1: *> \brief \b ZLA_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.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLA_GBRFSX_EXTENDED + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zla_gbrfsx_extended.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zla_gbrfsx_extended.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zla_gbrfsx_extended.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLA_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: * COMPLEX*16 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: *> ZLA_GBRFSX_EXTENDED improves the computed solution to a system of
51: *> linear equations by performing extra-precise iterative refinement
52: *> and provides error bounds and backward error estimates for the solution.
53: *> This subroutine is called by ZGBRFSX to perform iterative refinement.
54: *> In addition to normwise error bound, the code provides maximum
55: *> componentwise error bound if possible. See comments for ERR_BNDS_NORM
56: *> and ERR_BNDS_COMP for details of the error bounds. Note that this
57: *> subroutine is only resonsible for setting the second fields of
58: *> ERR_BNDS_NORM and ERR_BNDS_COMP.
59: *> \endverbatim
60: *
61: * Arguments:
62: * ==========
63: *
64: *> \param[in] PREC_TYPE
65: *> \verbatim
66: *> PREC_TYPE is INTEGER
67: *> Specifies the intermediate precision to be used in refinement.
68: *> The value is defined by ILAPREC(P) where P is a CHARACTER and
69: *> P = 'S': Single
70: *> = 'D': Double
71: *> = 'I': Indigenous
72: *> = 'X', 'E': Extra
73: *> \endverbatim
74: *>
75: *> \param[in] TRANS_TYPE
76: *> \verbatim
77: *> TRANS_TYPE is INTEGER
78: *> Specifies the transposition operation on A.
79: *> The value is defined by ILATRANS(T) where T is a CHARACTER and
80: *> T = 'N': No transpose
81: *> = 'T': Transpose
82: *> = 'C': Conjugate transpose
83: *> \endverbatim
84: *>
85: *> \param[in] N
86: *> \verbatim
87: *> N is INTEGER
88: *> The number of linear equations, i.e., the order of the
89: *> matrix A. N >= 0.
90: *> \endverbatim
91: *>
92: *> \param[in] KL
93: *> \verbatim
94: *> KL is INTEGER
95: *> The number of subdiagonals within the band of A. KL >= 0.
96: *> \endverbatim
97: *>
98: *> \param[in] KU
99: *> \verbatim
100: *> KU is INTEGER
101: *> The number of superdiagonals within the band of A. KU >= 0
102: *> \endverbatim
103: *>
104: *> \param[in] NRHS
105: *> \verbatim
106: *> NRHS is INTEGER
107: *> The number of right-hand-sides, i.e., the number of columns of the
108: *> matrix B.
109: *> \endverbatim
110: *>
111: *> \param[in] AB
112: *> \verbatim
113: *> AB is COMPLEX*16 array, dimension (LDAB,N)
114: *> On entry, the N-by-N matrix A.
115: *> \endverbatim
116: *>
117: *> \param[in] LDAB
118: *> \verbatim
119: *> LDAB is INTEGER
120: *> The leading dimension of the array A. LDAB >= max(1,N).
121: *> \endverbatim
122: *>
123: *> \param[in] AFB
124: *> \verbatim
125: *> AFB is COMPLEX*16 array, dimension (LDAF,N)
126: *> The factors L and U from the factorization
127: *> A = P*L*U as computed by ZGBTRF.
128: *> \endverbatim
129: *>
130: *> \param[in] LDAFB
131: *> \verbatim
132: *> LDAFB is INTEGER
133: *> The leading dimension of the array AF. LDAF >= max(1,N).
134: *> \endverbatim
135: *>
136: *> \param[in] IPIV
137: *> \verbatim
138: *> IPIV is INTEGER array, dimension (N)
139: *> The pivot indices from the factorization A = P*L*U
140: *> as computed by ZGBTRF; row i of the matrix was interchanged
141: *> with row IPIV(i).
142: *> \endverbatim
143: *>
144: *> \param[in] COLEQU
145: *> \verbatim
146: *> COLEQU is LOGICAL
147: *> If .TRUE. then column equilibration was done to A before calling
148: *> this routine. This is needed to compute the solution and error
149: *> bounds correctly.
150: *> \endverbatim
151: *>
152: *> \param[in] C
153: *> \verbatim
154: *> C is DOUBLE PRECISION array, dimension (N)
155: *> The column scale factors for A. If COLEQU = .FALSE., C
156: *> is not accessed. If C is input, each element of C should be a power
157: *> of the radix to ensure a reliable solution and error estimates.
