Annotation of rpl/lapack/lapack/zla_porfsx_extended.f, revision 1.3
1.1 bertrand 1: SUBROUTINE ZLA_PORFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA,
2: $ AF, LDAF, COLEQU, C, B, LDB, Y,
3: $ LDY, BERR_OUT, N_NORMS,
4: $ ERR_BNDS_NORM, ERR_BNDS_COMP, RES,
5: $ AYB, DY, Y_TAIL, RCOND, ITHRESH,
6: $ RTHRESH, DZ_UB, IGNORE_CWISE,
7: $ INFO )
8: *
9: * -- LAPACK routine (version 3.2.2) --
10: * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
11: * -- Jason Riedy of Univ. of California Berkeley. --
12: * -- June 2010 --
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, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
21: $ N_NORMS, ITHRESH
22: CHARACTER UPLO
23: LOGICAL COLEQU, IGNORE_CWISE
24: DOUBLE PRECISION RTHRESH, DZ_UB
25: * ..
26: * .. Array Arguments ..
27: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), 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: * ZLA_PORFSX_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 ZPORFSX 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: * UPLO (input) CHARACTER*1
59: * = 'U': Upper triangle of A is stored;
60: * = 'L': Lower triangle of A is stored.
61: *
62: * N (input) INTEGER
63: * The number of linear equations, i.e., the order of the
64: * matrix A. N >= 0.
65: *
66: * NRHS (input) INTEGER
67: * The number of right-hand-sides, i.e., the number of columns of the
68: * matrix B.
69: *
70: * A (input) COMPLEX*16 array, dimension (LDA,N)
71: * On entry, the N-by-N matrix A.
72: *
73: * LDA (input) INTEGER
74: * The leading dimension of the array A. LDA >= max(1,N).
75: *
76: * AF (input) COMPLEX*16 array, dimension (LDAF,N)
77: * The triangular factor U or L from the Cholesky factorization
78: * A = U**T*U or A = L*L**T, as computed by ZPOTRF.
79: *
80: * LDAF (input) INTEGER
81: * The leading dimension of the array AF. LDAF >= max(1,N).
82: *
83: * COLEQU (input) LOGICAL
84: * If .TRUE. then column equilibration was done to A before calling
85: * this routine. This is needed to compute the solution and error
86: * bounds correctly.
87: *
88: * C (input) DOUBLE PRECISION array, dimension (N)
89: * The column scale factors for A. If COLEQU = .FALSE., C
90: * is not accessed. If C is input, each element of C should be a power
91: * of the radix to ensure a reliable solution and error estimates.
92: * Scaling by powers of the radix does not cause rounding errors unless
93: * the result underflows or overflows. Rounding errors during scaling
94: * lead to refining with a matrix that is not equivalent to the
95: * input matrix, producing error estimates that may not be
96: * reliable.
97: *
98: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
99: * 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) COMPLEX*16 array, dimension
105: * (LDY,NRHS)
106: * On entry, the solution matrix X, as computed by ZPOTRS.
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 ZLA_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) COMPLEX*16 array, dimension (N)
220: * Workspace to hold the intermediate residual.
221: *
222: * AYB (input) DOUBLE PRECISION array, dimension (N)
223: * Workspace.
224: *
225: * DY (input) COMPLEX*16 PRECISION array, dimension (N)
226: * Workspace to hold the intermediate solution.
227: *
228: * Y_TAIL (input) COMPLEX*16 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 ZPOTRS had an illegal
272: * value
273: *
274: * =====================================================================
275: *
276: * .. Local Scalars ..
277: INTEGER UPLO2, CNT, I, J, X_STATE, Z_STATE,
278: $ Y_PREC_STATE
279: DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
280: $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
281: $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
282: $ EPS, HUGEVAL, INCR_THRESH
283: LOGICAL INCR_PREC
284: COMPLEX*16 ZDUM
285: * ..
286: * .. Parameters ..
287: INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
288: $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
289: $ EXTRA_Y
290: PARAMETER ( UNSTABLE_STATE = 0, WORKING_STATE = 1,
291: $ CONV_STATE = 2, NOPROG_STATE = 3 )
292: PARAMETER ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1,
293: $ EXTRA_Y = 2 )
294: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
295: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
296: INTEGER CMP_ERR_I, PIV_GROWTH_I
297: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
298: $ BERR_I = 3 )
299: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
300: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
301: $ PIV_GROWTH_I = 9 )
302: INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
303: $ LA_LINRX_CWISE_I
304: PARAMETER ( LA_LINRX_ITREF_I = 1,
305: $ LA_LINRX_ITHRESH_I = 2 )
306: PARAMETER ( LA_LINRX_CWISE_I = 3 )
307: INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
308: $ LA_LINRX_RCOND_I
309: PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
310: PARAMETER ( LA_LINRX_RCOND_I = 3 )
311: * ..
312: * .. External Functions ..
313: LOGICAL LSAME
314: EXTERNAL ILAUPLO
315: INTEGER ILAUPLO
316: * ..
