1: SUBROUTINE DLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
2: $ NRHS, AB, LDAB, AFB, LDAFB, IPIV,
3: $ COLEQU, C, B, LDB, Y, LDY,
4: $ BERR_OUT, N_NORMS, ERR_BNDS_NORM,
5: $ ERR_BNDS_COMP, RES, AYB, DY,
6: $ Y_TAIL, RCOND, ITHRESH, RTHRESH,
7: $ DZ_UB, IGNORE_CWISE, INFO )
8: *
9: * -- LAPACK routine (version 3.2.1) --
10: * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
11: * -- Jason Riedy of Univ. of California Berkeley. --
12: * -- April 2009 --
13: *
14: * -- LAPACK is a software package provided by Univ. of Tennessee, --
15: * -- Univ. of California Berkeley and NAG Ltd. --
16: *
17: IMPLICIT NONE
18: * ..
19: * .. Scalar Arguments ..
20: INTEGER INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
21: $ PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
22: LOGICAL COLEQU, IGNORE_CWISE
23: DOUBLE PRECISION RTHRESH, DZ_UB
24: * ..
25: * .. Array Arguments ..
26: INTEGER IPIV( * )
27: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
28: $ Y( LDY, * ), RES(*), DY(*), Y_TAIL(*)
29: DOUBLE PRECISION C( * ), AYB(*), RCOND, BERR_OUT(*),
30: $ ERR_BNDS_NORM( NRHS, * ),
31: $ ERR_BNDS_COMP( NRHS, * )
32: * ..
33: *
34: * Purpose
35: * =======
36: *
37: * DLA_GBRFSX_EXTENDED improves the computed solution to a system of
38: * linear equations by performing extra-precise iterative refinement
39: * and provides error bounds and backward error estimates for the solution.
40: * This subroutine is called by DGBRFSX to perform iterative refinement.
41: * In addition to normwise error bound, the code provides maximum
42: * componentwise error bound if possible. See comments for ERR_BNDS_NORM
43: * and ERR_BNDS_COMP for details of the error bounds. Note that this
44: * subroutine is only resonsible for setting the second fields of
45: * ERR_BNDS_NORM and ERR_BNDS_COMP.
46: *
47: * Arguments
48: * =========
49: *
50: * PREC_TYPE (input) INTEGER
51: * Specifies the intermediate precision to be used in refinement.
52: * The value is defined by ILAPREC(P) where P is a CHARACTER and
53: * P = 'S': Single
54: * = 'D': Double
55: * = 'I': Indigenous
56: * = 'X', 'E': Extra
57: *
58: * TRANS_TYPE (input) INTEGER
59: * Specifies the transposition operation on A.
60: * The value is defined by ILATRANS(T) where T is a CHARACTER and
61: * T = 'N': No transpose
62: * = 'T': Transpose
63: * = 'C': Conjugate transpose
64: *
65: * N (input) INTEGER
66: * The number of linear equations, i.e., the order of the
67: * matrix A. N >= 0.
68: *
69: * KL (input) INTEGER
70: * The number of subdiagonals within the band of A. KL >= 0.
71: *
72: * KU (input) INTEGER
73: * The number of superdiagonals within the band of A. KU >= 0
74: *
75: * NRHS (input) INTEGER
76: * The number of right-hand-sides, i.e., the number of columns of the
77: * matrix B.
78: *
79: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
80: * On entry, the N-by-N matrix A.
81: *
82: * LDA (input) INTEGER
83: * The leading dimension of the array A. LDA >= max(1,N).
84: *
85: * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
86: * The factors L and U from the factorization
87: * A = P*L*U as computed by DGBTRF.
88: *
89: * LDAF (input) INTEGER
90: * The leading dimension of the array AF. LDAF >= max(1,N).
91: *
92: * IPIV (input) INTEGER array, dimension (N)
93: * The pivot indices from the factorization A = P*L*U
94: * as computed by DGBTRF; row i of the matrix was interchanged
95: * with row IPIV(i).