158: *> Scaling by powers of the radix does not cause rounding errors unless
159: *> the result underflows or overflows. Rounding errors during scaling
160: *> lead to refining with a matrix that is not equivalent to the
161: *> input matrix, producing error estimates that may not be
162: *> reliable.
163: *> \endverbatim
164: *>
165: *> \param[in] B
166: *> \verbatim
167: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
168: *> The right-hand-side matrix B.
169: *> \endverbatim
170: *>
171: *> \param[in] LDB
172: *> \verbatim
173: *> LDB is INTEGER
174: *> The leading dimension of the array B. LDB >= max(1,N).
175: *> \endverbatim
176: *>
177: *> \param[in,out] Y
178: *> \verbatim
179: *> Y is COMPLEX*16 array, dimension (LDY,NRHS)
180: *> On entry, the solution matrix X, as computed by ZGBTRS.
181: *> On exit, the improved solution matrix Y.
182: *> \endverbatim
183: *>
184: *> \param[in] LDY
185: *> \verbatim
186: *> LDY is INTEGER
187: *> The leading dimension of the array Y. LDY >= max(1,N).
188: *> \endverbatim
189: *>
190: *> \param[out] BERR_OUT
191: *> \verbatim
192: *> BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
193: *> On exit, BERR_OUT(j) contains the componentwise relative backward
194: *> error for right-hand-side j from the formula
195: *> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
196: *> where abs(Z) is the componentwise absolute value of the matrix
197: *> or vector Z. This is computed by ZLA_LIN_BERR.
198: *> \endverbatim
199: *>
200: *> \param[in] N_NORMS
201: *> \verbatim
202: *> N_NORMS is INTEGER
203: *> Determines which error bounds to return (see ERR_BNDS_NORM
204: *> and ERR_BNDS_COMP).
205: *> If N_NORMS >= 1 return normwise error bounds.
206: *> If N_NORMS >= 2 return componentwise error bounds.
207: *> \endverbatim
208: *>
209: *> \param[in,out] ERR_BNDS_NORM
210: *> \verbatim
211: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension
212: *> (NRHS, N_ERR_BNDS)
213: *> For each right-hand side, this array contains information about
214: *> various error bounds and condition numbers corresponding to the
215: *> normwise relative error, which is defined as follows:
216: *>
217: *> Normwise relative error in the ith solution vector:
218: *> max_j (abs(XTRUE(j,i) - X(j,i)))
219: *> ------------------------------
220: *> max_j abs(X(j,i))
221: *>
222: *> The array is indexed by the type of error information as described
223: *> below. There currently are up to three pieces of information
224: *> returned.
225: *>
226: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
227: *> right-hand side.
228: *>
229: *> The second index in ERR_BNDS_NORM(:,err) contains the following
230: *> three fields:
231: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
232: *> reciprocal condition number is less than the threshold
233: *> sqrt(n) * slamch('Epsilon').
234: *>
235: *> err = 2 "Guaranteed" error bound: The estimated forward error,
236: *> almost certainly within a factor of 10 of the true error
237: *> so long as the next entry is greater than the threshold
238: *> sqrt(n) * slamch('Epsilon'). This error bound should only
239: *> be trusted if the previous boolean is true.
240: *>
241: *> err = 3 Reciprocal condition number: Estimated normwise
242: *> reciprocal condition number. Compared with the threshold
243: *> sqrt(n) * slamch('Epsilon') to determine if the error
244: *> estimate is "guaranteed". These reciprocal condition
245: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
246: *> appropriately scaled matrix Z.
247: *> Let Z = S*A, where S scales each row by a power of the
248: *> radix so all absolute row sums of Z are approximately 1.
249: *>
250: *> This subroutine is only responsible for setting the second field
251: *> above.
252: *> See Lapack Working Note 165 for further details and extra
253: *> cautions.
254: *> \endverbatim
255: *>
256: *> \param[in,out] ERR_BNDS_COMP
257: *> \verbatim
258: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension
259: *> (NRHS, N_ERR_BNDS)
260: *> For each right-hand side, this array contains information about
261: *> various error bounds and condition numbers corresponding to the
262: *> componentwise relative error, which is defined as follows:
263: *>
264: *> Componentwise relative error in the ith solution vector:
265: *> abs(XTRUE(j,i) - X(j,i))
266: *> max_j ----------------------
267: *> abs(X(j,i))
268: *>
269: *> The array is indexed by the right-hand side i (on which the
270: *> componentwise relative error depends), and the type of error
271: *> information as described below. There currently are up to three
272: *> pieces of information returned for each right-hand side. If
273: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
274: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
275: *> the first (:,N_ERR_BNDS) entries are returned.