317: * .. External Subroutines ..
318: EXTERNAL ZAXPY, ZCOPY, ZPOTRS, ZHEMV, BLAS_ZHEMV_X,
319: $ BLAS_ZHEMV2_X, ZLA_HEAMV, ZLA_WWADDW,
320: $ ZLA_LIN_BERR, DLAMCH
321: DOUBLE PRECISION DLAMCH
322: * ..
323: * .. Intrinsic Functions ..
324: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
325: * ..
326: * .. Statement Functions ..
327: DOUBLE PRECISION CABS1
328: * ..
329: * .. Statement Function Definitions ..
330: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
331: * ..
332: * .. Executable Statements ..
333: *
334: IF (INFO.NE.0) RETURN
335: EPS = DLAMCH( 'Epsilon' )
336: HUGEVAL = DLAMCH( 'Overflow' )
337: * Force HUGEVAL to Inf
338: HUGEVAL = HUGEVAL * HUGEVAL
339: * Using HUGEVAL may lead to spurious underflows.
340: INCR_THRESH = DBLE(N) * EPS
341:
342: IF (LSAME (UPLO, 'L')) THEN
343: UPLO2 = ILAUPLO( 'L' )
344: ELSE
345: UPLO2 = ILAUPLO( 'U' )
346: ENDIF
347:
348: DO J = 1, NRHS
349: Y_PREC_STATE = EXTRA_RESIDUAL
350: IF (Y_PREC_STATE .EQ. EXTRA_Y) THEN
351: DO I = 1, N
352: Y_TAIL( I ) = 0.0D+0
353: END DO
354: END IF
355:
356: DXRAT = 0.0D+0
357: DXRATMAX = 0.0D+0
358: DZRAT = 0.0D+0
359: DZRATMAX = 0.0D+0
360: FINAL_DX_X = HUGEVAL
361: FINAL_DZ_Z = HUGEVAL
362: PREVNORMDX = HUGEVAL
363: PREV_DZ_Z = HUGEVAL
364: DZ_Z = HUGEVAL
365: DX_X = HUGEVAL
366:
367: X_STATE = WORKING_STATE
368: Z_STATE = UNSTABLE_STATE
369: INCR_PREC = .FALSE.
370:
371: DO CNT = 1, ITHRESH
372: *
373: * Compute residual RES = B_s - op(A_s) * Y,
374: * op(A) = A, A**T, or A**H depending on TRANS (and type).
375: *
376: CALL ZCOPY( N, B( 1, J ), 1, RES, 1 )
377: IF (Y_PREC_STATE .EQ. BASE_RESIDUAL) THEN
378: CALL ZHEMV(UPLO, N, DCMPLX(-1.0D+0), A, LDA, Y(1,J), 1,
379: $ DCMPLX(1.0D+0), RES, 1)
380: ELSE IF (Y_PREC_STATE .EQ. EXTRA_RESIDUAL) THEN
381: CALL BLAS_ZHEMV_X(UPLO2, N, DCMPLX(-1.0D+0), A, LDA,
382: $ Y( 1, J ), 1, DCMPLX(1.0D+0), RES, 1, PREC_TYPE)
383: ELSE
384: CALL BLAS_ZHEMV2_X(UPLO2, N, DCMPLX(-1.0D+0), A, LDA,
385: $ Y(1, J), Y_TAIL, 1, DCMPLX(1.0D+0), RES, 1,
386: $ PREC_TYPE)
387: END IF
388:
389: ! XXX: RES is no longer needed.
390: CALL ZCOPY( N, RES, 1, DY, 1 )
391: CALL ZPOTRS( UPLO, N, 1, AF, LDAF, DY, N, INFO)
392: *
393: * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
394: *
395: NORMX = 0.0D+0
396: NORMY = 0.0D+0
397: NORMDX = 0.0D+0
398: DZ_Z = 0.0D+0
399: YMIN = HUGEVAL
400:
401: DO I = 1, N
402: YK = CABS1(Y(I, J))
403: DYK = CABS1(DY(I))
404:
405: IF (YK .NE. 0.0D+0) THEN
406: DZ_Z = MAX( DZ_Z, DYK / YK )
407: ELSE IF (DYK .NE. 0.0D+0) THEN
408: DZ_Z = HUGEVAL
409: END IF
410:
411: YMIN = MIN( YMIN, YK )
412:
413: NORMY = MAX( NORMY, YK )
414:
415: IF ( COLEQU ) THEN
416: NORMX = MAX(NORMX, YK * C(I))
417: NORMDX = MAX(NORMDX, DYK * C(I))
418: ELSE
419: NORMX = NORMY
420: NORMDX = MAX(NORMDX, DYK)
421: END IF
422: END DO
423:
424: IF (NORMX .NE. 0.0D+0) THEN
425: DX_X = NORMDX / NORMX
426: ELSE IF (NORMDX .EQ. 0.0D+0) THEN
427: DX_X = 0.0D+0
428: ELSE
429: DX_X = HUGEVAL
430: END IF
431:
432: DXRAT = NORMDX / PREVNORMDX
433: DZRAT = DZ_Z / PREV_DZ_Z
434: *
435: * Check termination criteria.
