001:       SUBROUTINE DLA_PORFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA,
002:      $                                AF, LDAF, COLEQU, C, B, LDB, Y,
003:      $                                LDY, BERR_OUT, N_NORMS,
004:      $                                ERR_BNDS_NORM, ERR_BNDS_COMP, RES,
005:      $                                AYB, DY, Y_TAIL, RCOND, ITHRESH,
006:      $                                RTHRESH, DZ_UB, IGNORE_CWISE,
007:      $                                INFO )
008: *
009: *     -- LAPACK routine (version 3.2.1)                                 --
010: *     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
011: *     -- Jason Riedy of Univ. of California Berkeley.                 --
012: *     -- April 2009                                                   --
013: *
014: *     -- LAPACK is a software package provided by Univ. of Tennessee, --
015: *     -- Univ. of California Berkeley and NAG Ltd.                    --
016: *
017:       IMPLICIT NONE
018: *     ..
019: *     .. Scalar Arguments ..
020:       INTEGER            INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
021:      $                   N_NORMS, ITHRESH
022:       CHARACTER          UPLO
023:       LOGICAL            COLEQU, IGNORE_CWISE
024:       DOUBLE PRECISION   RTHRESH, DZ_UB
025: *     ..
026: *     .. Array Arguments ..
027:       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
028:      $                   Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
029:       DOUBLE PRECISION   C( * ), AYB(*), RCOND, BERR_OUT( * ),
030:      $                   ERR_BNDS_NORM( NRHS, * ),
031:      $                   ERR_BNDS_COMP( NRHS, * )
032: *     ..
033: *
034: *  Purpose
035: *  =======
036: *
037: *  DLA_PORFSX_EXTENDED improves the computed solution to a system of
038: *  linear equations by performing extra-precise iterative refinement
039: *  and provides error bounds and backward error estimates for the solution.
040: *  This subroutine is called by DPORFSX to perform iterative refinement.
041: *  In addition to normwise error bound, the code provides maximum
042: *  componentwise error bound if possible. See comments for ERR_BNDS_NORM
043: *  and ERR_BNDS_COMP for details of the error bounds. Note that this
044: *  subroutine is only resonsible for setting the second fields of
045: *  ERR_BNDS_NORM and ERR_BNDS_COMP.
046: *
047: *  Arguments
048: *  =========
049: *
050: *     PREC_TYPE      (input) INTEGER
051: *     Specifies the intermediate precision to be used in refinement.
052: *     The value is defined by ILAPREC(P) where P is a CHARACTER and
053: *     P    = 'S':  Single
054: *          = 'D':  Double
055: *          = 'I':  Indigenous
056: *          = 'X', 'E':  Extra
057: *
058: *     UPLO    (input) CHARACTER*1
059: *       = 'U':  Upper triangle of A is stored;
060: *       = 'L':  Lower triangle of A is stored.
061: *
062: *     N              (input) INTEGER
063: *     The number of linear equations, i.e., the order of the
064: *     matrix A.  N >= 0.
065: *
066: *     NRHS           (input) INTEGER
067: *     The number of right-hand-sides, i.e., the number of columns of the
068: *     matrix B.
069: *
070: *     A              (input) DOUBLE PRECISION array, dimension (LDA,N)
071: *     On entry, the N-by-N matrix A.
072: *
073: *     LDA            (input) INTEGER
074: *     The leading dimension of the array A.  LDA >= max(1,N).
075: *
076: *     AF             (input) DOUBLE PRECISION array, dimension (LDAF,N)
077: *     The triangular factor U or L from the Cholesky factorization
078: *     A = U**T*U or A = L*L**T, as computed by DPOTRF.
079: *
080: *     LDAF           (input) INTEGER
081: *     The leading dimension of the array AF.  LDAF >= max(1,N).
082: *
083: *     COLEQU         (input) LOGICAL
084: *     If .TRUE. then column equilibration was done to A before calling
085: *     this routine. This is needed to compute the solution and error
086: *     bounds correctly.
