1*> \brief \b SLA_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. 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download SLA_GERFSX_EXTENDED + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sla_gerfsx_extended.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sla_gerfsx_extended.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sla_gerfsx_extended.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SLA_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, 25* AYB, DY, Y_TAIL, RCOND, ITHRESH, 26* RTHRESH, DZ_UB, IGNORE_CWISE, 27* INFO ) 28* 29* .. Scalar Arguments .. 30* INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, 31* $ TRANS_TYPE, N_NORMS, ITHRESH 32* LOGICAL COLEQU, IGNORE_CWISE 33* REAL RTHRESH, DZ_UB 34* .. 35* .. Array Arguments .. 36* INTEGER IPIV( * ) 37* REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 38* $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) 39* REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ), 40* $ ERRS_N( NRHS, * ), 41* $ ERRS_C( NRHS, * ) 42* .. 43* 44* 45*> \par Purpose: 46* ============= 47*> 48*> \verbatim 49*> 50*> SLA_GERFSX_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 SGERFSX to perform iterative refinement. 54*> In addition to normwise error bound, the code provides maximum 55*> componentwise error bound if possible. See comments for ERRS_N 56*> and ERRS_C for details of the error bounds. Note that this 57*> subroutine is only resonsible for setting the second fields of 58*> ERRS_N and ERRS_C. 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 P 69*> = 'S': Single 70*> = 'D': Double 71*> = 'I': Indigenous 72*> = 'X' or '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 T 80*> = '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] NRHS 93*> \verbatim 94*> NRHS is INTEGER 95*> The number of right-hand-sides, i.e., the number of columns of the 96*> matrix B. 97*> \endverbatim 98*> 99*> \param[in] A 100*> \verbatim 101*> A is REAL array, dimension (LDA,N) 102*> On entry, the N-by-N matrix A. 103*> \endverbatim 104*> 105*> \param[in] LDA 106*> \verbatim 107*> LDA is INTEGER 108*> The leading dimension of the array A. LDA >= max(1,N). 109*> \endverbatim 110*> 111*> \param[in] AF 112*> \verbatim 113*> AF is REAL array, dimension (LDAF,N) 114*> The factors L and U from the factorization 115*> A = P*L*U as computed by SGETRF. 116*> \endverbatim 117*> 118*> \param[in] LDAF 119*> \verbatim 120*> LDAF is INTEGER 121*> The leading dimension of the array AF. LDAF >= max(1,N). 122*> \endverbatim 123*> 124*> \param[in] IPIV 125*> \verbatim 126*> IPIV is INTEGER array, dimension (N) 127*> The pivot indices from the factorization A = P*L*U 128*> as computed by SGETRF; row i of the matrix was interchanged 129*> with row IPIV(i). 130*> \endverbatim 131*> 132*> \param[in] COLEQU 133*> \verbatim 134*> COLEQU is LOGICAL 135*> If .TRUE. then column equilibration was done to A before calling 136*> this routine. This is needed to compute the solution and error 137*> bounds correctly. 138*> \endverbatim 139*> 140*> \param[in] C 141*> \verbatim 142*> C is REAL array, dimension (N) 143*> The column scale factors for A. If COLEQU = .FALSE., C 144*> is not accessed. If C is input, each element of C should be a power 145*> of the radix to ensure a reliable solution and error estimates. 146*> Scaling by powers of the radix does not cause rounding errors unless 147*> the result underflows or overflows. Rounding errors during scaling 148*> lead to refining with a matrix that is not equivalent to the 149*> input matrix, producing error estimates that may not be 150*> reliable. 151*> \endverbatim 152*> 153*> \param[in] B 154*> \verbatim 155*> B is REAL array, dimension (LDB,NRHS) 156*> The right-hand-side matrix B. 157*> \endverbatim 158*> 159*> \param[in] LDB 160*> \verbatim 161*> LDB is INTEGER 162*> The leading dimension of the array B. LDB >= max(1,N). 163*> \endverbatim 164*> 165*> \param[in,out] Y 166*> \verbatim 167*> Y is REAL array, dimension (LDY,NRHS) 168*> On entry, the solution matrix X, as computed by SGETRS. 