Annotation of rpl/lapack/lapack/dla_gerfsx_extended.f, revision 1.16
1.8 bertrand 1: *> \brief \b DLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general 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: *
1.12 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.5 bertrand 7: *
8: *> \htmlonly
1.12 bertrand 9: *> Download DLA_GERFSX_EXTENDED + dependencies
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11: *> [TGZ]</a>
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14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dla_gerfsx_extended.f">
1.5 bertrand 15: *> [TXT]</a>
1.12 bertrand 16: *> \endhtmlonly
1.5 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, NRHS, A,
22: * LDA, AF, LDAF, IPIV, COLEQU, C, B,
23: * LDB, Y, LDY, BERR_OUT, N_NORMS,
24: * ERRS_N, ERRS_C, RES, AYB, DY,
25: * Y_TAIL, RCOND, ITHRESH, RTHRESH,
26: * DZ_UB, IGNORE_CWISE, INFO )
1.12 bertrand 27: *
1.5 bertrand 28: * .. Scalar Arguments ..
29: * INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
30: * $ TRANS_TYPE, N_NORMS, ITHRESH
31: * LOGICAL COLEQU, IGNORE_CWISE
32: * DOUBLE PRECISION RTHRESH, DZ_UB
33: * ..
34: * .. Array Arguments ..
35: * INTEGER IPIV( * )
36: * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
37: * $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
38: * DOUBLE PRECISION C( * ), AYB( * ), RCOND, BERR_OUT( * ),
39: * $ ERRS_N( NRHS, * ), ERRS_C( NRHS, * )
40: * ..
1.12 bertrand 41: *
1.5 bertrand 42: *
43: *> \par Purpose:
44: * =============
45: *>
46: *> \verbatim
47: *>
1.12 bertrand 48: *>
1.5 bertrand 49: *> DLA_GERFSX_EXTENDED improves the computed solution to a system of
50: *> linear equations by performing extra-precise iterative refinement
51: *> and provides error bounds and backward error estimates for the solution.
52: *> This subroutine is called by DGERFSX to perform iterative refinement.
53: *> In addition to normwise error bound, the code provides maximum
54: *> componentwise error bound if possible. See comments for ERRS_N
55: *> and ERRS_C for details of the error bounds. Note that this
56: *> subroutine is only resonsible for setting the second fields of
57: *> ERRS_N and ERRS_C.
58: *> \endverbatim
59: *
60: * Arguments:
61: * ==========
62: *
63: *> \param[in] PREC_TYPE
64: *> \verbatim
65: *> PREC_TYPE is INTEGER
66: *> Specifies the intermediate precision to be used in refinement.
1.16 ! bertrand 67: *> The value is defined by ILAPREC(P) where P is a CHARACTER and P
! 68: *> = 'S': Single
1.5 bertrand 69: *> = 'D': Double
70: *> = 'I': Indigenous
1.16 ! bertrand 71: *> = 'X' or 'E': Extra
1.5 bertrand 72: *> \endverbatim
73: *>
74: *> \param[in] TRANS_TYPE
75: *> \verbatim
76: *> TRANS_TYPE is INTEGER
77: *> Specifies the transposition operation on A.
1.16 ! bertrand 78: *> The value is defined by ILATRANS(T) where T is a CHARACTER and T
! 79: *> = 'N': No transpose
1.5 bertrand 80: *> = 'T': Transpose
81: *> = 'C': Conjugate transpose
82: *> \endverbatim
83: *>
84: *> \param[in] N
85: *> \verbatim
86: *> N is INTEGER
87: *> The number of linear equations, i.e., the order of the
88: *> matrix A. N >= 0.
89: *> \endverbatim
90: *>
91: *> \param[in] NRHS
92: *> \verbatim
93: *> NRHS is INTEGER
94: *> The number of right-hand-sides, i.e., the number of columns of the
95: *> matrix B.
96: *> \endverbatim
97: *>
98: *> \param[in] A
99: *> \verbatim
100: *> A is DOUBLE PRECISION array, dimension (LDA,N)
101: *> On entry, the N-by-N matrix A.