087: *
088: *     C              (input) DOUBLE PRECISION array, dimension (N)
089: *     The column scale factors for A. If COLEQU = .FALSE., C
090: *     is not accessed. If C is input, each element of C should be a power
091: *     of the radix to ensure a reliable solution and error estimates.
092: *     Scaling by powers of the radix does not cause rounding errors unless
093: *     the result underflows or overflows. Rounding errors during scaling
094: *     lead to refining with a matrix that is not equivalent to the
095: *     input matrix, producing error estimates that may not be
096: *     reliable.
097: *
098: *     B              (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
099: *     The right-hand-side matrix B.
100: *
101: *     LDB            (input) INTEGER
102: *     The leading dimension of the array B.  LDB >= max(1,N).
103: *
104: *     Y              (input/output) DOUBLE PRECISION array, dimension
105: *                    (LDY,NRHS)
106: *     On entry, the solution matrix X, as computed by DPOTRS.
107: *     On exit, the improved solution matrix Y.
108: *
109: *     LDY            (input) INTEGER
110: *     The leading dimension of the array Y.  LDY >= max(1,N).
111: *
112: *     BERR_OUT       (output) DOUBLE PRECISION array, dimension (NRHS)
113: *     On exit, BERR_OUT(j) contains the componentwise relative backward
114: *     error for right-hand-side j from the formula
115: *         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
116: *     where abs(Z) is the componentwise absolute value of the matrix
117: *     or vector Z. This is computed by DLA_LIN_BERR.
118: *
119: *     N_NORMS        (input) INTEGER
120: *     Determines which error bounds to return (see ERR_BNDS_NORM
121: *     and ERR_BNDS_COMP).
122: *     If N_NORMS >= 1 return normwise error bounds.
123: *     If N_NORMS >= 2 return componentwise error bounds.
124: *
125: *     ERR_BNDS_NORM  (input/output) DOUBLE PRECISION array, dimension
126: *                    (NRHS, N_ERR_BNDS)
127: *     For each right-hand side, this array contains information about
128: *     various error bounds and condition numbers corresponding to the
129: *     normwise relative error, which is defined as follows:
130: *
131: *     Normwise relative error in the ith solution vector:
132: *             max_j (abs(XTRUE(j,i) - X(j,i)))
133: *            ------------------------------
134: *                  max_j abs(X(j,i))
135: *
136: *     The array is indexed by the type of error information as described
137: *     below. There currently are up to three pieces of information
138: *     returned.
139: *
140: *     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
141: *     right-hand side.
142: *
143: *     The second index in ERR_BNDS_NORM(:,err) contains the following
144: *     three fields:
145: *     err = 1 "Trust/don't trust" boolean. Trust the answer if the
146: *              reciprocal condition number is less than the threshold
147: *              sqrt(n) * slamch('Epsilon').
148: *
149: *     err = 2 "Guaranteed" error bound: The estimated forward error,
150: *              almost certainly within a factor of 10 of the true error
151: *              so long as the next entry is greater than the threshold
152: *              sqrt(n) * slamch('Epsilon'). This error bound should only
153: *              be trusted if the previous boolean is true.
154: *
155: *     err = 3  Reciprocal condition number: Estimated normwise
156: *              reciprocal condition number.  Compared with the threshold
157: *              sqrt(n) * slamch('Epsilon') to determine if the error
158: *              estimate is "guaranteed". These reciprocal condition
159: *              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
160: *              appropriately scaled matrix Z.
161: *              Let Z = S*A, where S scales each row by a power of the
162: *              radix so all absolute row sums of Z are approximately 1.
163: *
164: *     This subroutine is only responsible for setting the second field
165: *     above.
166: *     See Lapack Working Note 165 for further details and extra
167: *     cautions.