96: *
97: * COLEQU (input) LOGICAL
98: * If .TRUE. then column equilibration was done to A before calling
99: * this routine. This is needed to compute the solution and error
100: * bounds correctly.
101: *
102: * C (input) DOUBLE PRECISION array, dimension (N)
103: * The column scale factors for A. If COLEQU = .FALSE., C
104: * is not accessed. If C is input, each element of C should be a power
105: * of the radix to ensure a reliable solution and error estimates.
106: * Scaling by powers of the radix does not cause rounding errors unless
107: * the result underflows or overflows. Rounding errors during scaling
108: * lead to refining with a matrix that is not equivalent to the
109: * input matrix, producing error estimates that may not be
110: * reliable.
111: *
112: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
113: * The right-hand-side matrix B.
114: *
115: * LDB (input) INTEGER
116: * The leading dimension of the array B. LDB >= max(1,N).
117: *
118: * Y (input/output) DOUBLE PRECISION array, dimension
119: * (LDY,NRHS)
120: * On entry, the solution matrix X, as computed by DGBTRS.
121: * On exit, the improved solution matrix Y.
122: *
123: * LDY (input) INTEGER
124: * The leading dimension of the array Y. LDY >= max(1,N).
125: *
126: * BERR_OUT (output) DOUBLE PRECISION array, dimension (NRHS)
127: * On exit, BERR_OUT(j) contains the componentwise relative backward
128: * error for right-hand-side j from the formula
129: * max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
130: * where abs(Z) is the componentwise absolute value of the matrix
131: * or vector Z. This is computed by DLA_LIN_BERR.
132: *
133: * N_NORMS (input) INTEGER
134: * Determines which error bounds to return (see ERR_BNDS_NORM
135: * and ERR_BNDS_COMP).
136: * If N_NORMS >= 1 return normwise error bounds.
137: * If N_NORMS >= 2 return componentwise error bounds.
138: *
139: * ERR_BNDS_NORM (input/output) DOUBLE PRECISION array, dimension
140: * (NRHS, N_ERR_BNDS)
141: * For each right-hand side, this array contains information about
142: * various error bounds and condition numbers corresponding to the
143: * normwise relative error, which is defined as follows:
144: *
145: * Normwise relative error in the ith solution vector:
146: * max_j (abs(XTRUE(j,i) - X(j,i)))
147: * ------------------------------
148: * max_j abs(X(j,i))
149: *
150: * The array is indexed by the type of error information as described
151: * below. There currently are up to three pieces of information
152: * returned.
153: *
154: * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
155: * right-hand side.
156: *
157: * The second index in ERR_BNDS_NORM(:,err) contains the following
158: * three fields:
159: * err = 1 "Trust/don't trust" boolean. Trust the answer if the
160: * reciprocal condition number is less than the threshold
161: * sqrt(n) * slamch('Epsilon').
162: *
163: * err = 2 "Guaranteed" error bound: The estimated forward error,
164: * almost certainly within a factor of 10 of the true error
165: * so long as the next entry is greater than the threshold
166: * sqrt(n) * slamch('Epsilon'). This error bound should only
167: * be trusted if the previous boolean is true.
168: *
169: * err = 3 Reciprocal condition number: Estimated normwise
170: * reciprocal condition number. Compared with the threshold
171: * sqrt(n) * slamch('Epsilon') to determine if the error
172: * estimate is "guaranteed". These reciprocal condition
173: * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
174: * appropriately scaled matrix Z.
175: * Let Z = S*A, where S scales each row by a power of the
176: * radix so all absolute row sums of Z are approximately 1.
177: *
178: * This subroutine is only responsible for setting the second field
179: * above.
180: * See Lapack Working Note 165 for further details and extra
181: * cautions.