102: *> \endverbatim
103: *>
104: *> \param[in] LDA
105: *> \verbatim
106: *> LDA is INTEGER
107: *> The leading dimension of the array A. LDA >= max(1,N).
108: *> \endverbatim
109: *>
110: *> \param[in] AF
111: *> \verbatim
112: *> AF is DOUBLE PRECISION array, dimension (LDAF,N)
113: *> The factors L and U from the factorization
114: *> A = P*L*U as computed by DGETRF.
115: *> \endverbatim
116: *>
117: *> \param[in] LDAF
118: *> \verbatim
119: *> LDAF is INTEGER
120: *> The leading dimension of the array AF. LDAF >= max(1,N).
121: *> \endverbatim
122: *>
123: *> \param[in] IPIV
124: *> \verbatim
125: *> IPIV is INTEGER array, dimension (N)
126: *> The pivot indices from the factorization A = P*L*U
127: *> as computed by DGETRF; row i of the matrix was interchanged
128: *> with row IPIV(i).
129: *> \endverbatim
130: *>
131: *> \param[in] COLEQU
132: *> \verbatim
133: *> COLEQU is LOGICAL
134: *> If .TRUE. then column equilibration was done to A before calling
135: *> this routine. This is needed to compute the solution and error
136: *> bounds correctly.
137: *> \endverbatim
138: *>
139: *> \param[in] C
140: *> \verbatim
141: *> C is DOUBLE PRECISION array, dimension (N)
142: *> The column scale factors for A. If COLEQU = .FALSE., C
143: *> is not accessed. If C is input, each element of C should be a power
144: *> of the radix to ensure a reliable solution and error estimates.
145: *> Scaling by powers of the radix does not cause rounding errors unless
146: *> the result underflows or overflows. Rounding errors during scaling
147: *> lead to refining with a matrix that is not equivalent to the
148: *> input matrix, producing error estimates that may not be
149: *> reliable.
150: *> \endverbatim
151: *>
152: *> \param[in] B
153: *> \verbatim
154: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
155: *> The right-hand-side matrix B.
156: *> \endverbatim
157: *>
158: *> \param[in] LDB
159: *> \verbatim
160: *> LDB is INTEGER
161: *> The leading dimension of the array B. LDB >= max(1,N).
162: *> \endverbatim
163: *>
164: *> \param[in,out] Y
165: *> \verbatim
1.14 bertrand 166: *> Y is DOUBLE PRECISION array, dimension (LDY,NRHS)
1.5 bertrand 167: *> On entry, the solution matrix X, as computed by DGETRS.
168: *> On exit, the improved solution matrix Y.
169: *> \endverbatim
170: *>
171: *> \param[in] LDY
172: *> \verbatim
173: *> LDY is INTEGER
174: *> The leading dimension of the array Y. LDY >= max(1,N).
175: *> \endverbatim
176: *>
177: *> \param[out] BERR_OUT
178: *> \verbatim
179: *> BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
180: *> On exit, BERR_OUT(j) contains the componentwise relative backward
181: *> error for right-hand-side j from the formula
182: *> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
183: *> where abs(Z) is the componentwise absolute value of the matrix
184: *> or vector Z. This is computed by DLA_LIN_BERR.
185: *> \endverbatim
186: *>
187: *> \param[in] N_NORMS
188: *> \verbatim
189: *> N_NORMS is INTEGER
190: *> Determines which error bounds to return (see ERRS_N
191: *> and ERRS_C).
192: *> If N_NORMS >= 1 return normwise error bounds.
193: *> If N_NORMS >= 2 return componentwise error bounds.
194: *> \endverbatim
195: *>
196: *> \param[in,out] ERRS_N
197: *> \verbatim
1.14 bertrand 198: *> ERRS_N is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
1.5 bertrand 199: *> For each right-hand side, this array contains information about
200: *> various error bounds and condition numbers corresponding to the
201: *> normwise relative error, which is defined as follows:
202: *>
203: *> Normwise relative error in the ith solution vector:
204: *> max_j (abs(XTRUE(j,i) - X(j,i)))
205: *> ------------------------------
206: *> max_j abs(X(j,i))
207: *>
208: *> The array is indexed by the type of error information as described
209: *> below. There currently are up to three pieces of information
210: *> returned.
