1*> \brief \b SLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite 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_SYRFSX_EXTENDED + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sla_syrfsx_extended.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sla_syrfsx_extended.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sla_syrfsx_extended.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SLA_SYRFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA, 22* AF, LDAF, IPIV, COLEQU, C, B, LDB, 23* Y, 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* INTEGER IPIV( * ) 38* REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 39* $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) 40* REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ), 41* $ ERR_BNDS_NORM( NRHS, * ), 42* $ ERR_BNDS_COMP( NRHS, * ) 43* .. 44* 45* 46*> \par Purpose: 47* ============= 48*> 49*> \verbatim 50*> 51*> 52*> SLA_SYRFSX_EXTENDED improves the computed solution to a system of 53*> linear equations by performing extra-precise iterative refinement 54*> and provides error bounds and backward error estimates for the solution. 55*> This subroutine is called by SSYRFSX to perform iterative refinement. 56*> In addition to normwise error bound, the code provides maximum 57*> componentwise error bound if possible. See comments for ERR_BNDS_NORM 58*> and ERR_BNDS_COMP for details of the error bounds. Note that this 59*> subroutine is only resonsible for setting the second fields of 60*> ERR_BNDS_NORM and ERR_BNDS_COMP. 61*> \endverbatim 62* 63* Arguments: 64* ========== 65* 66*> \param[in] PREC_TYPE 67*> \verbatim 68*> PREC_TYPE is INTEGER 69*> Specifies the intermediate precision to be used in refinement. 70*> The value is defined by ILAPREC(P) where P is a CHARACTER and P 71*> = 'S': Single 72*> = 'D': Double 73*> = 'I': Indigenous 74*> = 'X' or 'E': Extra 75*> \endverbatim 76*> 77*> \param[in] UPLO 78*> \verbatim 79*> UPLO is CHARACTER*1 80*> = 'U': Upper triangle of A is stored; 81*> = 'L': Lower triangle of A is stored. 82*> \endverbatim 83*> 84*> \param[in] N 85*> \verbatim 86*> N is INTEGER 87*> The number of linear equations, i.e., the order of the 88*> matrix A. N >= 0. 89*> \endverbatim 90*> 91*> \param[in] NRHS 92*> \verbatim 93*> NRHS is INTEGER 94*> The number of right-hand-sides, i.e., the number of columns of the 95*> matrix B. 96*> \endverbatim 97*> 98*> \param[in] A 99*> \verbatim 100*> A is REAL array, dimension (LDA,N) 101*> On entry, the N-by-N matrix A. 102*> \endverbatim 103*> 104*> \param[in] LDA 105*> \verbatim 106*> LDA is INTEGER 107*> The leading dimension of the array A. LDA >= max(1,N). 108*> \endverbatim 109*> 110*> \param[in] AF 111*> \verbatim 112*> AF is REAL array, dimension (LDAF,N) 113*> The block diagonal matrix D and the multipliers used to 114*> obtain the factor U or L as computed by SSYTRF. 115*> \endverbatim 116*> 117*> \param[in] LDAF 118*> \verbatim 119*> LDAF is INTEGER 120*> The leading dimension of the array AF. LDAF >= max(1,N). 121*> \endverbatim 122*> 123*> \param[in] IPIV 124*> \verbatim 125*> IPIV is INTEGER array, dimension (N) 126*> Details of the interchanges and the block structure of D 127*> as determined by SSYTRF. 128*> \endverbatim 129*> 130*> \param[in] COLEQU 131*> \verbatim 132*> COLEQU is LOGICAL 133*> If .TRUE. then column equilibration was done to A before calling 134*> this routine. This is needed to compute the solution and error 135*> bounds correctly. 