276: *>
277: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
278: *> right-hand side.
279: *>
280: *> The second index in ERR_BNDS_COMP(:,err) contains the following
281: *> three fields:
282: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
283: *> reciprocal condition number is less than the threshold
284: *> sqrt(n) * slamch('Epsilon').
285: *>
286: *> err = 2 "Guaranteed" error bound: The estimated forward error,
287: *> almost certainly within a factor of 10 of the true error
288: *> so long as the next entry is greater than the threshold
289: *> sqrt(n) * slamch('Epsilon'). This error bound should only
290: *> be trusted if the previous boolean is true.
291: *>
292: *> err = 3 Reciprocal condition number: Estimated componentwise
293: *> reciprocal condition number. Compared with the threshold
294: *> sqrt(n) * slamch('Epsilon') to determine if the error
295: *> estimate is "guaranteed". These reciprocal condition
296: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
297: *> appropriately scaled matrix Z.
298: *> Let Z = S*(A*diag(x)), where x is the solution for the
299: *> current right-hand side and S scales each row of
300: *> A*diag(x) by a power of the radix so all absolute row
301: *> sums of Z are approximately 1.
302: *>
303: *> This subroutine is only responsible for setting the second field
304: *> above.
305: *> See Lapack Working Note 165 for further details and extra
306: *> cautions.
307: *> \endverbatim
308: *>
309: *> \param[in] RES
310: *> \verbatim
311: *> RES is COMPLEX*16 array, dimension (N)
312: *> Workspace to hold the intermediate residual.
313: *> \endverbatim
314: *>
315: *> \param[in] AYB
316: *> \verbatim
317: *> AYB is DOUBLE PRECISION array, dimension (N)
318: *> Workspace.
319: *> \endverbatim
320: *>
321: *> \param[in] DY
322: *> \verbatim
323: *> DY is COMPLEX*16 array, dimension (N)
324: *> Workspace to hold the intermediate solution.
325: *> \endverbatim
326: *>
327: *> \param[in] Y_TAIL
328: *> \verbatim
329: *> Y_TAIL is COMPLEX*16 array, dimension (N)
330: *> Workspace to hold the trailing bits of the intermediate solution.
331: *> \endverbatim
332: *>
333: *> \param[in] RCOND
334: *> \verbatim
335: *> RCOND is DOUBLE PRECISION
336: *> Reciprocal scaled condition number. This is an estimate of the
337: *> reciprocal Skeel condition number of the matrix A after
338: *> equilibration (if done). If this is less than the machine
339: *> precision (in particular, if it is zero), the matrix is singular
340: *> to working precision. Note that the error may still be small even
341: *> if this number is very small and the matrix appears ill-
342: *> conditioned.
343: *> \endverbatim
344: *>
345: *> \param[in] ITHRESH
346: *> \verbatim
347: *> ITHRESH is INTEGER
348: *> The maximum number of residual computations allowed for
349: *> refinement. The default is 10. For 'aggressive' set to 100 to
350: *> permit convergence using approximate factorizations or
351: *> factorizations other than LU. If the factorization uses a
352: *> technique other than Gaussian elimination, the guarantees in
353: *> ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
354: *> \endverbatim
355: *>
356: *> \param[in] RTHRESH
357: *> \verbatim
358: *> RTHRESH is DOUBLE PRECISION
359: *> Determines when to stop refinement if the error estimate stops
360: *> decreasing. Refinement will stop when the next solution no longer
361: *> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
362: *> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
363: *> default value is 0.5. For 'aggressive' set to 0.9 to permit
364: *> convergence on extremely ill-conditioned matrices. See LAWN 165
365: *> for more details.
366: *> \endverbatim
367: *>
368: *> \param[in] DZ_UB
369: *> \verbatim
370: *> DZ_UB is DOUBLE PRECISION
371: *> Determines when to start considering componentwise convergence.
372: *> Componentwise convergence is only considered after each component
373: *> of the solution Y is stable, which we definte as the relative
374: *> change in each component being less than DZ_UB. The default value
375: *> is 0.25, requiring the first bit to be stable. See LAWN 165 for
376: *> more details.
377: *> \endverbatim
378: *>
379: *> \param[in] IGNORE_CWISE
380: *> \verbatim
381: *> IGNORE_CWISE is LOGICAL
382: *> If .TRUE. then ignore componentwise convergence. Default value
383: *> is .FALSE..