169*> On exit, the improved solution matrix Y. 170*> \endverbatim 171*> 172*> \param[in] LDY 173*> \verbatim 174*> LDY is INTEGER 175*> The leading dimension of the array Y. LDY >= max(1,N). 176*> \endverbatim 177*> 178*> \param[out] BERR_OUT 179*> \verbatim 180*> BERR_OUT is REAL array, dimension (NRHS) 181*> On exit, BERR_OUT(j) contains the componentwise relative backward 182*> error for right-hand-side j from the formula 183*> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) 184*> where abs(Z) is the componentwise absolute value of the matrix 185*> or vector Z. This is computed by SLA_LIN_BERR. 186*> \endverbatim 187*> 188*> \param[in] N_NORMS 189*> \verbatim 190*> N_NORMS is INTEGER 191*> Determines which error bounds to return (see ERRS_N 192*> and ERRS_C). 193*> If N_NORMS >= 1 return normwise error bounds. 194*> If N_NORMS >= 2 return componentwise error bounds. 195*> \endverbatim 196*> 197*> \param[in,out] ERRS_N 198*> \verbatim 199*> ERRS_N is REAL array, dimension (NRHS, N_ERR_BNDS) 200*> For each right-hand side, this array contains information about 201*> various error bounds and condition numbers corresponding to the 202*> normwise relative error, which is defined as follows: 203*> 204*> Normwise relative error in the ith solution vector: 205*> max_j (abs(XTRUE(j,i) - X(j,i))) 206*> ------------------------------ 207*> max_j abs(X(j,i)) 208*> 209*> The array is indexed by the type of error information as described 210*> below. There currently are up to three pieces of information 211*> returned. 212*> 213*> The first index in ERRS_N(i,:) corresponds to the ith 214*> right-hand side. 215*> 216*> The second index in ERRS_N(:,err) contains the following 217*> three fields: 218*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 219*> reciprocal condition number is less than the threshold 220*> sqrt(n) * slamch('Epsilon'). 221*> 222*> err = 2 "Guaranteed" error bound: The estimated forward error, 223*> almost certainly within a factor of 10 of the true error 224*> so long as the next entry is greater than the threshold 225*> sqrt(n) * slamch('Epsilon'). This error bound should only 226*> be trusted if the previous boolean is true. 227*> 228*> err = 3 Reciprocal condition number: Estimated normwise 229*> reciprocal condition number. Compared with the threshold 230*> sqrt(n) * slamch('Epsilon') to determine if the error 231*> estimate is "guaranteed". These reciprocal condition 232*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 233*> appropriately scaled matrix Z. 234*> Let Z = S*A, where S scales each row by a power of the 235*> radix so all absolute row sums of Z are approximately 1. 236*> 237*> This subroutine is only responsible for setting the second field 238*> above. 239*> See Lapack Working Note 165 for further details and extra 240*> cautions. 241*> \endverbatim 242*> 243*> \param[in,out] ERRS_C 244*> \verbatim 245*> ERRS_C is REAL array, dimension (NRHS, N_ERR_BNDS) 246*> For each right-hand side, this array contains information about 247*> various error bounds and condition numbers corresponding to the 248*> componentwise relative error, which is defined as follows: 249*> 250*> Componentwise relative error in the ith solution vector: 251*> abs(XTRUE(j,i) - X(j,i)) 252*> max_j ---------------------- 253*> abs(X(j,i)) 254*> 255*> The array is indexed by the right-hand side i (on which the 256*> componentwise relative error depends), and the type of error 257*> information as described below. There currently are up to three 258*> pieces of information returned for each right-hand side. If 259*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 260*> ERRS_C is not accessed. If N_ERR_BNDS < 3, then at most 261*> the first (:,N_ERR_BNDS) entries are returned. 262*> 263*> The first index in ERRS_C(i,:) corresponds to the ith 264*> right-hand side. 265*> 266*> The second index in ERRS_C(:,err) contains the following 267*> three fields: 268*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 269*> reciprocal condition number is less than the threshold 270*> sqrt(n) * slamch('Epsilon'). 