136*> \endverbatim 137*> 138*> \param[in] C 139*> \verbatim 140*> C is REAL array, dimension (N) 141*> The column scale factors for A. If COLEQU = .FALSE., C 142*> is not accessed. If C is input, each element of C should be a power 143*> of the radix to ensure a reliable solution and error estimates. 144*> Scaling by powers of the radix does not cause rounding errors unless 145*> the result underflows or overflows. Rounding errors during scaling 146*> lead to refining with a matrix that is not equivalent to the 147*> input matrix, producing error estimates that may not be 148*> reliable. 149*> \endverbatim 150*> 151*> \param[in] B 152*> \verbatim 153*> B is REAL array, dimension (LDB,NRHS) 154*> The right-hand-side matrix B. 155*> \endverbatim 156*> 157*> \param[in] LDB 158*> \verbatim 159*> LDB is INTEGER 160*> The leading dimension of the array B. LDB >= max(1,N). 161*> \endverbatim 162*> 163*> \param[in,out] Y 164*> \verbatim 165*> Y is REAL array, dimension (LDY,NRHS) 166*> On entry, the solution matrix X, as computed by SSYTRS. 167*> On exit, the improved solution matrix Y. 168*> \endverbatim 169*> 170*> \param[in] LDY 171*> \verbatim 172*> LDY is INTEGER 173*> The leading dimension of the array Y. LDY >= max(1,N). 174*> \endverbatim 175*> 176*> \param[out] BERR_OUT 177*> \verbatim 178*> BERR_OUT is REAL array, dimension (NRHS) 179*> On exit, BERR_OUT(j) contains the componentwise relative backward 180*> error for right-hand-side j from the formula 181*> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) 182*> where abs(Z) is the componentwise absolute value of the matrix 183*> or vector Z. This is computed by SLA_LIN_BERR. 184*> \endverbatim 185*> 186*> \param[in] N_NORMS 187*> \verbatim 188*> N_NORMS is INTEGER 189*> Determines which error bounds to return (see ERR_BNDS_NORM 190*> and ERR_BNDS_COMP). 191*> If N_NORMS >= 1 return normwise error bounds. 192*> If N_NORMS >= 2 return componentwise error bounds. 193*> \endverbatim 194*> 195*> \param[in,out] ERR_BNDS_NORM 196*> \verbatim 197*> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) 198*> For each right-hand side, this array contains information about 199*> various error bounds and condition numbers corresponding to the 200*> normwise relative error, which is defined as follows: 201*> 202*> Normwise relative error in the ith solution vector: 203*> max_j (abs(XTRUE(j,i) - X(j,i))) 204*> ------------------------------ 205*> max_j abs(X(j,i)) 206*> 207*> The array is indexed by the type of error information as described 208*> below. There currently are up to three pieces of information 209*> returned. 210*> 211*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 212*> right-hand side. 213*> 214*> The second index in ERR_BNDS_NORM(:,err) contains the following 215*> three fields: 216*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 217*> reciprocal condition number is less than the threshold 218*> sqrt(n) * slamch('Epsilon'). 219*> 220*> err = 2 "Guaranteed" error bound: The estimated forward error, 221*> almost certainly within a factor of 10 of the true error 222*> so long as the next entry is greater than the threshold 223*> sqrt(n) * slamch('Epsilon'). This error bound should only 224*> be trusted if the previous boolean is true. 225*> 226*> err = 3 Reciprocal condition number: Estimated normwise 227*> reciprocal condition number. Compared with the threshold 228*> sqrt(n) * slamch('Epsilon') to determine if the error 229*> estimate is "guaranteed". These reciprocal condition 230*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 231*> appropriately scaled matrix Z. 232*> Let Z = S*A, where S scales each row by a power of the 233*> radix so all absolute row sums of Z are approximately 1. 