j,i))
223: *>
224: *> The array is indexed by the type of error information as described
225: *> below. There currently are up to three pieces of information
226: *> returned.
227: *>
228: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
229: *> right-hand side.
230: *>
231: *> The second index in ERR_BNDS_NORM(:,err) contains the following
232: *> three fields:
233: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
234: *> reciprocal condition number is less than the threshold
235: *> sqrt(n) * slamch('Epsilon').
236: *>
237: *> err = 2 "Guaranteed" error bound: The estimated forward error,
238: *> almost certainly within a factor of 10 of the true error
239: *> so long as the next entry is greater than the threshold
240: *> sqrt(n) * slamch('Epsilon'). This error bound should only
241: *> be trusted if the previous boolean is true.
242: *>
243: *> err = 3 Reciprocal condition number: Estimated normwise
244: *> reciprocal condition number. Compared with the threshold
245: *> sqrt(n) * slamch('Epsilon') to determine if the error
246: *> estimate is "guaranteed". These reciprocal condition
247: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
248: *> appropriately scaled matrix Z.
249: *> Let Z = S*A, where S scales each row by a power of the
250: *> radix so all absolute row sums of Z are approximately 1.
251: *>
252: *> This subroutine is only responsible for setting the second field
253: *> above.
254: *> See Lapack Working Note 165 for further details and extra
255: *> cautions.
256: *> \endverbatim
257: *>
258: *> \param[in,out] ERR_BNDS_COMP
259: *> \verbatim
260: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension
261: *> (NRHS, N_ERR_BNDS)
262: *> For each right-hand side, this array contains information about
263: *> various error bounds and condition numbers corresponding to the
264: *> componentwise relative error, which is defined as follows:
265: *>
266: *> Componentwise relative error in the ith solution vector:
267: *> abs(XTRUE(j,i) - X(j,i))
268: *> max_j ----------------------
269: *> abs(X(j,i))
270: *>
271: *> The array is indexed by the right-hand side i (on which the
272: *> componentwise relative error depends), and the type of error
273: *> information as described below. There currently are up to three
274: *> pieces of information returned for each right-hand side. If
275: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
276: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
277: *> the first (:,N_ERR_BNDS) entries are returned.
278: *>
279: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
280: *> right-hand side.
281: *>
282: *> The second index in ERR_BNDS_COMP(:,err) contains the following
283: *> three fields:
284: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
285: *> reciprocal condition number is less than the threshold
286: *> sqrt(n) * slamch('Epsilon').
287: *>
288: *> err = 2 "Guaranteed" error bound: The estimated forward error,
289: *> almost certainly within a factor of 10 of the true error
290: *> so long as the next entry is greater than the threshold
291: *> sqrt(n) * slamch('Epsilon'). This error bound should only
292: *> be trusted if the previous boolean is true.
293: *>
294: *> err = 3 Reciprocal condition number: Estimated componentwise
295: *> reciprocal condition number. Compared with the threshold
296: *> sqrt(n) * slamch('Epsilon') to determine if the error
297: *> estimate is "guaranteed". These reciprocal condition
298: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
299: *> appropriately scaled matrix Z.
300: *> Let Z = S*(A*diag(x)), where x is the solution for the
301: *> current right-hand side and S scales each row of
302: *> A*diag(x) by a power of the radix so all absolute row
303: *> sums of Z are approximately 1.
304: *>
305: *> This subroutine is only responsible for setting the second field
306: *> above.
307: *> See Lapack Working Note 165 for further details and extra
308: *> cautions.
309: *> \endverbatim
310: *>
311: *> \param[in] RES
312: *> \verbatim
313: *> RES is DOUBLE PRECISION array, dimension (N)
314: *> Workspace to hold the intermediate residual.
315: *> \endverbatim
316: *>
317: *> \param[in] AYB
318: *> \verbatim
319: *> AYB is DOUBLE PRECISION array, dimension (N)
320: *> Workspace. This can be the same workspace passed for Y_TAIL.