271*> 272*> err = 2 "Guaranteed" error bound: The estimated forward error, 273*> almost certainly within a factor of 10 of the true error 274*> so long as the next entry is greater than the threshold 275*> sqrt(n) * slamch('Epsilon'). This error bound should only 276*> be trusted if the previous boolean is true. 277*> 278*> err = 3 Reciprocal condition number: Estimated componentwise 279*> reciprocal condition number. Compared with the threshold 280*> sqrt(n) * slamch('Epsilon') to determine if the error 281*> estimate is "guaranteed". These reciprocal condition 282*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 283*> appropriately scaled matrix Z. 284*> Let Z = S*(A*diag(x)), where x is the solution for the 285*> current right-hand side and S scales each row of 286*> A*diag(x) by a power of the radix so all absolute row 287*> sums of Z are approximately 1. 288*> 289*> This subroutine is only responsible for setting the second field 290*> above. 291*> See Lapack Working Note 165 for further details and extra 292*> cautions. 293*> \endverbatim 294*> 295*> \param[in] RES 296*> \verbatim 297*> RES is REAL array, dimension (N) 298*> Workspace to hold the intermediate residual. 299*> \endverbatim 300*> 301*> \param[in] AYB 302*> \verbatim 303*> AYB is REAL array, dimension (N) 304*> Workspace. This can be the same workspace passed for Y_TAIL. 305*> \endverbatim 306*> 307*> \param[in] DY 308*> \verbatim 309*> DY is REAL array, dimension (N) 310*> Workspace to hold the intermediate solution. 311*> \endverbatim 312*> 313*> \param[in] Y_TAIL 314*> \verbatim 315*> Y_TAIL is REAL array, dimension (N) 316*> Workspace to hold the trailing bits of the intermediate solution. 317*> \endverbatim 318*> 319*> \param[in] RCOND 320*> \verbatim 321*> RCOND is REAL 322*> Reciprocal scaled condition number. This is an estimate of the 323*> reciprocal Skeel condition number of the matrix A after 324*> equilibration (if done). If this is less than the machine 325*> precision (in particular, if it is zero), the matrix is singular 326*> to working precision. Note that the error may still be small even 327*> if this number is very small and the matrix appears ill- 328*> conditioned. 329*> \endverbatim 330*> 331*> \param[in] ITHRESH 332*> \verbatim 333*> ITHRESH is INTEGER 334*> The maximum number of residual computations allowed for 335*> refinement. The default is 10. For 'aggressive' set to 100 to 336*> permit convergence using approximate factorizations or 337*> factorizations other than LU. If the factorization uses a 338*> technique other than Gaussian elimination, the guarantees in 339*> ERRS_N and ERRS_C may no longer be trustworthy. 340*> \endverbatim 341*> 342*> \param[in] RTHRESH 343*> \verbatim 344*> RTHRESH is REAL 345*> Determines when to stop refinement if the error estimate stops 346*> decreasing. Refinement will stop when the next solution no longer 347*> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is 348*> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The 349*> default value is 0.5. For 'aggressive' set to 0.9 to permit 350*> convergence on extremely ill-conditioned matrices. See LAWN 165 351*> for more details. 352*> \endverbatim 353*> 354*> \param[in] DZ_UB 355*> \verbatim 356*> DZ_UB is REAL 357*> Determines when to start considering componentwise convergence. 358*> Componentwise convergence is only considered after each component 359*> of the solution Y is stable, which we define as the relative 360*> change in each component being less than DZ_UB. The default value 361*> is 0.25, requiring the first bit to be stable. See LAWN 165 for 362*> more details. 363*> \endverbatim 364*> 365*> \param[in] IGNORE_CWISE 366*> \verbatim 367*> IGNORE_CWISE is LOGICAL 368*> If .TRUE. then ignore componentwise convergence. Default value 369*> is .FALSE.. 370*> \endverbatim 371*> 372*> \param[out] INFO 373*> \verbatim 374*> INFO is INTEGER 375*> = 0: Successful exit. 376*> < 0: if INFO = -i, the ith argument to SGETRS had an illegal 377*> value 378*> \endverbatim 379* 380* Authors: 381* ======== 382* 383*> \author Univ. of Tennessee 384*> \author Univ. of California Berkeley 385*> \author Univ. of Colorado Denver 386*> \author NAG Ltd. 387* 388*> \ingroup realGEcomputational 389* 390* ===================================================================== 391 SUBROUTINE SLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, NRHS, A, 392 $ LDA, AF, LDAF, IPIV, COLEQU, C, B, 393 $ LDB, Y, LDY, BERR_OUT, N_NORMS, 394 $ ERRS_N, ERRS_C, RES, 395 $ AYB, DY, Y_TAIL, RCOND, ITHRESH, 396 $ RTHRESH, DZ_UB, IGNORE_CWISE, 397 $ INFO ) 398* 399* -- LAPACK computational routine -- 400* -- LAPACK is a software package provided by Univ. of Tennessee, -- 401* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 402* 403* .. Scalar Arguments .. 404 INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, 405 $ TRANS_TYPE, N_NORMS, ITHRESH 406 LOGICAL COLEQU, IGNORE_CWISE 407 REAL RTHRESH, DZ_UB 408* .. 409* .. Array Arguments .. 410 INTEGER IPIV( * ) 411 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 412 $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) 413 REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ), 414 $ ERRS_N( NRHS, * ), 415 $ 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 REAL 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 SAXPY, SCOPY, SGETRS, SGEMV, BLAS_SGEMV_X, 457 $ BLAS_SGEMV2_X, SLA_GEAMV, SLA_WWADDW, SLAMCH, 458 $ CHLA_TRANSTYPE, SLA_LIN_BERR 459 REAL SLAMCH 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 = SLAMCH( 'Epsilon' ) 470 HUGEVAL = SLAMCH( 'Overflow' ) 471* Force HUGEVAL to Inf 472 HUGEVAL = HUGEVAL * HUGEVAL 473* Using HUGEVAL may lead to spurious underflows. 474 INCR_THRESH = REAL( 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 ) = 0.0 481 END DO 482 END IF 483 484 DXRAT = 0.0 485 DXRATMAX = 0.0 486 DZRAT = 0.0 487 DZRATMAX = 0.0 488 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 SCOPY( N, B( 1, J ), 1, RES, 1 ) 505 IF ( Y_PREC_STATE .EQ. BASE_RESIDUAL ) THEN 506 CALL SGEMV( TRANS, N, N, -1.0, A, LDA, Y( 1, J ), 1, 507 $ 1.0, RES, 1 ) 508 ELSE IF ( Y_PREC_STATE .EQ. EXTRA_RESIDUAL ) THEN 509 CALL BLAS_SGEMV_X( TRANS_TYPE, N, N, -1.0, A, LDA, 510 $ Y( 1, J ), 1, 1.0, RES, 1, PREC_TYPE ) 511 ELSE 512 CALL BLAS_SGEMV2_X( TRANS_TYPE, N, N, -1.0, A, LDA, 513 $ Y( 1, J ), Y_TAIL, 1, 1.0, RES, 1, PREC_TYPE ) 514 END IF 515 516! XXX: RES is no longer needed. 517 CALL SCOPY( N, RES, 1, DY, 1 ) 518 CALL SGETRS( 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 = 0.0 523 NORMY = 0.0 524 NORMDX = 0.0 525 DZ_Z = 0.0 526 YMIN = HUGEVAL 527* 528 DO I = 1, N 529 YK = ABS( Y( I, J ) ) 530 DYK = ABS( DY( I ) ) 531 532 IF ( YK .NE. 0.0 ) THEN 533 DZ_Z = MAX( DZ_Z, DYK / YK ) 534 ELSE IF ( DYK .NE. 0.0 ) 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. 0.0 ) THEN 552 DX_X = NORMDX / NORMX 553 ELSE IF ( NORMDX .EQ. 0.0 ) THEN 554 DX_X = 0.0 555 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 = 0.0 596 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 ) = 0.0 625 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 SAXPY( N, 1.0, DY, 1, Y( 1, J ), 1 ) 635 ELSE 636 CALL SLA_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 ) = 652 $ FINAL_DX_X / (1 - DXRATMAX) 653 END IF 654 IF ( N_NORMS .GE. 2 ) THEN 655 ERRS_C( J, LA_LINRX_ERR_I ) = 656 $ FINAL_DZ_Z / (1 - DZRATMAX) 657 END IF 658* 659* Compute componentwise relative backward error from formula 660* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) 661* where abs(Z) is the componentwise absolute value of the matrix 662* or vector Z. 663* 664* Compute residual RES = B_s - op(A_s) * Y, 665* op(A) = A, A**T, or A**H depending on TRANS (and type). 666* 667 CALL SCOPY( N, B( 1, J ), 1, RES, 1 ) 668 CALL SGEMV( TRANS, N, N, -1.0, A, LDA, Y(1,J), 1, 1.0, RES, 1 ) 669 670 DO I = 1, N 671 AYB( I ) = ABS( B( I, J ) ) 672 END DO 673* 674* Compute abs(op(A_s))*abs(Y) + abs(B_s). 675* 676 CALL SLA_GEAMV ( TRANS_TYPE, N, N, 1.0, 677 $ A, LDA, Y(1, J), 1, 1.0, AYB, 1 ) 678 679 CALL SLA_LIN_BERR ( N, N, 1, RES, AYB, BERR_OUT( J ) ) 680* 681* End of loop for each RHS. 682* 683 END DO 684* 685 RETURN 686* 687* End of SLA_GERFSX_EXTENDED 688* 689 END 690