234*> 235*> This subroutine is only responsible for setting the second field 236*> above. 237*> See Lapack Working Note 165 for further details and extra 238*> cautions. 239*> \endverbatim 240*> 241*> \param[in,out] ERR_BNDS_COMP 242*> \verbatim 243*> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) 244*> For each right-hand side, this array contains information about 245*> various error bounds and condition numbers corresponding to the 246*> componentwise relative error, which is defined as follows: 247*> 248*> Componentwise relative error in the ith solution vector: 249*> abs(XTRUE(j,i) - X(j,i)) 250*> max_j ---------------------- 251*> abs(X(j,i)) 252*> 253*> The array is indexed by the right-hand side i (on which the 254*> componentwise relative error depends), and the type of error 255*> information as described below. There currently are up to three 256*> pieces of information returned for each right-hand side. If 257*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 258*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most 259*> the first (:,N_ERR_BNDS) entries are returned. 260*> 261*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 262*> right-hand side. 263*> 264*> The second index in ERR_BNDS_COMP(:,err) contains the following 265*> three fields: 266*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 267*> reciprocal condition number is less than the threshold 268*> sqrt(n) * slamch('Epsilon'). 269*> 270*> err = 2 "Guaranteed" error bound: The estimated forward error, 271*> almost certainly within a factor of 10 of the true error 272*> so long as the next entry is greater than the threshold 273*> sqrt(n) * slamch('Epsilon'). This error bound should only 274*> be trusted if the previous boolean is true. 275*> 276*> err = 3 Reciprocal condition number: Estimated componentwise 277*> reciprocal condition number. Compared with the threshold 278*> sqrt(n) * slamch('Epsilon') to determine if the error 279*> estimate is "guaranteed". These reciprocal condition 280*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 281*> appropriately scaled matrix Z. 282*> Let Z = S*(A*diag(x)), where x is the solution for the 283*> current right-hand side and S scales each row of 284*> A*diag(x) by a power of the radix so all absolute row 285*> sums of Z are approximately 1. 286*> 287*> This subroutine is only responsible for setting the second field 288*> above. 289*> See Lapack Working Note 165 for further details and extra 290*> cautions. 291*> \endverbatim 292*> 293*> \param[in] RES 294*> \verbatim 295*> RES is REAL array, dimension (N) 296*> Workspace to hold the intermediate residual. 297*> \endverbatim 298*> 299*> \param[in] AYB 300*> \verbatim 301*> AYB is REAL array, dimension (N) 302*> Workspace. This can be the same workspace passed for Y_TAIL. 303*> \endverbatim 304*> 305*> \param[in] DY 306*> \verbatim 307*> DY is REAL array, dimension (N) 308*> Workspace to hold the intermediate solution. 309*> \endverbatim 310*> 311*> \param[in] Y_TAIL 312*> \verbatim 313*> Y_TAIL is REAL array, dimension (N) 314*> Workspace to hold the trailing bits of the intermediate solution. 