384: *> \endverbatim
385: *>
386: *> \param[out] INFO
387: *> \verbatim
388: *> INFO is INTEGER
389: *> = 0: Successful exit.
390: *> < 0: if INFO = -i, the ith argument to ZGBTRS had an illegal
391: *> value
392: *> \endverbatim
393: *
394: * Authors:
395: * ========
396: *
397: *> \author Univ. of Tennessee
398: *> \author Univ. of California Berkeley
399: *> \author Univ. of Colorado Denver
400: *> \author NAG Ltd.
401: *
402: *> \date December 2016
403: *
404: *> \ingroup complex16GBcomputational
405: *
406: * =====================================================================
407: SUBROUTINE ZLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
408: $ NRHS, AB, LDAB, AFB, LDAFB, IPIV,
409: $ COLEQU, C, B, LDB, Y, LDY,
410: $ BERR_OUT, N_NORMS, ERR_BNDS_NORM,
411: $ ERR_BNDS_COMP, RES, AYB, DY,
412: $ Y_TAIL, RCOND, ITHRESH, RTHRESH,
413: $ DZ_UB, IGNORE_CWISE, INFO )
414: *
415: * -- LAPACK computational routine (version 3.7.0) --
416: * -- LAPACK is a software package provided by Univ. of Tennessee, --
417: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
418: * December 2016
419: *
420: * .. Scalar Arguments ..
421: INTEGER INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
422: $ PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
423: LOGICAL COLEQU, IGNORE_CWISE
424: DOUBLE PRECISION RTHRESH, DZ_UB
425: * ..
426: * .. Array Arguments ..
427: INTEGER IPIV( * )
428: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
429: $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
430: DOUBLE PRECISION C( * ), AYB(*), RCOND, BERR_OUT( * ),
431: $ ERR_BNDS_NORM( NRHS, * ),
432: $ ERR_BNDS_COMP( NRHS, * )
433: * ..
434: *
435: * =====================================================================
436: *
437: * .. Local Scalars ..
438: CHARACTER TRANS
439: INTEGER CNT, I, J, M, X_STATE, Z_STATE, Y_PREC_STATE
440: DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
441: $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
442: $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
443: $ EPS, HUGEVAL, INCR_THRESH
444: LOGICAL INCR_PREC
445: COMPLEX*16 ZDUM
446: * ..
447: * .. Parameters ..
448: INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
449: $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
450: $ EXTRA_Y
451: PARAMETER ( UNSTABLE_STATE = 0, WORKING_STATE = 1,
452: $ CONV_STATE = 2, NOPROG_STATE = 3 )
453: PARAMETER ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1,
454: $ EXTRA_Y = 2 )
455: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
456: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
457: INTEGER CMP_ERR_I, PIV_GROWTH_I
458: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
459: $ BERR_I = 3 )
460: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
461: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
462: $ PIV_GROWTH_I = 9 )
463: INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
464: $ LA_LINRX_CWISE_I
465: PARAMETER ( LA_LINRX_ITREF_I = 1,
466: $ LA_LINRX_ITHRESH_I = 2 )
467: PARAMETER ( LA_LINRX_CWISE_I = 3 )
468: INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
469: $ LA_LINRX_RCOND_I
470: PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
471: PARAMETER ( LA_LINRX_RCOND_I = 3 )
472: * ..
473: * .. External Subroutines ..
474: EXTERNAL ZAXPY, ZCOPY, ZGBTRS, ZGBMV, BLAS_ZGBMV_X,
475: $ BLAS_ZGBMV2_X, ZLA_GBAMV, ZLA_WWADDW, DLAMCH,
476: $ CHLA_TRANSTYPE, ZLA_LIN_BERR
477: DOUBLE PRECISION DLAMCH
478: CHARACTER CHLA_TRANSTYPE
479: * ..
480: * .. Intrinsic Functions..
481: INTRINSIC ABS, MAX, MIN
482: * ..
483: * .. Statement Functions ..
484: DOUBLE PRECISION CABS1
485: * ..
486: * .. Statement Function Definitions ..
487: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
488: * ..
489: * .. Executable Statements ..
490: *
491: IF (INFO.NE.0) RETURN
492: TRANS = CHLA_TRANSTYPE(TRANS_TYPE)
493: EPS = DLAMCH( 'Epsilon' )
494: HUGEVAL = DLAMCH( 'Overflow' )
495: * Force HUGEVAL to Inf
496: HUGEVAL = HUGEVAL * HUGEVAL
497: * Using HUGEVAL may lead to spurious underflows.