168: *
169: *     ERR_BNDS_COMP  (input/output) DOUBLE PRECISION array, dimension
170: *                    (NRHS, N_ERR_BNDS)
171: *     For each right-hand side, this array contains information about
172: *     various error bounds and condition numbers corresponding to the
173: *     componentwise relative error, which is defined as follows:
174: *
175: *     Componentwise relative error in the ith solution vector:
176: *                    abs(XTRUE(j,i) - X(j,i))
177: *             max_j ----------------------
178: *                         abs(X(j,i))
179: *
180: *     The array is indexed by the right-hand side i (on which the
181: *     componentwise relative error depends), and the type of error
182: *     information as described below. There currently are up to three
183: *     pieces of information returned for each right-hand side. If
184: *     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
185: *     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
186: *     the first (:,N_ERR_BNDS) entries are returned.
187: *
188: *     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
189: *     right-hand side.
190: *
191: *     The second index in ERR_BNDS_COMP(:,err) contains the following
192: *     three fields:
193: *     err = 1 "Trust/don't trust" boolean. Trust the answer if the
194: *              reciprocal condition number is less than the threshold
195: *              sqrt(n) * slamch('Epsilon').
196: *
197: *     err = 2 "Guaranteed" error bound: The estimated forward error,
198: *              almost certainly within a factor of 10 of the true error
199: *              so long as the next entry is greater than the threshold
200: *              sqrt(n) * slamch('Epsilon'). This error bound should only
201: *              be trusted if the previous boolean is true.
202: *
203: *     err = 3  Reciprocal condition number: Estimated componentwise
204: *              reciprocal condition number.  Compared with the threshold
205: *              sqrt(n) * slamch('Epsilon') to determine if the error
206: *              estimate is "guaranteed". These reciprocal condition
207: *              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
208: *              appropriately scaled matrix Z.
209: *              Let Z = S*(A*diag(x)), where x is the solution for the
210: *              current right-hand side and S scales each row of
211: *              A*diag(x) by a power of the radix so all absolute row
212: *              sums of Z are approximately 1.
213: *
214: *     This subroutine is only responsible for setting the second field
215: *     above.
216: *     See Lapack Working Note 165 for further details and extra
217: *     cautions.
218: *
219: *     RES            (input) DOUBLE PRECISION array, dimension (N)
220: *     Workspace to hold the intermediate residual.
221: *
222: *     AYB            (input) DOUBLE PRECISION array, dimension (N)
223: *     Workspace. This can be the same workspace passed for Y_TAIL.
224: *
225: *     DY             (input) DOUBLE PRECISION array, dimension (N)
226: *     Workspace to hold the intermediate solution.
227: *
228: *     Y_TAIL         (input) DOUBLE PRECISION array, dimension (N)
229: *     Workspace to hold the trailing bits of the intermediate solution.
230: *
231: *     RCOND          (input) DOUBLE PRECISION
232: *     Reciprocal scaled condition number.  This is an estimate of the
233: *     reciprocal Skeel condition number of the matrix A after
234: *     equilibration (if done).  If this is less than the machine
235: *     precision (in particular, if it is zero), the matrix is singular
236: *     to working precision.  Note that the error may still be small even
237: *     if this number is very small and the matrix appears ill-
238: *     conditioned.
239: *
240: *     ITHRESH        (input) INTEGER
241: *     The maximum number of residual computations allowed for
242: *     refinement. The default is 10. For 'aggressive' set to 100 to
243: *     permit convergence using approximate factorizations or
244: *     factorizations other than LU. If the factorization uses a
245: *     technique other than Gaussian elimination, the guarantees in
246: *     ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
247: *
248: *     RTHRESH        (input) DOUBLE PRECISION
249: *     Determines when to stop refinement if the error estimate stops
250: *     decreasing. Refinement will stop when the next solution no longer
251: *     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
252: *     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
253: *     default value is 0.5. For 'aggressive' set to 0.9 to permit
254: *     convergence on extremely ill-conditioned matrices. See LAWN 165
255: *     for more details.
256: *
257: *     DZ_UB          (input) DOUBLE PRECISION
258: *     Determines when to start considering componentwise convergence.
259: *     Componentwise convergence is only considered after each component
260: *     of the solution Y is stable, which we definte as the relative
261: *     change in each component being less than DZ_UB. The default value
262: *     is 0.25, requiring the first bit to be stable. See LAWN 165 for
263: *     more details.