315*> \endverbatim 316*> 317*> \param[in] RCOND 318*> \verbatim 319*> RCOND is REAL 320*> Reciprocal scaled condition number. This is an estimate of the 321*> reciprocal Skeel condition number of the matrix A after 322*> equilibration (if done). If this is less than the machine 323*> precision (in particular, if it is zero), the matrix is singular 324*> to working precision. Note that the error may still be small even 325*> if this number is very small and the matrix appears ill- 326*> conditioned. 327*> \endverbatim 328*> 329*> \param[in] ITHRESH 330*> \verbatim 331*> ITHRESH is INTEGER 332*> The maximum number of residual computations allowed for 333*> refinement. The default is 10. For 'aggressive' set to 100 to 334*> permit convergence using approximate factorizations or 335*> factorizations other than LU. If the factorization uses a 336*> technique other than Gaussian elimination, the guarantees in 337*> ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. 338*> \endverbatim 339*> 340*> \param[in] RTHRESH 341*> \verbatim 342*> RTHRESH is REAL 343*> Determines when to stop refinement if the error estimate stops 344*> decreasing. Refinement will stop when the next solution no longer 345*> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is 346*> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The 347*> default value is 0.5. For 'aggressive' set to 0.9 to permit 348*> convergence on extremely ill-conditioned matrices. See LAWN 165 349*> for more details. 350*> \endverbatim 351*> 352*> \param[in] DZ_UB 353*> \verbatim 354*> DZ_UB is REAL 355*> Determines when to start considering componentwise convergence. 356*> Componentwise convergence is only considered after each component 357*> of the solution Y is stable, which we definte as the relative 358*> change in each component being less than DZ_UB. The default value 359*> is 0.25, requiring the first bit to be stable. See LAWN 165 for 360*> more details. 361*> \endverbatim 362*> 363*> \param[in] IGNORE_CWISE 364*> \verbatim 365*> IGNORE_CWISE is LOGICAL 366*> If .TRUE. then ignore componentwise convergence. Default value 367*> is .FALSE.. 368*> \endverbatim 369*> 370*> \param[out] INFO 371*> \verbatim 372*> INFO is INTEGER 373*> = 0: Successful exit. 374*> < 0: if INFO = -i, the ith argument to SLA_SYRFSX_EXTENDED had an illegal 375*> value 376*> \endverbatim 377* 378* Authors: 379* ======== 380* 381*> \author Univ. of Tennessee 382*> \author Univ. of California Berkeley 383*> \author Univ. of Colorado Denver 384*> \author NAG Ltd. 385* 386*> \date December 2016 387* 388*> \ingroup realSYcomputational 389* 390* ===================================================================== 391 SUBROUTINE SLA_SYRFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA, 392 $ AF, LDAF, IPIV, COLEQU, C, B, LDB, 393 $ Y, LDY, BERR_OUT, N_NORMS, 394 $ ERR_BNDS_NORM, ERR_BNDS_COMP, RES, 395 $ AYB, DY, Y_TAIL, RCOND, ITHRESH, 396 $ RTHRESH, DZ_UB, IGNORE_CWISE, 397 $ INFO ) 398* 399* -- LAPACK computational routine (version 3.7.0) -- 400* -- LAPACK is a software package provided by Univ. of Tennessee, -- 401* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 402* December 2016 403* 404* .. Scalar Arguments .. 405 INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, 406 $ N_NORMS, ITHRESH 407 CHARACTER UPLO 408 LOGICAL COLEQU, IGNORE_CWISE 409 REAL RTHRESH, DZ_UB 410* .. 411* .. Array Arguments .. 412 INTEGER IPIV( * ) 413 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 414 $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) 415 REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ), 416 $ ERR_BNDS_NORM( NRHS, * ), 417 $ ERR_BNDS_COMP( NRHS, * ) 418* .. 