211: *>
212: *> The first index in ERRS_N(i,:) corresponds to the ith
213: *> right-hand side.
214: *>
215: *> The second index in ERRS_N(:,err) contains the following
216: *> three fields:
217: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
218: *> reciprocal condition number is less than the threshold
219: *> sqrt(n) * slamch('Epsilon').
220: *>
221: *> err = 2 "Guaranteed" error bound: The estimated forward error,
222: *> almost certainly within a factor of 10 of the true error
223: *> so long as the next entry is greater than the threshold
224: *> sqrt(n) * slamch('Epsilon'). This error bound should only
225: *> be trusted if the previous boolean is true.
226: *>
227: *> err = 3 Reciprocal condition number: Estimated normwise
228: *> reciprocal condition number. Compared with the threshold
229: *> sqrt(n) * slamch('Epsilon') to determine if the error
230: *> estimate is "guaranteed". These reciprocal condition
231: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
232: *> appropriately scaled matrix Z.
233: *> Let Z = S*A, where S scales each row by a power of the
234: *> radix so all absolute row sums of Z are approximately 1.
235: *>
236: *> This subroutine is only responsible for setting the second field
237: *> above.
238: *> See Lapack Working Note 165 for further details and extra
239: *> cautions.
240: *> \endverbatim
241: *>
242: *> \param[in,out] ERRS_C
243: *> \verbatim
1.14 bertrand 244: *> ERRS_C is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
1.5 bertrand 245: *> For each right-hand side, this array contains information about
246: *> various error bounds and condition numbers corresponding to the
247: *> componentwise relative error, which is defined as follows:
248: *>
249: *> Componentwise relative error in the ith solution vector:
250: *> abs(XTRUE(j,i) - X(j,i))
251: *> max_j ----------------------
252: *> abs(X(j,i))
253: *>
254: *> The array is indexed by the right-hand side i (on which the
255: *> componentwise relative error depends), and the type of error
256: *> information as described below. There currently are up to three
257: *> pieces of information returned for each right-hand side. If
258: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
1.16 ! bertrand 259: *> ERRS_C is not accessed. If N_ERR_BNDS < 3, then at most
1.5 bertrand 260: *> the first (:,N_ERR_BNDS) entries are returned.
261: *>
262: *> The first index in ERRS_C(i,:) corresponds to the ith
263: *> right-hand side.
264: *>
265: *> The second index in ERRS_C(:,err) contains the following
266: *> three fields:
267: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
268: *> reciprocal condition number is less than the threshold
269: *> sqrt(n) * slamch('Epsilon').
270: *>
271: *> err = 2 "Guaranteed" error bound: The estimated forward error,
272: *> almost certainly within a factor of 10 of the true error
273: *> so long as the next entry is greater than the threshold
274: *> sqrt(n) * slamch('Epsilon'). This error bound should only
275: *> be trusted if the previous boolean is true.
276: *>
277: *> err = 3 Reciprocal condition number: Estimated componentwise
278: *> reciprocal condition number. Compared with the threshold
279: *> sqrt(n) * slamch('Epsilon') to determine if the error
280: *> estimate is "guaranteed". These reciprocal condition
281: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
282: *> appropriately scaled matrix Z.
283: *> Let Z = S*(A*diag(x)), where x is the solution for the
284: *> current right-hand side and S scales each row of
285: *> A*diag(x) by a power of the radix so all absolute row
286: *> sums of Z are approximately 1.
287: *>
288: *> This subroutine is only responsible for setting the second field
289: *> above.
290: *> See Lapack Working Note 165 for further details and extra
291: *> cautions.
292: *> \endverbatim
293: *>
294: *> \param[in] RES
295: *> \verbatim
296: *> RES is DOUBLE PRECISION array, dimension (N)
297: *> Workspace to hold the intermediate residual.