264: *
265: *     IGNORE_CWISE   (input) LOGICAL
266: *     If .TRUE. then ignore componentwise convergence. Default value
267: *     is .FALSE..
268: *
269: *     INFO           (output) INTEGER
270: *       = 0:  Successful exit.
271: *       < 0:  if INFO = -i, the ith argument to DPOTRS had an illegal
272: *             value
273: *
274: *  =====================================================================
275: *
276: *     .. Local Scalars ..
277:       INTEGER            UPLO2, CNT, I, J, X_STATE, Z_STATE
278:       DOUBLE PRECISION   YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
279:      $                   DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
280:      $                   DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
281:      $                   EPS, HUGEVAL, INCR_THRESH
282:       LOGICAL            INCR_PREC
283: *     ..
284: *     .. Parameters ..
285:       INTEGER           UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
286:      $                  NOPROG_STATE, Y_PREC_STATE, BASE_RESIDUAL,
287:      $                  EXTRA_RESIDUAL, EXTRA_Y
288:       PARAMETER         ( UNSTABLE_STATE = 0, WORKING_STATE = 1,
289:      $                  CONV_STATE = 2, NOPROG_STATE = 3 )
290:       PARAMETER         ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1,
291:      $                  EXTRA_Y = 2 )
292:       INTEGER            FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
293:       INTEGER            RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
294:       INTEGER            CMP_ERR_I, PIV_GROWTH_I
295:       PARAMETER          ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
296:      $                   BERR_I = 3 )
297:       PARAMETER          ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
298:       PARAMETER          ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
299:      $                   PIV_GROWTH_I = 9 )
300:       INTEGER            LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
301:      $                   LA_LINRX_CWISE_I
302:       PARAMETER          ( LA_LINRX_ITREF_I = 1,
303:      $                   LA_LINRX_ITHRESH_I = 2 )
304:       PARAMETER          ( LA_LINRX_CWISE_I = 3 )
305:       INTEGER            LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
306:      $                   LA_LINRX_RCOND_I
307:       PARAMETER          ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
308:       PARAMETER          ( LA_LINRX_RCOND_I = 3 )
309: *     ..
310: *     .. External Functions ..
311:       LOGICAL            LSAME
312:       EXTERNAL           ILAUPLO
313:       INTEGER            ILAUPLO
314: *     ..
315: *     .. External Subroutines ..
316:       EXTERNAL          DAXPY, DCOPY, DPOTRS, DSYMV, BLAS_DSYMV_X,
317:      $                  BLAS_DSYMV2_X, DLA_SYAMV, DLA_WWADDW,
318:      $                  DLA_LIN_BERR
319:       DOUBLE PRECISION   DLAMCH
320: *     ..
321: *     .. Intrinsic Functions ..
322:       INTRINSIC         ABS, MAX, MIN
323: *     ..
324: *     .. Executable Statements ..
325: *
326:       IF (INFO.NE.0) RETURN
327:       EPS = DLAMCH( 'Epsilon' )
328:       HUGEVAL = DLAMCH( 'Overflow' )
329: *     Force HUGEVAL to Inf
330:       HUGEVAL = HUGEVAL * HUGEVAL
331: *     Using HUGEVAL may lead to spurious underflows.
332:       INCR_THRESH = DBLE( N ) * EPS
333: 
334:       IF ( LSAME ( UPLO, 'L' ) ) THEN
335:          UPLO2 = ILAUPLO( 'L' )
336:       ELSE
337:          UPLO2 = ILAUPLO( 'U' )
338:       ENDIF
339: 
340:       DO J = 1, NRHS
341:          Y_PREC_STATE = EXTRA_RESIDUAL
342:          IF ( Y_PREC_STATE .EQ. EXTRA_Y ) THEN
343:             DO I = 1, N
344:                Y_TAIL( I ) = 0.0D+0
345:             END DO
346:          END IF
347: 
348:          DXRAT = 0.0D+0
349:          DXRATMAX = 0.0D+0
350:          DZRAT = 0.0D+0
351:          DZRATMAX = 0.0D+0
352:          FINAL_DX_X = HUGEVAL
353:          FINAL_DZ_Z = HUGEVAL
354:          PREVNORMDX = HUGEVAL
355:          PREV_DZ_Z = HUGEVAL
356:          DZ_Z = HUGEVAL
357:          DX_X = HUGEVAL
358: 
359:          X_STATE = WORKING_STATE
360:          Z_STATE = UNSTABLE_STATE
361:          INCR_PREC = .FALSE.