182: *
183: * ERR_BNDS_COMP (input/output) DOUBLE PRECISION array, dimension
184: * (NRHS, N_ERR_BNDS)
185: * For each right-hand side, this array contains information about
186: * various error bounds and condition numbers corresponding to the
187: * componentwise relative error, which is defined as follows:
188: *
189: * Componentwise relative error in the ith solution vector:
190: * abs(XTRUE(j,i) - X(j,i))
191: * max_j ----------------------
192: * abs(X(j,i))
193: *
194: * The array is indexed by the right-hand side i (on which the
195: * componentwise relative error depends), and the type of error
196: * information as described below. There currently are up to three
197: * pieces of information returned for each right-hand side. If
198: * componentwise accuracy is not requested (PARAMS(3) = 0.0), then
199: * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
200: * the first (:,N_ERR_BNDS) entries are returned.
201: *
202: * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
203: * right-hand side.
204: *
205: * The second index in ERR_BNDS_COMP(:,err) contains the following
206: * three fields:
207: * err = 1 "Trust/don't trust" boolean. Trust the answer if the
208: * reciprocal condition number is less than the threshold
209: * sqrt(n) * slamch('Epsilon').
210: *
211: * err = 2 "Guaranteed" error bound: The estimated forward error,
212: * almost certainly within a factor of 10 of the true error
213: * so long as the next entry is greater than the threshold
214: * sqrt(n) * slamch('Epsilon'). This error bound should only
215: * be trusted if the previous boolean is true.
216: *
217: * err = 3 Reciprocal condition number: Estimated componentwise
218: * reciprocal condition number. Compared with the threshold
219: * sqrt(n) * slamch('Epsilon') to determine if the error
220: * estimate is "guaranteed". These reciprocal condition
221: * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
222: * appropriately scaled matrix Z.
223: * Let Z = S*(A*diag(x)), where x is the solution for the
224: * current right-hand side and S scales each row of
225: * A*diag(x) by a power of the radix so all absolute row
226: * sums of Z are approximately 1.
227: *
228: * This subroutine is only responsible for setting the second field
229: * above.
230: * See Lapack Working Note 165 for further details and extra
231: * cautions.
232: *
233: * RES (input) DOUBLE PRECISION array, dimension (N)
234: * Workspace to hold the intermediate residual.
235: *
236: * AYB (input) DOUBLE PRECISION array, dimension (N)
237: * Workspace. This can be the same workspace passed for Y_TAIL.
238: *
239: * DY (input) DOUBLE PRECISION array, dimension (N)
240: * Workspace to hold the intermediate solution.
241: *
242: * Y_TAIL (input) DOUBLE PRECISION array, dimension (N)
243: * Workspace to hold the trailing bits of the intermediate solution.
244: *
245: * RCOND (input) DOUBLE PRECISION
246: * Reciprocal scaled condition number. This is an estimate of the
247: * reciprocal Skeel condition number of the matrix A after
248: * equilibration (if done). If this is less than the machine
249: * precision (in particular, if it is zero), the matrix is singular
250: * to working precision. Note that the error may still be small even
251: * if this number is very small and the matrix appears ill-
252: * conditioned.
253: *
254: * ITHRESH (input) INTEGER
255: * The maximum number of residual computations allowed for
256: * refinement. The default is 10. For 'aggressive' set to 100 to
257: * permit convergence using approximate factorizations or
258: * factorizations other than LU. If the factorization uses a
259: * technique other than Gaussian elimination, the guarantees in
260: * ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
261: *
262: * RTHRESH (input) DOUBLE PRECISION
263: * Determines when to stop refinement if the error estimate stops
264: * decreasing. Refinement will stop when the next solution no longer
265: * satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
266: * the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
267: * default value is 0.5. For 'aggressive' set to 0.9 to permit
268: * convergence on extremely ill-conditioned matrices. See LAWN 165
269: * for more details.
270: *
271: * DZ_UB (input) DOUBLE PRECISION
272: * Determines when to start considering componentwise convergence.