298: *> \endverbatim
299: *>
300: *> \param[in] AYB
301: *> \verbatim
302: *> AYB is DOUBLE PRECISION array, dimension (N)
303: *> Workspace. This can be the same workspace passed for Y_TAIL.
304: *> \endverbatim
305: *>
306: *> \param[in] DY
307: *> \verbatim
308: *> DY is DOUBLE PRECISION array, dimension (N)
309: *> Workspace to hold the intermediate solution.
310: *> \endverbatim
311: *>
312: *> \param[in] Y_TAIL
313: *> \verbatim
314: *> Y_TAIL is DOUBLE PRECISION array, dimension (N)
315: *> Workspace to hold the trailing bits of the intermediate solution.
316: *> \endverbatim
317: *>
318: *> \param[in] RCOND
319: *> \verbatim
320: *> RCOND is DOUBLE PRECISION
321: *> Reciprocal scaled condition number. This is an estimate of the
322: *> reciprocal Skeel condition number of the matrix A after
323: *> equilibration (if done). If this is less than the machine
324: *> precision (in particular, if it is zero), the matrix is singular
325: *> to working precision. Note that the error may still be small even
326: *> if this number is very small and the matrix appears ill-
327: *> conditioned.
328: *> \endverbatim
329: *>
330: *> \param[in] ITHRESH
331: *> \verbatim
332: *> ITHRESH is INTEGER
333: *> The maximum number of residual computations allowed for
334: *> refinement. The default is 10. For 'aggressive' set to 100 to
335: *> permit convergence using approximate factorizations or
336: *> factorizations other than LU. If the factorization uses a
337: *> technique other than Gaussian elimination, the guarantees in
338: *> ERRS_N and ERRS_C may no longer be trustworthy.
339: *> \endverbatim
340: *>
341: *> \param[in] RTHRESH
342: *> \verbatim
343: *> RTHRESH is DOUBLE PRECISION
344: *> Determines when to stop refinement if the error estimate stops
345: *> decreasing. Refinement will stop when the next solution no longer
346: *> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
347: *> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
348: *> default value is 0.5. For 'aggressive' set to 0.9 to permit
349: *> convergence on extremely ill-conditioned matrices. See LAWN 165
350: *> for more details.
351: *> \endverbatim
352: *>
353: *> \param[in] DZ_UB
354: *> \verbatim
355: *> DZ_UB is DOUBLE PRECISION
356: *> Determines when to start considering componentwise convergence.
357: *> Componentwise convergence is only considered after each component
358: *> of the solution Y is stable, which we definte as the relative
359: *> change in each component being less than DZ_UB. The default value
360: *> is 0.25, requiring the first bit to be stable. See LAWN 165 for
361: *> more details.
362: *> \endverbatim
363: *>
364: *> \param[in] IGNORE_CWISE
365: *> \verbatim
366: *> IGNORE_CWISE is LOGICAL
367: *> If .TRUE. then ignore componentwise convergence. Default value
368: *> is .FALSE..
369: *> \endverbatim
370: *>
371: *> \param[out] INFO
372: *> \verbatim
373: *> INFO is INTEGER
374: *> = 0: Successful exit.
375: *> < 0: if INFO = -i, the ith argument to DGETRS had an illegal
376: *> value
377: *> \endverbatim
378: *
379: * Authors:
380: * ========
381: *
1.12 bertrand 382: *> \author Univ. of Tennessee
383: *> \author Univ. of California Berkeley
384: *> \author Univ. of Colorado Denver
385: *> \author NAG Ltd.