362: 
363:          DO CNT = 1, ITHRESH
364: *
365: *         Compute residual RES = B_s - op(A_s) * Y,
366: *             op(A) = A, A**T, or A**H depending on TRANS (and type).
367: *
368:             CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
369:             IF ( Y_PREC_STATE .EQ. BASE_RESIDUAL ) THEN
370:                CALL DSYMV( UPLO, N, -1.0D+0, A, LDA, Y(1,J), 1,
371:      $              1.0D+0, RES, 1 )
372:             ELSE IF ( Y_PREC_STATE .EQ. EXTRA_RESIDUAL ) THEN
373:                CALL BLAS_DSYMV_X( UPLO2, N, -1.0D+0, A, LDA,
374:      $              Y( 1, J ), 1, 1.0D+0, RES, 1, PREC_TYPE )
375:             ELSE
376:                CALL BLAS_DSYMV2_X(UPLO2, N, -1.0D+0, A, LDA,
377:      $              Y(1, J), Y_TAIL, 1, 1.0D+0, RES, 1, PREC_TYPE)
378:             END IF
379: 
380: !         XXX: RES is no longer needed.
381:             CALL DCOPY( N, RES, 1, DY, 1 )
382:             CALL DPOTRS( UPLO, N, NRHS, AF, LDAF, DY, N, INFO )
383: *
384: *         Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
385: *
386:             NORMX = 0.0D+0
387:             NORMY = 0.0D+0
388:             NORMDX = 0.0D+0
389:             DZ_Z = 0.0D+0
390:             YMIN = HUGEVAL
391: 
392:             DO I = 1, N
393:                YK = ABS( Y( I, J ) )
394:                DYK = ABS( DY( I ) )
395: 
396:                IF ( YK .NE. 0.0D+0 ) THEN
397:                   DZ_Z = MAX( DZ_Z, DYK / YK )
398:                ELSE IF ( DYK .NE. 0.0D+0 ) THEN
399:                   DZ_Z = HUGEVAL
400:                END IF
401: 
402:                YMIN = MIN( YMIN, YK )
403: 
404:                NORMY = MAX( NORMY, YK )
405: 
406:                IF ( COLEQU ) THEN
407:                   NORMX = MAX( NORMX, YK * C( I ) )
408:                   NORMDX = MAX( NORMDX, DYK * C( I ) )
409:                ELSE
410:                   NORMX = NORMY
411:                   NORMDX = MAX( NORMDX, DYK )
412:                END IF
413:             END DO
414: 
415:             IF ( NORMX .NE. 0.0D+0 ) THEN
416:                DX_X = NORMDX / NORMX
417:             ELSE IF ( NORMDX .EQ. 0.0D+0 ) THEN
418:                DX_X = 0.0D+0
419:             ELSE
420:                DX_X = HUGEVAL
421:             END IF
422: 
423:             DXRAT = NORMDX / PREVNORMDX
424:             DZRAT = DZ_Z / PREV_DZ_Z
425: *
426: *         Check termination criteria.
427: *
428:             IF ( YMIN*RCOND .LT. INCR_THRESH*NORMY
429:      $           .AND. Y_PREC_STATE .LT. EXTRA_Y )
430:      $           INCR_PREC = .TRUE.
431: 
432:             IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH )
433:      $           X_STATE = WORKING_STATE
434:             IF ( X_STATE .EQ. WORKING_STATE ) THEN
435:                IF ( DX_X .LE. EPS ) THEN
436:                   X_STATE = CONV_STATE
437:                ELSE IF ( DXRAT .GT. RTHRESH ) THEN
438:                   IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
439:                      INCR_PREC = .TRUE.