321: *> \endverbatim
322: *>
323: *> \param[in] DY
324: *> \verbatim
325: *> DY is DOUBLE PRECISION array, dimension (N)
326: *> Workspace to hold the intermediate solution.
327: *> \endverbatim
328: *>
329: *> \param[in] Y_TAIL
330: *> \verbatim
331: *> Y_TAIL is DOUBLE PRECISION array, dimension (N)
332: *> Workspace to hold the trailing bits of the intermediate solution.
333: *> \endverbatim
334: *>
335: *> \param[in] RCOND
336: *> \verbatim
337: *> RCOND is DOUBLE PRECISION
338: *> Reciprocal scaled condition number. This is an estimate of the
339: *> reciprocal Skeel condition number of the matrix A after
340: *> equilibration (if done). If this is less than the machine
341: *> precision (in particular, if it is zero), the matrix is singular
342: *> to working precision. Note that the error may still be small even
343: *> if this number is very small and the matrix appears ill-
344: *> conditioned.
345: *> \endverbatim
346: *>
347: *> \param[in] ITHRESH
348: *> \verbatim
349: *> ITHRESH is INTEGER
350: *> The maximum number of residual computations allowed for
351: *> refinement. The default is 10. For 'aggressive' set to 100 to
352: *> permit convergence using approximate factorizations or
353: *> factorizations other than LU. If the factorization uses a
354: *> technique other than Gaussian elimination, the guarantees in
355: *> ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
356: *> \endverbatim
357: *>
358: *> \param[in] RTHRESH
359: *> \verbatim
360: *> RTHRESH is DOUBLE PRECISION
361: *> Determines when to stop refinement if the error estimate stops
362: *> decreasing. Refinement will stop when the next solution no longer
363: *> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
364: *> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
365: *> default value is 0.5. For 'aggressive' set to 0.9 to permit
366: *> convergence on extremely ill-conditioned matrices. See LAWN 165
367: *> for more details.
368: *> \endverbatim
369: *>
370: *> \param[in] DZ_UB
371: *> \verbatim
372: *> DZ_UB is DOUBLE PRECISION
373: *> Determines when to start considering componentwise convergence.
374: *> Componentwise convergence is only considered after each component
375: *> of the solution Y is stable, which we definte as the relative
376: *> change in each component being less than DZ_UB. The default value
377: *> is 0.25, requiring the first bit to be stable. See LAWN 165 for
378: *> more details.
379: *> \endverbatim
380: *>
381: *> \param[in] IGNORE_CWISE
382: *> \verbatim
383: *> IGNORE_CWISE is LOGICAL
384: *> If .TRUE. then ignore componentwise convergence. Default value
385: *> is .FALSE..
386: *> \endverbatim
387: *>
388: *> \param[out] INFO
389: *> \verbatim
390: *> INFO is INTEGER
391: *> = 0: Successful exit.
392: *> < 0: if INFO = -i, the ith argument to DGBTRS had an illegal
393: *> value
394: *> \endverbatim
395: *
396: * Authors:
397: * ========
398: *
399: *> \author Univ. of Tennessee
400: *> \author Univ. of California Berkeley
401: *> \author Univ. of Colorado Denver
402: *> \author NAG Ltd.
403: *
1.8 bertrand 404: *> \date September 2012
1.5 bertrand 405: *
406: *> \ingroup doubleGBcomputational
407: *
408: * =====================================================================
1.1 bertrand 409: SUBROUTINE DLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
410: $ NRHS, AB, LDAB, AFB, LDAFB, IPIV,
411: $ COLEQU, C, B, LDB, Y, LDY,
412: $ BERR_OUT, N_NORMS, ERR_BNDS_NORM,
413: $ ERR_BNDS_COMP, RES, AYB, DY,
414: $ Y_TAIL, RCOND, ITHRESH, RTHRESH,
415: $ DZ_UB, IGNORE_CWISE, INFO )
416: *
1.8 bertrand 417: * -- LAPACK computational routine (version 3.4.2) --
1.5 bertrand 418: * -- LAPACK is a software package provided by Univ. of Tennessee, --
419: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 bertrand 420: * September 2012
1.1 bertrand 421: *
422: * .. Scalar Arguments ..
423: INTEGER INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
424: $ PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
425: LOGICAL COLEQU, IGNORE_CWISE
426: DOUBLE PRECISION RTHRESH, DZ_UB
427: * ..
428: * .. Array Arguments ..
429: INTEGER IPIV( * )
430: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
431: $ Y( LDY, * ), RES(*), DY(*), Y_TAIL(*)
432: DOUBLE PRECISION C( * ), AYB(*), RCOND, BERR_OUT(*),
433: $ ERR_BNDS_NORM( NRHS, * ),
434: $ ERR_BNDS_COMP( NRHS, * )
435: * ..
436: *
437: * =====================================================================
438: *
439: * .. Local Scalars ..
440: CHARACTER TRANS
441: INTEGER CNT, I, J, M, X_STATE, Z_STATE, Y_PREC_STATE
442: DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
443: $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
444: $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
445: $ EPS, HUGEVAL, INCR_THRESH
446: LOGICAL INCR_PREC
447: * ..
448: * .. Parameters ..
449: INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
450: $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
451: $ EXTRA_Y
452: PARAMETER ( UNSTABLE_STATE = 0, WORKING_STATE = 1,
453: $ CONV_STATE = 2, NOPROG_STATE = 3 )
454: PARAMETER ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1,
455: $ EXTRA_Y = 2 )
456: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
457: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
458: INTEGER CMP_ERR_I, PIV_GROWTH_I
459: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
460: $ BERR_I = 3 )
461: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
462: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
463: $ PIV_GROWTH_I = 9 )
464: INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
465: $ LA_LINRX_CWISE_I
466: PARAMETER ( LA_LINRX_ITREF_I = 1,
467: $ LA_LINRX_ITHRESH_I = 2 )
468: PARAMETER ( LA_LINRX_CWISE_I = 3 )
469: INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
470: $ LA_LINRX_RCOND_I
471: PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
472: PARAMETER ( LA_LINRX_RCOND_I = 3 )
473: * ..
474: * .. External Subroutines ..
475: EXTERNAL DAXPY, DCOPY, DGBTRS, DGBMV, BLAS_DGBMV_X,
476: $ BLAS_DGBMV2_X, DLA_GBAMV, DLA_WWADDW, DLAMCH,
477: $ CHLA_TRANSTYPE, DLA_LIN_BERR
478: DOUBLE PRECISION DLAMCH
479: CHARACTER CHLA_TRANSTYPE
480: * ..
481: * .. Intrinsic Functions ..
482: INTRINSIC ABS, MAX, MIN
483: * ..
484: * .. Executable Statements ..
485: *
486: IF (INFO.NE.0) RETURN
487: TRANS = CHLA_TRANSTYPE(TRANS_TYPE)
488: EPS = DLAMCH( 'Epsilon' )
489: HUGEVAL = DLAMCH( 'Overflow' )
490: * Force HUGEVAL to Inf
491: HUGEVAL = HUGEVAL * HUGEVAL
492: * Using HUGEVAL may lead to spurious underflows.
493: INCR_THRESH = DBLE( N ) * EPS
494: M = KL+KU+1
495:
496: DO J = 1, NRHS
497: Y_PREC_STATE = EXTRA_RESIDUAL
498: IF ( Y_PREC_STATE .EQ. EXTRA_Y ) THEN
499: DO I = 1, N
500: Y_TAIL( I ) = 0.0D+0
501: END DO
502: END IF
503:
504: DXRAT = 0.0D+0
505: DXRATMAX = 0.0D+0
506: DZRAT = 0.0D+0
507: DZRATMAX = 0.0D+0
508: FINAL_DX_X = HUGEVAL
509: FINAL_DZ_Z = HUGEVAL
510: PREVNORMDX = HUGEVAL
511: PREV_DZ_Z = HUGEVAL
512: DZ_Z = HUGEVAL
513: DX_X = HUGEVAL
514:
515: X_STATE = WORKING_STATE
516: Z_STATE = UNSTABLE_STATE
517: INCR_PREC = .FALSE.