1.5 bertrand 386: *
1.14 bertrand 387: *> \date June 2017
1.5 bertrand 388: *
389: *> \ingroup doubleGEcomputational
390: *
391: * =====================================================================
1.1 bertrand 392: SUBROUTINE DLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, NRHS, A,
393: $ LDA, AF, LDAF, IPIV, COLEQU, C, B,
394: $ LDB, Y, LDY, BERR_OUT, N_NORMS,
395: $ ERRS_N, ERRS_C, RES, AYB, DY,
396: $ Y_TAIL, RCOND, ITHRESH, RTHRESH,
397: $ DZ_UB, IGNORE_CWISE, INFO )
398: *
1.14 bertrand 399: * -- LAPACK computational routine (version 3.7.1) --
1.5 bertrand 400: * -- LAPACK is a software package provided by Univ. of Tennessee, --
401: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.14 bertrand 402: * June 2017
1.1 bertrand 403: *
404: * .. Scalar Arguments ..
405: INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
406: $ TRANS_TYPE, N_NORMS, ITHRESH
407: LOGICAL COLEQU, IGNORE_CWISE
408: DOUBLE PRECISION RTHRESH, DZ_UB
409: * ..
410: * .. Array Arguments ..
411: INTEGER IPIV( * )
412: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
413: $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
414: DOUBLE PRECISION C( * ), AYB( * ), RCOND, BERR_OUT( * ),
415: $ ERRS_N( NRHS, * ), ERRS_C( NRHS, * )
416: * ..
417: *
418: * =====================================================================
419: *
420: * .. Local Scalars ..
421: CHARACTER TRANS
422: INTEGER CNT, I, J, X_STATE, Z_STATE, Y_PREC_STATE
423: DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
424: $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
425: $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
426: $ EPS, HUGEVAL, INCR_THRESH
427: LOGICAL INCR_PREC
428: * ..
429: * .. Parameters ..
430: INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
431: $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
432: $ EXTRA_Y
433: PARAMETER ( UNSTABLE_STATE = 0, WORKING_STATE = 1,
434: $ CONV_STATE = 2, NOPROG_STATE = 3 )
435: PARAMETER ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1,
436: $ EXTRA_Y = 2 )
437: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
438: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
439: INTEGER CMP_ERR_I, PIV_GROWTH_I
440: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
441: $ BERR_I = 3 )
442: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
443: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
444: $ PIV_GROWTH_I = 9 )
445: INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
446: $ LA_LINRX_CWISE_I
447: PARAMETER ( LA_LINRX_ITREF_I = 1,
448: $ LA_LINRX_ITHRESH_I = 2 )
449: PARAMETER ( LA_LINRX_CWISE_I = 3 )
450: INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
451: $ LA_LINRX_RCOND_I
452: PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
453: PARAMETER ( LA_LINRX_RCOND_I = 3 )
454: * ..
455: * .. External Subroutines ..
456: EXTERNAL DAXPY, DCOPY, DGETRS, DGEMV, BLAS_DGEMV_X,
457: $ BLAS_DGEMV2_X, DLA_GEAMV, DLA_WWADDW, DLAMCH,
458: $ CHLA_TRANSTYPE, DLA_LIN_BERR
459: DOUBLE PRECISION DLAMCH
460: CHARACTER CHLA_TRANSTYPE
461: * ..
462: * .. Intrinsic Functions ..
463: INTRINSIC ABS, MAX, MIN
464: * ..
465: * .. Executable Statements ..
466: *
467: IF ( INFO.NE.0 ) RETURN
468: TRANS = CHLA_TRANSTYPE(TRANS_TYPE)
469: EPS = DLAMCH( 'Epsilon' )
470: HUGEVAL = DLAMCH( 'Overflow' )
471: * Force HUGEVAL to Inf
472: HUGEVAL = HUGEVAL * HUGEVAL
473: * Using HUGEVAL may lead to spurious underflows.
474: INCR_THRESH = DBLE( N ) * EPS
475: *
476: DO J = 1, NRHS
477: Y_PREC_STATE = EXTRA_RESIDUAL
478: IF ( Y_PREC_STATE .EQ. EXTRA_Y ) THEN
479: DO I = 1, N
480: Y_TAIL( I ) = 0.0D+0
481: END DO
482: END IF
483:
484: DXRAT = 0.0D+0
485: DXRATMAX = 0.0D+0
486: DZRAT = 0.0D+0
487: DZRATMAX = 0.0D+0
488: FINAL_DX_X = HUGEVAL
489: FINAL_DZ_Z = HUGEVAL
490: PREVNORMDX = HUGEVAL
491: PREV_DZ_Z = HUGEVAL
492: DZ_Z = HUGEVAL
493: DX_X = HUGEVAL
494:
495: X_STATE = WORKING_STATE
496: Z_STATE = UNSTABLE_STATE
497: INCR_PREC = .FALSE.