440:                   ELSE
441:                      X_STATE = NOPROG_STATE
442:                   END IF
443:                ELSE
444:                   IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT
445:                END IF
446:                IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X
447:             END IF
448: 
449:             IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB )
450:      $           Z_STATE = WORKING_STATE
451:             IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH )
452:      $           Z_STATE = WORKING_STATE
453:             IF ( Z_STATE .EQ. WORKING_STATE ) THEN
454:                IF ( DZ_Z .LE. EPS ) THEN
455:                   Z_STATE = CONV_STATE
456:                ELSE IF ( DZ_Z .GT. DZ_UB ) THEN
457:                   Z_STATE = UNSTABLE_STATE
458:                   DZRATMAX = 0.0D+0
459:                   FINAL_DZ_Z = HUGEVAL
460:                ELSE IF ( DZRAT .GT. RTHRESH ) THEN
461:                   IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
462:                      INCR_PREC = .TRUE.
463:                   ELSE
464:                      Z_STATE = NOPROG_STATE
465:                   END IF
466:                ELSE
467:                   IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT
468:                END IF
469:                IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
470:             END IF
471: 
472:             IF ( X_STATE.NE.WORKING_STATE.AND.
473:      $           ( IGNORE_CWISE.OR.Z_STATE.NE.WORKING_STATE ) )
474:      $           GOTO 666
475: 
476:             IF ( INCR_PREC ) THEN
477:                INCR_PREC = .FALSE.
478:                Y_PREC_STATE = Y_PREC_STATE + 1
479:                DO I = 1, N
480:                   Y_TAIL( I ) = 0.0D+0
481:                END DO
482:             END IF
483: 
484:             PREVNORMDX = NORMDX
485:             PREV_DZ_Z = DZ_Z
486: *
487: *           Update soluton.
488: *
489:             IF (Y_PREC_STATE .LT. EXTRA_Y) THEN
490:                CALL DAXPY( N, 1.0D+0, DY, 1, Y(1,J), 1 )
491:             ELSE
492:                CALL DLA_WWADDW( N, Y( 1, J ), Y_TAIL, DY )
493:             END IF
494: 
495:          END DO
496: *        Target of "IF (Z_STOP .AND. X_STOP)".  Sun's f77 won't EXIT.
497:  666     CONTINUE
498: *
499: *     Set final_* when cnt hits ithresh.
500: *
501:          IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X
502:          IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
503: *
504: *     Compute error bounds.
505: *
506:          IF ( N_NORMS .GE. 1 ) THEN
507:             ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) =
508:      $           FINAL_DX_X / (1 - DXRATMAX)
509:          END IF
510:          IF ( N_NORMS .GE. 2 ) THEN
511:             ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) =
512:      $           FINAL_DZ_Z / (1 - DZRATMAX)
513:          END IF
514: *
515: *     Compute componentwise relative backward error from formula
516: *         max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
517: *     where abs(Z) is the componentwise absolute value of the matrix
518: *     or vector Z.
519: *
520: *        Compute residual RES = B_s - op(A_s) * Y,
521: *            op(A) = A, A**T, or A**H depending on TRANS (and type).
522: *
523:          CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
524:          CALL DSYMV( UPLO, N, -1.0D+0, A, LDA, Y(1,J), 1, 1.0D+0, RES,
525:      $     1 )
526: 
527:          DO I = 1, N
528:             AYB( I ) = ABS( B( I, J ) )
529:          END DO
530: *
531: *     Compute abs(op(A_s))*abs(Y) + abs(B_s).
532: *
533:          CALL DLA_SYAMV( UPLO2, N, 1.0D+0,
534:      $        A, LDA, Y(1, J), 1, 1.0D+0, AYB, 1 )
535: 
536:          CALL DLA_LIN_BERR( N, N, 1, RES, AYB, BERR_OUT( J ) )
537: *
538: *     End of loop for each RHS.
539: *
540:       END DO
541: *
542:       RETURN
543:       END
544: