1*> \brief \b SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite 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_PORFSX_EXTENDED + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sla_porfsx_extended.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sla_porfsx_extended.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sla_porfsx_extended.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SLA_PORFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA, 22* AF, LDAF, COLEQU, C, B, LDB, Y, 23* LDY, BERR_OUT, N_NORMS, 24* ERR_BNDS_NORM, ERR_BNDS_COMP, 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* $ N_NORMS, ITHRESH 32* CHARACTER UPLO 33* LOGICAL COLEQU, IGNORE_CWISE 34* REAL RTHRESH, DZ_UB 35* .. 36* .. Array Arguments .. 37* REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 38* $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) 39* REAL C( * ), AYB(*), RCOND, BERR_OUT( * ), 40* $ ERR_BNDS_NORM( NRHS, * ), 41* $ ERR_BNDS_COMP( NRHS, * ) 42* .. 43* 44* 45*> \par Purpose: 46* ============= 47*> 48*> \verbatim 49*> 50*> SLA_PORFSX_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 SPORFSX to perform iterative refinement. 54*> In addition to normwise error bound, the code provides maximum 55*> componentwise error bound if possible. See comments for ERR_BNDS_NORM 56*> and ERR_BNDS_COMP for details of the error bounds. Note that this 57*> subroutine is only resonsible for setting the second fields of 58*> ERR_BNDS_NORM and ERR_BNDS_COMP. 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 69*> P = 'S': Single 70*> = 'D': Double 71*> = 'I': Indigenous 72*> = 'X', 'E': Extra 73*> \endverbatim 74*> 75*> \param[in] UPLO 76*> \verbatim 77*> UPLO is CHARACTER*1 78*> = 'U': Upper triangle of A is stored; 79*> = 'L': Lower triangle of A is stored. 80*> \endverbatim 81*> 82*> \param[in] N 83*> \verbatim 84*> N is INTEGER 85*> The number of linear equations, i.e., the order of the 86*> matrix A. N >= 0. 87*> \endverbatim 88*> 89*> \param[in] NRHS 90*> \verbatim 91*> NRHS is INTEGER 92*> The number of right-hand-sides, i.e., the number of columns of the 93*> matrix B. 94*> \endverbatim 95*> 96*> \param[in] A 97*> \verbatim 98*> A is REAL array, dimension (LDA,N) 99*> On entry, the N-by-N matrix A. 100*> \endverbatim 101*> 102*> \param[in] LDA 103*> \verbatim 104*> LDA is INTEGER 105*> The leading dimension of the array A. LDA >= max(1,N). 106*> \endverbatim 107*> 108*> \param[in] AF 109*> \verbatim 110*> AF is REAL array, dimension (LDAF,N) 111*> The triangular factor U or L from the Cholesky factorization 112*> A = U**T*U or A = L*L**T, as computed by SPOTRF. 113*> \endverbatim 114*> 115*> \param[in] LDAF 116*> \verbatim 117*> LDAF is INTEGER 118*> The leading dimension of the array AF. LDAF >= max(1,N). 119*> \endverbatim 120*> 121*> \param[in] COLEQU 122*> \verbatim 123*> COLEQU is LOGICAL 124*> If .TRUE. then column equilibration was done to A before calling 125*> this routine. This is needed to compute the solution and error 126*> bounds correctly. 127*> \endverbatim 128*> 129*> \param[in] C 130*> \verbatim 131*> C is REAL array, dimension (N) 132*> The column scale factors for A. If COLEQU = .FALSE., C 133*> is not accessed. If C is input, each element of C should be a power 134*> of the radix to ensure a reliable solution and error estimates. 135*> Scaling by powers of the radix does not cause rounding errors unless 136*> the result underflows or overflows. Rounding errors during scaling 137*> lead to refining with a matrix that is not equivalent to the 138*> input matrix, producing error estimates that may not be 139*> reliable. 