419* 420* ===================================================================== 421* 422* .. Local Scalars .. 423 INTEGER UPLO2, CNT, I, J, X_STATE, Z_STATE 424 REAL YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT, 425 $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX, 426 $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z, 427 $ EPS, HUGEVAL, INCR_THRESH 428 LOGICAL INCR_PREC, UPPER 429* .. 430* .. Parameters .. 431 INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE, 432 $ NOPROG_STATE, Y_PREC_STATE, BASE_RESIDUAL, 433 $ EXTRA_RESIDUAL, EXTRA_Y 434 PARAMETER ( UNSTABLE_STATE = 0, WORKING_STATE = 1, 435 $ CONV_STATE = 2, NOPROG_STATE = 3 ) 436 PARAMETER ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1, 437 $ EXTRA_Y = 2 ) 438 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 439 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 440 INTEGER CMP_ERR_I, PIV_GROWTH_I 441 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 442 $ BERR_I = 3 ) 443 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 444 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 445 $ PIV_GROWTH_I = 9 ) 446 INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I, 447 $ LA_LINRX_CWISE_I 448 PARAMETER ( LA_LINRX_ITREF_I = 1, 449 $ LA_LINRX_ITHRESH_I = 2 ) 450 PARAMETER ( LA_LINRX_CWISE_I = 3 ) 451 INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I, 452 $ LA_LINRX_RCOND_I 453 PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 ) 454 PARAMETER ( LA_LINRX_RCOND_I = 3 ) 455* .. 456* .. External Functions .. 457 LOGICAL LSAME 458 EXTERNAL ILAUPLO 459 INTEGER ILAUPLO 460* .. 461* .. External Subroutines .. 462 EXTERNAL SAXPY, SCOPY, SSYTRS, SSYMV, BLAS_SSYMV_X, 463 $ BLAS_SSYMV2_X, SLA_SYAMV, SLA_WWADDW, 464 $ SLA_LIN_BERR 465 REAL SLAMCH 466* .. 467* .. Intrinsic Functions .. 468 INTRINSIC ABS, MAX, MIN 469* .. 470* .. Executable Statements .. 471* 472 INFO = 0 473 UPPER = LSAME( UPLO, 'U' ) 474 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 475 INFO = -2 476 ELSE IF( N.LT.0 ) THEN 477 INFO = -3 478 ELSE IF( NRHS.LT.0 ) THEN 479 INFO = -4 480 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 481 INFO = -6 482 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 483 INFO = -8 484 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 485 INFO = -13 486 ELSE IF( LDY.LT.MAX( 1, N ) ) THEN 487 INFO = -15 488 END IF 489 IF( INFO.NE.0 ) THEN 490 CALL XERBLA( 'SLA_SYRFSX_EXTENDED', -INFO ) 491 RETURN 492 END IF 493 EPS = SLAMCH( 'Epsilon' ) 494 HUGEVAL = SLAMCH( 'Overflow' ) 495* Force HUGEVAL to Inf 496 HUGEVAL = HUGEVAL * HUGEVAL 497* Using HUGEVAL may lead to spurious underflows. 498 INCR_THRESH = REAL( N )*EPS 499 500 IF ( LSAME ( UPLO, 'L' ) ) THEN 501 UPLO2 = ILAUPLO( 'L' ) 502 ELSE 503 UPLO2 = ILAUPLO( 'U' ) 504 ENDIF 505 506 DO J = 1, NRHS 507 Y_PREC_STATE = EXTRA_RESIDUAL 508 IF ( Y_PREC_STATE .EQ. EXTRA_Y ) THEN 509 DO I = 1, N 510 Y_TAIL( I ) = 0.0 511 END DO 512 END IF 513 514 DXRAT = 0.0 515 DXRATMAX = 0.0 516 DZRAT = 0.0 517 DZRATMAX = 0.0 518 FINAL_DX_X = HUGEVAL 519 FINAL_DZ_Z = HUGEVAL 520 PREVNORMDX = HUGEVAL 521 PREV_DZ_Z = HUGEVAL 522 DZ_Z = HUGEVAL 523 DX_X = HUGEVAL 524 525 X_STATE = WORKING_STATE 526 Z_STATE = UNSTABLE_STATE 527 INCR_PREC = .FALSE. 528 529 DO CNT = 1, ITHRESH 530* 531* Compute residual RES = B_s - op(A_s) * Y, 532* op(A) = A, A**T, or A**H depending on TRANS (and type). 