518:
519: DO CNT = 1, ITHRESH
520: *
521: * Compute residual RES = B_s - op(A_s) * Y,
522: * op(A) = A, A**T, or A**H depending on TRANS (and type).
523: *
524: CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
525: IF ( Y_PREC_STATE .EQ. BASE_RESIDUAL ) THEN
526: CALL DGBMV( TRANS, M, N, KL, KU, -1.0D+0, AB, LDAB,
527: $ Y( 1, J ), 1, 1.0D+0, RES, 1 )
528: ELSE IF ( Y_PREC_STATE .EQ. EXTRA_RESIDUAL ) THEN
529: CALL BLAS_DGBMV_X( TRANS_TYPE, N, N, KL, KU,
530: $ -1.0D+0, AB, LDAB, Y( 1, J ), 1, 1.0D+0, RES, 1,
531: $ PREC_TYPE )
532: ELSE
533: CALL BLAS_DGBMV2_X( TRANS_TYPE, N, N, KL, KU, -1.0D+0,
534: $ AB, LDAB, Y( 1, J ), Y_TAIL, 1, 1.0D+0, RES, 1,
535: $ PREC_TYPE )
536: END IF
537:
538: ! XXX: RES is no longer needed.
539: CALL DCOPY( N, RES, 1, DY, 1 )
540: CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, DY, N,
541: $ INFO )
542: *
543: * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
544: *
545: NORMX = 0.0D+0
546: NORMY = 0.0D+0
547: NORMDX = 0.0D+0
548: DZ_Z = 0.0D+0
549: YMIN = HUGEVAL
550:
551: DO I = 1, N
552: YK = ABS( Y( I, J ) )
553: DYK = ABS( DY( I ) )
554:
555: IF ( YK .NE. 0.0D+0 ) THEN
556: DZ_Z = MAX( DZ_Z, DYK / YK )
557: ELSE IF ( DYK .NE. 0.0D+0 ) THEN
558: DZ_Z = HUGEVAL
559: END IF
560:
561: YMIN = MIN( YMIN, YK )
562:
563: NORMY = MAX( NORMY, YK )
564:
565: IF ( COLEQU ) THEN
566: NORMX = MAX( NORMX, YK * C( I ) )
567: NORMDX = MAX( NORMDX, DYK * C( I ) )
568: ELSE
569: NORMX = NORMY
570: NORMDX = MAX( NORMDX, DYK )
571: END IF
572: END DO
573:
574: IF ( NORMX .NE. 0.0D+0 ) THEN
575: DX_X = NORMDX / NORMX
576: ELSE IF ( NORMDX .EQ. 0.0D+0 ) THEN
577: DX_X = 0.0D+0
578: ELSE
579: DX_X = HUGEVAL
580: END IF
581:
582: DXRAT = NORMDX / PREVNORMDX
583: DZRAT = DZ_Z / PREV_DZ_Z
584: *
585: * Check termination criteria.
586: *
587: IF ( .NOT.IGNORE_CWISE
588: $ .AND. YMIN*RCOND .LT. INCR_THRESH*NORMY
589: $ .AND. Y_PREC_STATE .LT. EXTRA_Y )
590: $ INCR_PREC = .TRUE.
591:
592: IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH )
593: $ X_STATE = WORKING_STATE
594: IF ( X_STATE .EQ. WORKING_STATE ) THEN
595: IF ( DX_X .LE. EPS ) THEN
596: X_STATE = CONV_STATE
597: ELSE IF ( DXRAT .GT. RTHRESH ) THEN
598: IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
599: INCR_PREC = .TRUE.