140*> \endverbatim 141*> 142*> \param[in] B 143*> \verbatim 144*> B is REAL array, dimension (LDB,NRHS) 145*> The right-hand-side matrix B. 146*> \endverbatim 147*> 148*> \param[in] LDB 149*> \verbatim 150*> LDB is INTEGER 151*> The leading dimension of the array B. LDB >= max(1,N). 152*> \endverbatim 153*> 154*> \param[in,out] Y 155*> \verbatim 156*> Y is REAL array, dimension (LDY,NRHS) 157*> On entry, the solution matrix X, as computed by SPOTRS. 158*> On exit, the improved solution matrix Y. 159*> \endverbatim 160*> 161*> \param[in] LDY 162*> \verbatim 163*> LDY is INTEGER 164*> The leading dimension of the array Y. LDY >= max(1,N). 165*> \endverbatim 166*> 167*> \param[out] BERR_OUT 168*> \verbatim 169*> BERR_OUT is REAL array, dimension (NRHS) 170*> On exit, BERR_OUT(j) contains the componentwise relative backward 171*> error for right-hand-side j from the formula 172*> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) 173*> where abs(Z) is the componentwise absolute value of the matrix 174*> or vector Z. This is computed by SLA_LIN_BERR. 175*> \endverbatim 176*> 177*> \param[in] N_NORMS 178*> \verbatim 179*> N_NORMS is INTEGER 180*> Determines which error bounds to return (see ERR_BNDS_NORM 181*> and ERR_BNDS_COMP). 182*> If N_NORMS >= 1 return normwise error bounds. 183*> If N_NORMS >= 2 return componentwise error bounds. 184*> \endverbatim 185*> 186*> \param[in,out] ERR_BNDS_NORM 187*> \verbatim 188*> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) 189*> For each right-hand side, this array contains information about 190*> various error bounds and condition numbers corresponding to the 191*> normwise relative error, which is defined as follows: 192*> 193*> Normwise relative error in the ith solution vector: 194*> max_j (abs(XTRUE(j,i) - X(j,i))) 195*> ------------------------------ 196*> max_j abs(X(j,i)) 197*> 198*> The array is indexed by the type of error information as described 199*> below. There currently are up to three pieces of information 200*> returned. 201*> 202*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 203*> right-hand side. 204*> 205*> The second index in ERR_BNDS_NORM(:,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 normwise 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, where S scales each row by a power of the 224*> radix so all absolute row sums of Z are approximately 1. 225*> 226*> This subroutine is only responsible for setting the second field 227*> above. 228*> See Lapack Working Note 165 for further details and extra 229*> cautions. 230*> \endverbatim 231*> 232*> \param[in,out] ERR_BNDS_COMP 233*> \verbatim 234*> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) 235*> For each right-hand side, this array contains information about 236*> various error bounds and condition numbers corresponding to the 237*> componentwise relative error, which is defined as follows: 238*> 239*> Componentwise relative error in the ith solution vector: 240*> abs(XTRUE(j,i) - X(j,i)) 241*> max_j ---------------------- 242*> abs(X(j,i)) 243*> 244*> The array is indexed by the right-hand side i (on which the 245*> componentwise relative error depends), and the type of error 246*> information as described below. There currently are up to three 247*> pieces of information returned for each right-hand side. If 248*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 249*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most 250*> the first (:,N_ERR_BNDS) entries are returned. 251*> 252*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 253*> right-hand side. 254*> 255*> The second index in ERR_BNDS_COMP(:,err) contains the following 256*> three fields: 257*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 258*> reciprocal condition number is less than the threshold 259*> sqrt(n) * slamch('Epsilon'). 