498:
499: DO CNT = 1, ITHRESH
500: *
501: * Compute residual RES = B_s - op(A_s) * Y,
502: * op(A) = A, A**T, or A**H depending on TRANS (and type).
503: *
504: CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
505: IF ( Y_PREC_STATE .EQ. BASE_RESIDUAL ) THEN
506: CALL DGEMV( TRANS, N, N, -1.0D+0, A, LDA, Y( 1, J ), 1,
507: $ 1.0D+0, RES, 1 )
508: ELSE IF ( Y_PREC_STATE .EQ. EXTRA_RESIDUAL ) THEN
509: CALL BLAS_DGEMV_X( TRANS_TYPE, N, N, -1.0D+0, A, LDA,
510: $ Y( 1, J ), 1, 1.0D+0, RES, 1, PREC_TYPE )
511: ELSE
512: CALL BLAS_DGEMV2_X( TRANS_TYPE, N, N, -1.0D+0, A, LDA,
513: $ Y( 1, J ), Y_TAIL, 1, 1.0D+0, RES, 1, PREC_TYPE )
514: END IF
515:
516: ! XXX: RES is no longer needed.
517: CALL DCOPY( N, RES, 1, DY, 1 )
518: CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, DY, N, INFO )
519: *
520: * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
521: *
522: NORMX = 0.0D+0
523: NORMY = 0.0D+0
524: NORMDX = 0.0D+0
525: DZ_Z = 0.0D+0
526: YMIN = HUGEVAL
527: *
528: DO I = 1, N
529: YK = ABS( Y( I, J ) )
530: DYK = ABS( DY( I ) )
531:
532: IF ( YK .NE. 0.0D+0 ) THEN
533: DZ_Z = MAX( DZ_Z, DYK / YK )
534: ELSE IF ( DYK .NE. 0.0D+0 ) THEN
535: DZ_Z = HUGEVAL
536: END IF
537:
538: YMIN = MIN( YMIN, YK )
539:
540: NORMY = MAX( NORMY, YK )
541:
542: IF ( COLEQU ) THEN
543: NORMX = MAX( NORMX, YK * C( I ) )
544: NORMDX = MAX( NORMDX, DYK * C( I ) )
545: ELSE
546: NORMX = NORMY
547: NORMDX = MAX( NORMDX, DYK )
548: END IF
549: END DO
550:
551: IF ( NORMX .NE. 0.0D+0 ) THEN
552: DX_X = NORMDX / NORMX
553: ELSE IF ( NORMDX .EQ. 0.0D+0 ) THEN
554: DX_X = 0.0D+0
555: ELSE
556: DX_X = HUGEVAL
557: END IF
558:
559: DXRAT = NORMDX / PREVNORMDX
560: DZRAT = DZ_Z / PREV_DZ_Z
561: *
562: * Check termination criteria
563: *
564: IF (.NOT.IGNORE_CWISE
565: $ .AND. YMIN*RCOND .LT. INCR_THRESH*NORMY
566: $ .AND. Y_PREC_STATE .LT. EXTRA_Y)
567: $ INCR_PREC = .TRUE.
568:
569: IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH )
570: $ X_STATE = WORKING_STATE
571: IF ( X_STATE .EQ. WORKING_STATE ) THEN
572: IF ( DX_X .LE. EPS ) THEN
573: X_STATE = CONV_STATE
574: ELSE IF ( DXRAT .GT. RTHRESH ) THEN
575: IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
576: INCR_PREC = .TRUE.