273: * Componentwise convergence is only considered after each component
274: * of the solution Y is stable, which we definte as the relative
275: * change in each component being less than DZ_UB. The default value
276: * is 0.25, requiring the first bit to be stable. See LAWN 165 for
277: * more details.
278: *
279: * IGNORE_CWISE (input) LOGICAL
280: * If .TRUE. then ignore componentwise convergence. Default value
281: * is .FALSE..
282: *
283: * INFO (output) INTEGER
284: * = 0: Successful exit.
285: * < 0: if INFO = -i, the ith argument to DGBTRS had an illegal
286: * value
287: *
288: * =====================================================================
289: *
290: * .. Local Scalars ..
291: CHARACTER TRANS
292: INTEGER CNT, I, J, M, X_STATE, Z_STATE, Y_PREC_STATE
293: DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
294: $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
295: $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
296: $ EPS, HUGEVAL, INCR_THRESH
297: LOGICAL INCR_PREC
298: * ..
299: * .. Parameters ..
300: INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
301: $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
302: $ EXTRA_Y
303: PARAMETER ( UNSTABLE_STATE = 0, WORKING_STATE = 1,
304: $ CONV_STATE = 2, NOPROG_STATE = 3 )
305: PARAMETER ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1,
306: $ EXTRA_Y = 2 )
307: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
308: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
309: INTEGER CMP_ERR_I, PIV_GROWTH_I
310: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
311: $ BERR_I = 3 )
312: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
313: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
314: $ PIV_GROWTH_I = 9 )
315: INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
316: $ LA_LINRX_CWISE_I
317: PARAMETER ( LA_LINRX_ITREF_I = 1,
318: $ LA_LINRX_ITHRESH_I = 2 )
319: PARAMETER ( LA_LINRX_CWISE_I = 3 )
320: INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
321: $ LA_LINRX_RCOND_I
322: PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
323: PARAMETER ( LA_LINRX_RCOND_I = 3 )
324: * ..
325: * .. External Subroutines ..
326: EXTERNAL DAXPY, DCOPY, DGBTRS, DGBMV, BLAS_DGBMV_X,
327: $ BLAS_DGBMV2_X, DLA_GBAMV, DLA_WWADDW, DLAMCH,
328: $ CHLA_TRANSTYPE, DLA_LIN_BERR
329: DOUBLE PRECISION DLAMCH
330: CHARACTER CHLA_TRANSTYPE
331: * ..
332: * .. Intrinsic Functions ..
333: INTRINSIC ABS, MAX, MIN
334: * ..
335: * .. Executable Statements ..
336: *
337: IF (INFO.NE.0) RETURN
338: TRANS = CHLA_TRANSTYPE(TRANS_TYPE)
339: EPS = DLAMCH( 'Epsilon' )
340: HUGEVAL = DLAMCH( 'Overflow' )
341: * Force HUGEVAL to Inf
342: HUGEVAL = HUGEVAL * HUGEVAL
343: * Using HUGEVAL may lead to spurious underflows.
344: INCR_THRESH = DBLE( N ) * EPS
345: M = KL+KU+1
346:
347: DO J = 1, NRHS
348: Y_PREC_STATE = EXTRA_RESIDUAL
349: IF ( Y_PREC_STATE .EQ. EXTRA_Y ) THEN
350: DO I = 1, N
351: Y_TAIL( I ) = 0.0D+0
352: END DO
353: END IF
354:
355: DXRAT = 0.0D+0
356: DXRATMAX = 0.0D+0
357: DZRAT = 0.0D+0
358: DZRATMAX = 0.0D+0
359: FINAL_DX_X = HUGEVAL
360: FINAL_DZ_Z = HUGEVAL
361: PREVNORMDX = HUGEVAL
362: PREV_DZ_Z = HUGEVAL
363: DZ_Z = HUGEVAL
364: DX_X = HUGEVAL
365:
366: X_STATE = WORKING_STATE
367: Z_STATE = UNSTABLE_STATE
368: INCR_PREC = .FALSE.