260*> 261*> err = 2 "Guaranteed" error bound: The estimated forward error, 262*> almost certainly within a factor of 10 of the true error 263*> so long as the next entry is greater than the threshold 264*> sqrt(n) * slamch('Epsilon'). This error bound should only 265*> be trusted if the previous boolean is true. 266*> 267*> err = 3 Reciprocal condition number: Estimated componentwise 268*> reciprocal condition number. Compared with the threshold 269*> sqrt(n) * slamch('Epsilon') to determine if the error 270*> estimate is "guaranteed". These reciprocal condition 271*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 272*> appropriately scaled matrix Z. 273*> Let Z = S*(A*diag(x)), where x is the solution for the 274*> current right-hand side and S scales each row of 275*> A*diag(x) by a power of the radix so all absolute row 276*> sums of Z are approximately 1. 277*> 278*> This subroutine is only responsible for setting the second field 279*> above. 280*> See Lapack Working Note 165 for further details and extra 281*> cautions. 282*> \endverbatim 283*> 284*> \param[in] RES 285*> \verbatim 286*> RES is REAL array, dimension (N) 287*> Workspace to hold the intermediate residual. 288*> \endverbatim 289*> 290*> \param[in] AYB 291*> \verbatim 292*> AYB is REAL array, dimension (N) 293*> Workspace. This can be the same workspace passed for Y_TAIL. 294*> \endverbatim 295*> 296*> \param[in] DY 297*> \verbatim 298*> DY is REAL array, dimension (N) 299*> Workspace to hold the intermediate solution. 300*> \endverbatim 301*> 302*> \param[in] Y_TAIL 303*> \verbatim 304*> Y_TAIL is REAL array, dimension (N) 305*> Workspace to hold the trailing bits of the intermediate solution. 306*> \endverbatim 307*> 308*> \param[in] RCOND 309*> \verbatim 310*> RCOND is REAL 311*> Reciprocal scaled condition number. This is an estimate of the 312*> reciprocal Skeel condition number of the matrix A after 313*> equilibration (if done). If this is less than the machine 314*> precision (in particular, if it is zero), the matrix is singular 315*> to working precision. Note that the error may still be small even 316*> if this number is very small and the matrix appears ill- 317*> conditioned. 318*> \endverbatim 319*> 320*> \param[in] ITHRESH 321*> \verbatim 322*> ITHRESH is INTEGER 323*> The maximum number of residual computations allowed for 324*> refinement. The default is 10. For 'aggressive' set to 100 to 325*> permit convergence using approximate factorizations or 326*> factorizations other than LU. If the factorization uses a 327*> technique other than Gaussian elimination, the guarantees in 328*> ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. 329*> \endverbatim 330*> 331*> \param[in] RTHRESH 332*> \verbatim 333*> RTHRESH is REAL 334*> Determines when to stop refinement if the error estimate stops 335*> decreasing. Refinement will stop when the next solution no longer 336*> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is 337*> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The 338*> default value is 0.5. For 'aggressive' set to 0.9 to permit 339*> convergence on extremely ill-conditioned matrices. See LAWN 165 340*> for more details. 341*> \endverbatim 342*> 343*> \param[in] DZ_UB 344*> \verbatim 345*> DZ_UB is REAL 346*> Determines when to start considering componentwise convergence. 347*> Componentwise convergence is only considered after each component 348*> of the solution Y is stable, which we definte as the relative 349*> change in each component being less than DZ_UB. The default value 350*> is 0.25, requiring the first bit to be stable. See LAWN 165 for 351*> more details. 