577: ELSE
578: X_STATE = NOPROG_STATE
579: END IF
580: ELSE
581: IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT
582: END IF
583: IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X
584: END IF
585:
586: IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB )
587: $ Z_STATE = WORKING_STATE
588: IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH )
589: $ Z_STATE = WORKING_STATE
590: IF ( Z_STATE .EQ. WORKING_STATE ) THEN
591: IF ( DZ_Z .LE. EPS ) THEN
592: Z_STATE = CONV_STATE
593: ELSE IF ( DZ_Z .GT. DZ_UB ) THEN
594: Z_STATE = UNSTABLE_STATE
595: DZRATMAX = 0.0D+0
596: FINAL_DZ_Z = HUGEVAL
597: ELSE IF ( DZRAT .GT. RTHRESH ) THEN
598: IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
599: INCR_PREC = .TRUE.
600: ELSE
601: Z_STATE = NOPROG_STATE
602: END IF
603: ELSE
604: IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT
605: END IF
606: IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
607: END IF
608: *
609: * Exit if both normwise and componentwise stopped working,
610: * but if componentwise is unstable, let it go at least two
611: * iterations.
612: *
613: IF ( X_STATE.NE.WORKING_STATE ) THEN
614: IF ( IGNORE_CWISE) GOTO 666
615: IF ( Z_STATE.EQ.NOPROG_STATE .OR. Z_STATE.EQ.CONV_STATE )
616: $ GOTO 666
617: IF ( Z_STATE.EQ.UNSTABLE_STATE .AND. CNT.GT.1 ) GOTO 666
618: END IF
619:
620: IF ( INCR_PREC ) THEN
621: INCR_PREC = .FALSE.
622: Y_PREC_STATE = Y_PREC_STATE + 1
623: DO I = 1, N
624: Y_TAIL( I ) = 0.0D+0
625: END DO
626: END IF
627:
628: PREVNORMDX = NORMDX
629: PREV_DZ_Z = DZ_Z
630: *
631: * Update soluton.
632: *
633: IF ( Y_PREC_STATE .LT. EXTRA_Y ) THEN
634: CALL DAXPY( N, 1.0D+0, DY, 1, Y( 1, J ), 1 )
635: ELSE
636: CALL DLA_WWADDW( N, Y( 1, J ), Y_TAIL, DY )
637: END IF
638:
639: END DO
640: * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
641: 666 CONTINUE
642: *
643: * Set final_* when cnt hits ithresh.
644: *
645: IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X
646: IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
647: *
648: * Compute error bounds
649: *
650: IF (N_NORMS .GE. 1) THEN
651: ERRS_N( J, LA_LINRX_ERR_I ) = FINAL_DX_X / (1 - DXRATMAX)
652: END IF
653: IF ( N_NORMS .GE. 2 ) THEN
654: ERRS_C( J, LA_LINRX_ERR_I ) = FINAL_DZ_Z / (1 - DZRATMAX)
655: END IF
656: *
657: * Compute componentwise relative backward error from formula
658: * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
659: * where abs(Z) is the componentwise absolute value of the matrix
660: * or vector Z.
661: *
662: * Compute residual RES = B_s - op(A_s) * Y,
663: * op(A) = A, A**T, or A**H depending on TRANS (and type).
664: *
665: CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
1.12 bertrand 666: CALL DGEMV( TRANS, N, N, -1.0D+0, A, LDA, Y(1,J), 1, 1.0D+0,
1.1 bertrand 667: $ RES, 1 )
668:
669: DO I = 1, N
670: AYB( I ) = ABS( B( I, J ) )
671: END DO
672: *
673: * Compute abs(op(A_s))*abs(Y) + abs(B_s).
674: *
675: CALL DLA_GEAMV ( TRANS_TYPE, N, N, 1.0D+0,
676: $ A, LDA, Y(1, J), 1, 1.0D+0, AYB, 1 )
677:
678: CALL DLA_LIN_BERR ( N, N, 1, RES, AYB, BERR_OUT( J ) )
679: *
680: * End of loop for each RHS.
681: *
682: END DO
683: *
684: RETURN
685: END
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