369:
370: DO CNT = 1, ITHRESH
371: *
372: * Compute residual RES = B_s - op(A_s) * Y,
373: * op(A) = A, A**T, or A**H depending on TRANS (and type).
374: *
375: CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
376: IF ( Y_PREC_STATE .EQ. BASE_RESIDUAL ) THEN
377: CALL DGBMV( TRANS, M, N, KL, KU, -1.0D+0, AB, LDAB,
378: $ Y( 1, J ), 1, 1.0D+0, RES, 1 )
379: ELSE IF ( Y_PREC_STATE .EQ. EXTRA_RESIDUAL ) THEN
380: CALL BLAS_DGBMV_X( TRANS_TYPE, N, N, KL, KU,
381: $ -1.0D+0, AB, LDAB, Y( 1, J ), 1, 1.0D+0, RES, 1,
382: $ PREC_TYPE )
383: ELSE
384: CALL BLAS_DGBMV2_X( TRANS_TYPE, N, N, KL, KU, -1.0D+0,
385: $ AB, LDAB, Y( 1, J ), Y_TAIL, 1, 1.0D+0, RES, 1,
386: $ PREC_TYPE )
387: END IF
388:
389: ! XXX: RES is no longer needed.
390: CALL DCOPY( N, RES, 1, DY, 1 )
391: CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, DY, N,
392: $ INFO )
393: *
394: * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
395: *
396: NORMX = 0.0D+0
397: NORMY = 0.0D+0
398: NORMDX = 0.0D+0
399: DZ_Z = 0.0D+0
400: YMIN = HUGEVAL
401:
402: DO I = 1, N
403: YK = ABS( Y( I, J ) )
404: DYK = ABS( DY( I ) )
405:
406: IF ( YK .NE. 0.0D+0 ) THEN
407: DZ_Z = MAX( DZ_Z, DYK / YK )
408: ELSE IF ( DYK .NE. 0.0D+0 ) THEN
409: DZ_Z = HUGEVAL
410: END IF
411:
412: YMIN = MIN( YMIN, YK )
413:
414: NORMY = MAX( NORMY, YK )
415:
416: IF ( COLEQU ) THEN
417: NORMX = MAX( NORMX, YK * C( I ) )
418: NORMDX = MAX( NORMDX, DYK * C( I ) )
419: ELSE
420: NORMX = NORMY
421: NORMDX = MAX( NORMDX, DYK )
422: END IF
423: END DO
424:
425: IF ( NORMX .NE. 0.0D+0 ) THEN
426: DX_X = NORMDX / NORMX
427: ELSE IF ( NORMDX .EQ. 0.0D+0 ) THEN
428: DX_X = 0.0D+0
429: ELSE
430: DX_X = HUGEVAL
431: END IF
432:
433: DXRAT = NORMDX / PREVNORMDX
434: DZRAT = DZ_Z / PREV_DZ_Z
435: *
436: * Check termination criteria.
437: *
438: IF ( .NOT.IGNORE_CWISE
439: $ .AND. YMIN*RCOND .LT. INCR_THRESH*NORMY
440: $ .AND. Y_PREC_STATE .LT. EXTRA_Y )
441: $ INCR_PREC = .TRUE.
442:
443: IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH )
444: $ X_STATE = WORKING_STATE
445: IF ( X_STATE .EQ. WORKING_STATE ) THEN
446: IF ( DX_X .LE. EPS ) THEN
447: X_STATE = CONV_STATE
448: ELSE IF ( DXRAT .GT. RTHRESH ) THEN
449: IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
450: INCR_PREC = .TRUE.