352*> \endverbatim 353*> 354*> \param[in] IGNORE_CWISE 355*> \verbatim 356*> IGNORE_CWISE is LOGICAL 357*> If .TRUE. then ignore componentwise convergence. Default value 358*> is .FALSE.. 359*> \endverbatim 360*> 361*> \param[out] INFO 362*> \verbatim 363*> INFO is INTEGER 364*> = 0: Successful exit. 365*> < 0: if INFO = -i, the ith argument to SPOTRS had an illegal 366*> value 367*> \endverbatim 368* 369* Authors: 370* ======== 371* 372*> \author Univ. of Tennessee 373*> \author Univ. of California Berkeley 374*> \author Univ. of Colorado Denver 375*> \author NAG Ltd. 376* 377*> \date September 2012 378* 379*> \ingroup realPOcomputational 380* 381* ===================================================================== 382 SUBROUTINE SLA_PORFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA, 383 $ AF, LDAF, COLEQU, C, B, LDB, Y, 384 $ LDY, BERR_OUT, N_NORMS, 385 $ ERR_BNDS_NORM, ERR_BNDS_COMP, RES, 386 $ AYB, DY, Y_TAIL, RCOND, ITHRESH, 387 $ RTHRESH, DZ_UB, IGNORE_CWISE, 388 $ INFO ) 389* 390* -- LAPACK computational routine (version 3.4.2) -- 391* -- LAPACK is a software package provided by Univ. of Tennessee, -- 392* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 393* September 2012 394* 395* .. Scalar Arguments .. 396 INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, 397 $ N_NORMS, ITHRESH 398 CHARACTER UPLO 399 LOGICAL COLEQU, IGNORE_CWISE 400 REAL RTHRESH, DZ_UB 401* .. 402* .. Array Arguments .. 403 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 404 $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) 405 REAL C( * ), AYB(*), RCOND, BERR_OUT( * ), 406 $ ERR_BNDS_NORM( NRHS, * ), 407 $ ERR_BNDS_COMP( NRHS, * ) 408* .. 409* 410* ===================================================================== 411* 412* .. Local Scalars .. 413 INTEGER UPLO2, CNT, I, J, X_STATE, Z_STATE 414 REAL YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT, 415 $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX, 416 $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z, 417 $ EPS, HUGEVAL, INCR_THRESH 418 LOGICAL INCR_PREC 419* .. 420* .. Parameters .. 421 INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE, 422 $ NOPROG_STATE, Y_PREC_STATE, BASE_RESIDUAL, 423 $ EXTRA_RESIDUAL, EXTRA_Y 424 PARAMETER ( UNSTABLE_STATE = 0, WORKING_STATE = 1, 425 $ CONV_STATE = 2, NOPROG_STATE = 3 ) 426 PARAMETER ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1, 427 $ EXTRA_Y = 2 ) 428 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 429 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 430 INTEGER CMP_ERR_I, PIV_GROWTH_I 431 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 432 $ BERR_I = 3 ) 433 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 434 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 435 $ PIV_GROWTH_I = 9 ) 436 INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I, 437 $ LA_LINRX_CWISE_I 438 PARAMETER ( LA_LINRX_ITREF_I = 1, 439 $ LA_LINRX_ITHRESH_I = 2 ) 440 PARAMETER ( LA_LINRX_CWISE_I = 3 ) 441 INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I, 442 $ LA_LINRX_RCOND_I 443 PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 ) 444 PARAMETER ( LA_LINRX_RCOND_I = 3 ) 445* .. 446* .. External Functions .. 447 LOGICAL LSAME 448 EXTERNAL ILAUPLO 449 INTEGER ILAUPLO 450* .. 451* .. External Subroutines .. 452 EXTERNAL SAXPY, SCOPY, SPOTRS, SSYMV, BLAS_SSYMV_X, 453 $ BLAS_SSYMV2_X, SLA_SYAMV, SLA_WWADDW, 454 $ SLA_LIN_BERR 455 REAL SLAMCH 456* .. 457* .. Intrinsic Functions .. 458 INTRINSIC ABS, MAX, MIN 459* .. 460* .. Executable Statements .. 461* 462 IF (INFO.NE.0) RETURN 463 EPS = SLAMCH( 'Epsilon' ) 464 HUGEVAL = SLAMCH( 'Overflow' ) 465* Force HUGEVAL to Inf 466 HUGEVAL = HUGEVAL * HUGEVAL 467* Using HUGEVAL may lead to spurious underflows. 