498: INCR_THRESH = DBLE( N ) * EPS
499: M = KL+KU+1
500:
501: DO J = 1, NRHS
502: Y_PREC_STATE = EXTRA_RESIDUAL
503: IF ( Y_PREC_STATE .EQ. EXTRA_Y ) THEN
504: DO I = 1, N
505: Y_TAIL( I ) = 0.0D+0
506: END DO
507: END IF
508:
509: DXRAT = 0.0D+0
510: DXRATMAX = 0.0D+0
511: DZRAT = 0.0D+0
512: DZRATMAX = 0.0D+0
513: FINAL_DX_X = HUGEVAL
514: FINAL_DZ_Z = HUGEVAL
515: PREVNORMDX = HUGEVAL
516: PREV_DZ_Z = HUGEVAL
517: DZ_Z = HUGEVAL
518: DX_X = HUGEVAL
519:
520: X_STATE = WORKING_STATE
521: Z_STATE = UNSTABLE_STATE
522: INCR_PREC = .FALSE.
523:
524: DO CNT = 1, ITHRESH
525: *
526: * Compute residual RES = B_s - op(A_s) * Y,
527: * op(A) = A, A**T, or A**H depending on TRANS (and type).
528: *
529: CALL ZCOPY( N, B( 1, J ), 1, RES, 1 )
530: IF ( Y_PREC_STATE .EQ. BASE_RESIDUAL ) THEN
531: CALL ZGBMV( TRANS, M, N, KL, KU, (-1.0D+0,0.0D+0), AB,
532: $ LDAB, Y( 1, J ), 1, (1.0D+0,0.0D+0), RES, 1 )
533: ELSE IF ( Y_PREC_STATE .EQ. EXTRA_RESIDUAL ) THEN
534: CALL BLAS_ZGBMV_X( TRANS_TYPE, N, N, KL, KU,
535: $ (-1.0D+0,0.0D+0), AB, LDAB, Y( 1, J ), 1,
536: $ (1.0D+0,0.0D+0), RES, 1, PREC_TYPE )
537: ELSE
538: CALL BLAS_ZGBMV2_X( TRANS_TYPE, N, N, KL, KU,
539: $ (-1.0D+0,0.0D+0), AB, LDAB, Y( 1, J ), Y_TAIL, 1,
540: $ (1.0D+0,0.0D+0), RES, 1, PREC_TYPE )
541: END IF
542:
543: ! XXX: RES is no longer needed.
544: CALL ZCOPY( N, RES, 1, DY, 1 )
545: CALL ZGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, DY, N,
546: $ INFO )
547: *
548: * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
549: *
550: NORMX = 0.0D+0
551: NORMY = 0.0D+0
552: NORMDX = 0.0D+0
553: DZ_Z = 0.0D+0
554: YMIN = HUGEVAL
555:
556: DO I = 1, N
557: YK = CABS1( Y( I, J ) )
558: DYK = CABS1( DY( I ) )
559:
560: IF (YK .NE. 0.0D+0) THEN
561: DZ_Z = MAX( DZ_Z, DYK / YK )
562: ELSE IF ( DYK .NE. 0.0D+0 ) THEN
563: DZ_Z = HUGEVAL
564: END IF
565:
566: YMIN = MIN( YMIN, YK )
567:
568: NORMY = MAX( NORMY, YK )
569:
570: IF ( COLEQU ) THEN
571: NORMX = MAX( NORMX, YK * C( I ) )
572: NORMDX = MAX(NORMDX, DYK * C(I))
573: ELSE
574: NORMX = NORMY
575: NORMDX = MAX( NORMDX, DYK )
576: END IF
577: END DO
578:
579: IF ( NORMX .NE. 0.0D+0 ) THEN
580: DX_X = NORMDX / NORMX
581: ELSE IF ( NORMDX .EQ. 0.0D+0 ) THEN
582: DX_X = 0.0D+0
583: ELSE
584: DX_X = HUGEVAL
585: END IF
586:
587: DXRAT = NORMDX / PREVNORMDX
588: DZRAT = DZ_Z / PREV_DZ_Z
589: *
590: * Check termination criteria.
591: *
592: IF (.NOT.IGNORE_CWISE
593: $ .AND. YMIN*RCOND .LT. INCR_THRESH*NORMY
594: $ .AND. Y_PREC_STATE .LT. EXTRA_Y )
595: $ INCR_PREC = .TRUE.