451: ELSE
452: X_STATE = NOPROG_STATE
453: END IF
454: ELSE
455: IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT
456: END IF
457: IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X
458: END IF
459:
460: IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB )
461: $ Z_STATE = WORKING_STATE
462: IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH )
463: $ Z_STATE = WORKING_STATE
464: IF ( Z_STATE .EQ. WORKING_STATE ) THEN
465: IF ( DZ_Z .LE. EPS ) THEN
466: Z_STATE = CONV_STATE
467: ELSE IF ( DZ_Z .GT. DZ_UB ) THEN
468: Z_STATE = UNSTABLE_STATE
469: DZRATMAX = 0.0D+0
470: FINAL_DZ_Z = HUGEVAL
471: ELSE IF ( DZRAT .GT. RTHRESH ) THEN
472: IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
473: INCR_PREC = .TRUE.
474: ELSE
475: Z_STATE = NOPROG_STATE
476: END IF
477: ELSE
478: IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT
479: END IF
480: IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
481: END IF
482: *
483: * Exit if both normwise and componentwise stopped working,
484: * but if componentwise is unstable, let it go at least two
485: * iterations.
486: *
487: IF ( X_STATE.NE.WORKING_STATE ) THEN
488: IF ( IGNORE_CWISE ) GOTO 666
489: IF ( Z_STATE.EQ.NOPROG_STATE .OR. Z_STATE.EQ.CONV_STATE )
490: $ GOTO 666
491: IF ( Z_STATE.EQ.UNSTABLE_STATE .AND. CNT.GT.1 ) GOTO 666
492: END IF
493:
494: IF ( INCR_PREC ) THEN
495: INCR_PREC = .FALSE.
496: Y_PREC_STATE = Y_PREC_STATE + 1
497: DO I = 1, N
498: Y_TAIL( I ) = 0.0D+0
499: END DO
500: END IF
501:
502: PREVNORMDX = NORMDX
503: PREV_DZ_Z = DZ_Z
504: *
505: * Update soluton.
506: *
507: IF (Y_PREC_STATE .LT. EXTRA_Y) THEN
508: CALL DAXPY( N, 1.0D+0, DY, 1, Y(1,J), 1 )
509: ELSE
510: CALL DLA_WWADDW( N, Y(1,J), Y_TAIL, DY )
511: END IF
512:
513: END DO
514: * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
515: 666 CONTINUE
516: *
517: * Set final_* when cnt hits ithresh.
518: *
519: IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X
520: IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
521: *
522: * Compute error bounds.
523: *
524: IF ( N_NORMS .GE. 1 ) THEN
525: ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) =
526: $ FINAL_DX_X / (1 - DXRATMAX)
527: END IF
528: IF (N_NORMS .GE. 2) THEN
529: ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) =
530: $ FINAL_DZ_Z / (1 - DZRATMAX)
531: END IF
532: *
533: * Compute componentwise relative backward error from formula
534: * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
535: * where abs(Z) is the componentwise absolute value of the matrix
536: * or vector Z.
537: *
538: * Compute residual RES = B_s - op(A_s) * Y,
539: * op(A) = A, A**T, or A**H depending on TRANS (and type).
540: *
541: CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
542: CALL DGBMV(TRANS, N, N, KL, KU, -1.0D+0, AB, LDAB, Y(1,J),
543: $ 1, 1.0D+0, RES, 1 )
544:
545: DO I = 1, N
546: AYB( I ) = ABS( B( I, J ) )
547: END DO
548: *
549: * Compute abs(op(A_s))*abs(Y) + abs(B_s).
550: *
551: CALL DLA_GBAMV( TRANS_TYPE, N, N, KL, KU, 1.0D+0,
552: $ AB, LDAB, Y(1, J), 1, 1.0D+0, AYB, 1 )
553:
554: CALL DLA_LIN_BERR( N, N, 1, RES, AYB, BERR_OUT( J ) )
555: *
556: * End of loop for each RHS
557: *
558: END DO
559: *
560: RETURN
561: END
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