468 INCR_THRESH = REAL( N ) * EPS 469 470 IF ( LSAME ( UPLO, 'L' ) ) THEN 471 UPLO2 = ILAUPLO( 'L' ) 472 ELSE 473 UPLO2 = ILAUPLO( 'U' ) 474 ENDIF 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 SSYMV( UPLO, 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_SSYMV_X( UPLO2, N, -1.0, A, LDA, 510 $ Y( 1, J ), 1, 1.0, RES, 1, PREC_TYPE ) 511 ELSE 512 CALL BLAS_SSYMV2_X(UPLO2, 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 SPOTRS( UPLO, N, 1, AF, LDAF, 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 ( YMIN*RCOND .LT. INCR_THRESH*NORMY 565 $ .AND. Y_PREC_STATE .LT. EXTRA_Y ) 566 $ INCR_PREC = .TRUE. 567 568 IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH ) 569 $ X_STATE = WORKING_STATE 570 IF ( X_STATE .EQ. WORKING_STATE ) THEN 571 IF ( DX_X .LE. EPS ) THEN 572 X_STATE = CONV_STATE 573 ELSE IF ( DXRAT .GT. RTHRESH ) THEN 574 IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN 575 INCR_PREC = .TRUE. 576 ELSE 577 X_STATE = NOPROG_STATE 578 END IF 579 ELSE 580 IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT 581 END IF 582 IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X 583 END IF 584 585 IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB ) 586 $ Z_STATE = WORKING_STATE 587 IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH ) 588 $ Z_STATE = WORKING_STATE 589 IF ( Z_STATE .EQ. WORKING_STATE ) THEN 590 IF ( DZ_Z .LE. EPS ) THEN 591 Z_STATE = CONV_STATE 592 ELSE IF ( DZ_Z .GT. DZ_UB ) THEN 593 Z_STATE = UNSTABLE_STATE 594 DZRATMAX = 0.0 595 FINAL_DZ_Z = HUGEVAL 596 ELSE IF ( DZRAT .GT. RTHRESH ) THEN 597 IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN 598 INCR_PREC = .TRUE. 599 ELSE 600 Z_STATE = NOPROG_STATE 601 END IF 602 ELSE 603 IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT 604 END IF 605 IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z 606 END IF 607 608 IF ( X_STATE.NE.WORKING_STATE.AND. 609 $ ( IGNORE_CWISE.OR.Z_STATE.NE.WORKING_STATE ) ) 610 $ GOTO 666 611 612 IF ( INCR_PREC ) THEN 613 INCR_PREC = .FALSE. 614 Y_PREC_STATE = Y_PREC_STATE + 1 615 DO I = 1, N 616 Y_TAIL( I ) = 0.0 617 END DO 618 END IF 619 620 PREVNORMDX = NORMDX 621 PREV_DZ_Z = DZ_Z 622* 623* Update soluton. 624* 625 IF (Y_PREC_STATE .LT. EXTRA_Y) THEN 626 CALL SAXPY( N, 1.0, DY, 1, Y(1,J), 1 ) 627 ELSE 628 CALL SLA_WWADDW( N, Y( 1, J ), Y_TAIL, DY ) 629 END IF 630 631 END DO 632* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT. 633 666 CONTINUE 634* 635* Set final_* when cnt hits ithresh. 636* 637 IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X 638 IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z 639* 640* Compute error bounds. 641* 642 IF ( N_NORMS .GE. 1 ) THEN 643 ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 644 $ FINAL_DX_X / (1 - DXRATMAX) 645 END IF 646 IF ( N_NORMS .GE. 2 ) THEN 647 ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 648 $ FINAL_DZ_Z / (1 - DZRATMAX) 649 END IF 650* 651* Compute componentwise relative backward error from formula 652* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) 653* where abs(Z) is the componentwise absolute value of the matrix 654* or vector Z. 655* 656* Compute residual RES = B_s - op(A_s) * Y, 657* op(A) = A, A**T, or A**H depending on TRANS (and type). 658* 659 CALL SCOPY( N, B( 1, J ), 1, RES, 1 ) 660 CALL SSYMV( UPLO, N, -1.0, A, LDA, Y(1,J), 1, 1.0, RES, 1 ) 661 662 DO I = 1, N 663 AYB( I ) = ABS( B( I, J ) ) 664 END DO 665* 666* Compute abs(op(A_s))*abs(Y) + abs(B_s). 667* 668 CALL SLA_SYAMV( UPLO2, N, 1.0, 669 $ A, LDA, Y(1, J), 1, 1.0, AYB, 1 ) 670 671 CALL SLA_LIN_BERR( N, N, 1, RES, AYB, BERR_OUT( J ) ) 672* 673* End of loop for each RHS. 674* 675 END DO 676* 677 RETURN 678 END 679