600: ELSE
601: X_STATE = NOPROG_STATE
602: END IF
603: ELSE
604: IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT
605: END IF
606: IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X
607: END IF
608:
609: IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB )
610: $ Z_STATE = WORKING_STATE
611: IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH )
612: $ Z_STATE = WORKING_STATE
613: IF ( Z_STATE .EQ. WORKING_STATE ) THEN
614: IF ( DZ_Z .LE. EPS ) THEN
615: Z_STATE = CONV_STATE
616: ELSE IF ( DZ_Z .GT. DZ_UB ) THEN
617: Z_STATE = UNSTABLE_STATE
618: DZRATMAX = 0.0D+0
619: FINAL_DZ_Z = HUGEVAL
620: ELSE IF ( DZRAT .GT. RTHRESH ) THEN
621: IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
622: INCR_PREC = .TRUE.
623: ELSE
624: Z_STATE = NOPROG_STATE
625: END IF
626: ELSE
627: IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT
628: END IF
629: IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
630: END IF
631: *
632: * Exit if both normwise and componentwise stopped working,
633: * but if componentwise is unstable, let it go at least two
634: * iterations.
635: *
636: IF ( X_STATE.NE.WORKING_STATE ) THEN
637: IF ( IGNORE_CWISE ) GOTO 666
638: IF ( Z_STATE.EQ.NOPROG_STATE .OR. Z_STATE.EQ.CONV_STATE )
639: $ GOTO 666
640: IF ( Z_STATE.EQ.UNSTABLE_STATE .AND. CNT.GT.1 ) GOTO 666
641: END IF
642:
643: IF ( INCR_PREC ) THEN
644: INCR_PREC = .FALSE.
645: Y_PREC_STATE = Y_PREC_STATE + 1
646: DO I = 1, N
647: Y_TAIL( I ) = 0.0D+0
648: END DO
649: END IF
650:
651: PREVNORMDX = NORMDX
652: PREV_DZ_Z = DZ_Z
653: *
654: * Update soluton.
655: *
656: IF (Y_PREC_STATE .LT. EXTRA_Y) THEN
657: CALL DAXPY( N, 1.0D+0, DY, 1, Y(1,J), 1 )
658: ELSE
659: CALL DLA_WWADDW( N, Y(1,J), Y_TAIL, DY )
660: END IF
661:
662: END DO
663: * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
664: 666 CONTINUE
665: *
666: * Set final_* when cnt hits ithresh.
667: *
668: IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X
669: IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
670: *
671: * Compute error bounds.
672: *
673: IF ( N_NORMS .GE. 1 ) THEN
674: ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) =
675: $ FINAL_DX_X / (1 - DXRATMAX)
676: END IF
677: IF (N_NORMS .GE. 2) THEN
678: ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) =
679: $ FINAL_DZ_Z / (1 - DZRATMAX)
680: END IF
681: *
682: * Compute componentwise relative backward error from formula
683: * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
684: * where abs(Z) is the componentwise absolute value of the matrix
685: * or vector Z.
686: *
687: * Compute residual RES = B_s - op(A_s) * Y,
688: * op(A) = A, A**T, or A**H depending on TRANS (and type).
689: *
690: CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
691: CALL DGBMV(TRANS, N, N, KL, KU, -1.0D+0, AB, LDAB, Y(1,J),
692: $ 1, 1.0D+0, RES, 1 )
693:
694: DO I = 1, N
695: AYB( I ) = ABS( B( I, J ) )
696: END DO
697: *
698: * Compute abs(op(A_s))*abs(Y) + abs(B_s).
699: *
700: CALL DLA_GBAMV( TRANS_TYPE, N, N, KL, KU, 1.0D+0,
701: $ AB, LDAB, Y(1, J), 1, 1.0D+0, AYB, 1 )
702:
703: CALL DLA_LIN_BERR( N, N, 1, RES, AYB, BERR_OUT( J ) )
704: *
705: * End of loop for each RHS
706: *
707: END DO
708: *
709: RETURN
710: END
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