596:
597: IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH )
598: $ X_STATE = WORKING_STATE
599: IF ( X_STATE .EQ. WORKING_STATE ) THEN
600: IF ( DX_X .LE. EPS ) THEN
601: X_STATE = CONV_STATE
602: ELSE IF ( DXRAT .GT. RTHRESH ) THEN
603: IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
604: INCR_PREC = .TRUE.
605: ELSE
606: X_STATE = NOPROG_STATE
607: END IF
608: ELSE
609: IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT
610: END IF
611: IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X
612: END IF
613:
614: IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB )
615: $ Z_STATE = WORKING_STATE
616: IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH )
617: $ Z_STATE = WORKING_STATE
618: IF ( Z_STATE .EQ. WORKING_STATE ) THEN
619: IF ( DZ_Z .LE. EPS ) THEN
620: Z_STATE = CONV_STATE
621: ELSE IF ( DZ_Z .GT. DZ_UB ) THEN
622: Z_STATE = UNSTABLE_STATE
623: DZRATMAX = 0.0D+0
624: FINAL_DZ_Z = HUGEVAL
625: ELSE IF ( DZRAT .GT. RTHRESH ) THEN
626: IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
627: INCR_PREC = .TRUE.
628: ELSE
629: Z_STATE = NOPROG_STATE
630: END IF
631: ELSE
632: IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT
633: END IF
634: IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
635: END IF
636: *
637: * Exit if both normwise and componentwise stopped working,
638: * but if componentwise is unstable, let it go at least two
639: * iterations.
640: *
641: IF ( X_STATE.NE.WORKING_STATE ) THEN
642: IF ( IGNORE_CWISE ) GOTO 666
643: IF ( Z_STATE.EQ.NOPROG_STATE .OR. Z_STATE.EQ.CONV_STATE )
644: $ GOTO 666
645: IF ( Z_STATE.EQ.UNSTABLE_STATE .AND. CNT.GT.1 ) GOTO 666
646: END IF
647:
648: IF ( INCR_PREC ) THEN
649: INCR_PREC = .FALSE.
650: Y_PREC_STATE = Y_PREC_STATE + 1
651: DO I = 1, N
652: Y_TAIL( I ) = 0.0D+0
653: END DO
654: END IF
655:
656: PREVNORMDX = NORMDX
657: PREV_DZ_Z = DZ_Z
658: *
659: * Update soluton.
660: *
661: IF ( Y_PREC_STATE .LT. EXTRA_Y ) THEN
662: CALL ZAXPY( N, (1.0D+0,0.0D+0), DY, 1, Y(1,J), 1 )
663: ELSE
664: CALL ZLA_WWADDW( N, Y(1,J), Y_TAIL, DY )
665: END IF
666:
667: END DO
668: * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
669: 666 CONTINUE
670: *
671: * Set final_* when cnt hits ithresh.
672: *
673: IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X
674: IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
675: *
676: * Compute error bounds.
677: *
678: IF ( N_NORMS .GE. 1 ) THEN
679: ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) =
680: $ FINAL_DX_X / (1 - DXRATMAX)
681: END IF
682: IF ( N_NORMS .GE. 2 ) THEN
683: ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) =
684: $ FINAL_DZ_Z / (1 - DZRATMAX)
685: END IF
686: *
687: * Compute componentwise relative backward error from formula
688: * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
689: * where abs(Z) is the componentwise absolute value of the matrix
690: * or vector Z.
691: *
692: * Compute residual RES = B_s - op(A_s) * Y,
693: * op(A) = A, A**T, or A**H depending on TRANS (and type).
694: *
695: CALL ZCOPY( N, B( 1, J ), 1, RES, 1 )
696: CALL ZGBMV( TRANS, N, N, KL, KU, (-1.0D+0,0.0D+0), AB, LDAB,
697: $ Y(1,J), 1, (1.0D+0,0.0D+0), RES, 1 )
698:
699: DO I = 1, N
700: AYB( I ) = CABS1( B( I, J ) )
701: END DO
702: *
703: * Compute abs(op(A_s))*abs(Y) + abs(B_s).
704: *
705: CALL ZLA_GBAMV( TRANS_TYPE, N, N, KL, KU, 1.0D+0,
706: $ AB, LDAB, Y(1, J), 1, 1.0D+0, AYB, 1 )
707:
708: CALL ZLA_LIN_BERR( N, N, 1, RES, AYB, BERR_OUT( J ) )
709: *
710: * End of loop for each RHS.
711: *
712: END DO
713: *
714: RETURN
715: END
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