436: *
437: IF (YMIN*RCOND .LT. INCR_THRESH*NORMY
438: $ .AND. Y_PREC_STATE .LT. EXTRA_Y)
439: $ INCR_PREC = .TRUE.
440:
441: IF (X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH)
442: $ X_STATE = WORKING_STATE
443: IF (X_STATE .EQ. WORKING_STATE) THEN
444: IF (DX_X .LE. EPS) THEN
445: X_STATE = CONV_STATE
446: ELSE IF (DXRAT .GT. RTHRESH) THEN
447: IF (Y_PREC_STATE .NE. EXTRA_Y) THEN
448: INCR_PREC = .TRUE.
449: ELSE
450: X_STATE = NOPROG_STATE
451: END IF
452: ELSE
453: IF (DXRAT .GT. DXRATMAX) DXRATMAX = DXRAT
454: END IF
455: IF (X_STATE .GT. WORKING_STATE) FINAL_DX_X = DX_X
456: END IF
457:
458: IF (Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB)
459: $ Z_STATE = WORKING_STATE
460: IF (Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH)
461: $ Z_STATE = WORKING_STATE
462: IF (Z_STATE .EQ. WORKING_STATE) THEN
463: IF (DZ_Z .LE. EPS) THEN
464: Z_STATE = CONV_STATE
465: ELSE IF (DZ_Z .GT. DZ_UB) THEN
466: Z_STATE = UNSTABLE_STATE
467: DZRATMAX = 0.0D+0
468: FINAL_DZ_Z = HUGEVAL
469: ELSE IF (DZRAT .GT. RTHRESH) THEN
470: IF (Y_PREC_STATE .NE. EXTRA_Y) THEN
471: INCR_PREC = .TRUE.
472: ELSE
473: Z_STATE = NOPROG_STATE
474: END IF
475: ELSE
476: IF (DZRAT .GT. DZRATMAX) DZRATMAX = DZRAT
477: END IF
478: IF (Z_STATE .GT. WORKING_STATE) FINAL_DZ_Z = DZ_Z
479: END IF
480:
481: IF ( X_STATE.NE.WORKING_STATE.AND.
482: $ (IGNORE_CWISE.OR.Z_STATE.NE.WORKING_STATE) )
483: $ GOTO 666
484:
485: IF (INCR_PREC) THEN
486: INCR_PREC = .FALSE.
487: Y_PREC_STATE = Y_PREC_STATE + 1
488: DO I = 1, N
489: Y_TAIL( I ) = 0.0D+0
490: END DO
491: END IF
492:
493: PREVNORMDX = NORMDX
494: PREV_DZ_Z = DZ_Z
495: *
496: * Update soluton.
497: *
498: IF (Y_PREC_STATE .LT. EXTRA_Y) THEN
499: CALL ZAXPY( N, DCMPLX(1.0D+0), DY, 1, Y(1,J), 1 )
500: ELSE
501: CALL ZLA_WWADDW(N, Y(1,J), Y_TAIL, DY)
502: END IF
503:
504: END DO
505: * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
506: 666 CONTINUE
507: *
508: * Set final_* when cnt hits ithresh.
509: *
510: IF (X_STATE .EQ. WORKING_STATE) FINAL_DX_X = DX_X
511: IF (Z_STATE .EQ. WORKING_STATE) FINAL_DZ_Z = DZ_Z
512: *
513: * Compute error bounds.
514: *
515: IF (N_NORMS .GE. 1) THEN
516: ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) =
517: $ FINAL_DX_X / (1 - DXRATMAX)
518: END IF
519: IF (N_NORMS .GE. 2) THEN
520: ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) =
521: $ FINAL_DZ_Z / (1 - DZRATMAX)
522: END IF
523: *
524: * Compute componentwise relative backward error from formula
525: * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
526: * where abs(Z) is the componentwise absolute value of the matrix
527: * or vector Z.
528: *
529: * Compute residual RES = B_s - op(A_s) * Y,
530: * op(A) = A, A**T, or A**H depending on TRANS (and type).
531: *
532: CALL ZCOPY( N, B( 1, J ), 1, RES, 1 )
533: CALL ZHEMV(UPLO, N, DCMPLX(-1.0D+0), A, LDA, Y(1,J), 1,
534: $ DCMPLX(1.0D+0), RES, 1)
535:
536: DO I = 1, N
537: AYB( I ) = CABS1( B( I, J ) )
538: END DO
539: *
540: * Compute abs(op(A_s))*abs(Y) + abs(B_s).
541: *
542: CALL ZLA_HEAMV (UPLO2, N, 1.0D+0,
543: $ A, LDA, Y(1, J), 1, 1.0D+0, AYB, 1)
544:
545: CALL ZLA_LIN_BERR (N, N, 1, RES, AYB, BERR_OUT(J))
546: *
547: * End of loop for each RHS.
548: *
549: END DO
550: *
551: RETURN
552: END
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