533* 534 CALL SCOPY( N, B( 1, J ), 1, RES, 1 ) 535 IF (Y_PREC_STATE .EQ. BASE_RESIDUAL) THEN 536 CALL SSYMV( UPLO, N, -1.0, A, LDA, Y(1,J), 1, 537 $ 1.0, RES, 1 ) 538 ELSE IF (Y_PREC_STATE .EQ. EXTRA_RESIDUAL) THEN 539 CALL BLAS_SSYMV_X( UPLO2, N, -1.0, A, LDA, 540 $ Y( 1, J ), 1, 1.0, RES, 1, PREC_TYPE ) 541 ELSE 542 CALL BLAS_SSYMV2_X(UPLO2, N, -1.0, A, LDA, 543 $ Y(1, J), Y_TAIL, 1, 1.0, RES, 1, PREC_TYPE) 544 END IF 545 546! XXX: RES is no longer needed. 547 CALL SCOPY( N, RES, 1, DY, 1 ) 548 CALL SSYTRS( UPLO, N, 1, AF, LDAF, IPIV, DY, N, INFO ) 549* 550* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT. 551* 552 NORMX = 0.0 553 NORMY = 0.0 554 NORMDX = 0.0 555 DZ_Z = 0.0 556 YMIN = HUGEVAL 557 558 DO I = 1, N 559 YK = ABS( Y( I, J ) ) 560 DYK = ABS( DY( I ) ) 561 562 IF ( YK .NE. 0.0 ) THEN 563 DZ_Z = MAX( DZ_Z, DYK / YK ) 564 ELSE IF ( DYK .NE. 0.0 ) THEN 565 DZ_Z = HUGEVAL 566 END IF 567 568 YMIN = MIN( YMIN, YK ) 569 570 NORMY = MAX( NORMY, YK ) 571 572 IF ( COLEQU ) THEN 573 NORMX = MAX( NORMX, YK * C( I ) ) 574 NORMDX = MAX( NORMDX, DYK * C( I ) ) 575 ELSE 576 NORMX = NORMY 577 NORMDX = MAX(NORMDX, DYK) 578 END IF 579 END DO 580 581 IF ( NORMX .NE. 0.0 ) THEN 582 DX_X = NORMDX / NORMX 583 ELSE IF ( NORMDX .EQ. 0.0 ) THEN 584 DX_X = 0.0 585 ELSE 586 DX_X = HUGEVAL 587 END IF 588 589 DXRAT = NORMDX / PREVNORMDX 590 DZRAT = DZ_Z / PREV_DZ_Z 591* 592* Check termination criteria. 593* 594 IF ( YMIN*RCOND .LT. INCR_THRESH*NORMY 595 $ .AND. Y_PREC_STATE .LT. EXTRA_Y ) 596 $ INCR_PREC = .TRUE. 597 598 IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH ) 599 $ X_STATE = WORKING_STATE 600 IF ( X_STATE .EQ. WORKING_STATE ) THEN 601 IF ( DX_X .LE. EPS ) THEN 602 X_STATE = CONV_STATE 603 ELSE IF ( DXRAT .GT. RTHRESH ) THEN 604 IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN 605 INCR_PREC = .TRUE. 606 ELSE 607 X_STATE = NOPROG_STATE 608 END IF 609 ELSE 610 IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT 611 END IF 612 IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X 613 END IF 614 615 IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB ) 616 $ Z_STATE = WORKING_STATE 617 IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH ) 618 $ Z_STATE = WORKING_STATE 619 IF ( Z_STATE .EQ. WORKING_STATE ) THEN 620 IF ( DZ_Z .LE. EPS ) THEN 621 Z_STATE = CONV_STATE 622 ELSE IF ( DZ_Z .GT. DZ_UB ) THEN 623 Z_STATE = UNSTABLE_STATE 624 DZRATMAX = 0.0 625 FINAL_DZ_Z = HUGEVAL 626 ELSE IF ( DZRAT .GT. RTHRESH ) THEN 627 IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN 628 INCR_PREC = .TRUE. 629 ELSE 630 Z_STATE = NOPROG_STATE 631 END IF 632 ELSE 633 IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT 634 END IF 635 IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z 636 END IF 637 638 IF ( X_STATE.NE.WORKING_STATE.AND. 639 $ ( IGNORE_CWISE.OR.Z_STATE.NE.WORKING_STATE ) ) 640 $ GOTO 666 641 642 IF ( INCR_PREC ) THEN 643 INCR_PREC = .FALSE. 644 Y_PREC_STATE = Y_PREC_STATE + 1 645 DO I = 1, N 646 Y_TAIL( I ) = 0.0 647 END DO 648 END IF 649 650 PREVNORMDX = NORMDX 651 PREV_DZ_Z = DZ_Z 652* 653* Update soluton. 654* 655 IF (Y_PREC_STATE .LT. EXTRA_Y) THEN 656 CALL SAXPY( N, 1.0, DY, 1, Y(1,J), 1 ) 657 ELSE 658 CALL SLA_WWADDW( N, Y(1,J), Y_TAIL, DY ) 659 END IF 660 661 END DO 662* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT. 663 666 CONTINUE 664* 665* Set final_* when cnt hits ithresh. 666* 667 IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X 668 IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z 669* 670* Compute error bounds. 671* 672 IF ( N_NORMS .GE. 1 ) THEN 673 ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 674 $ FINAL_DX_X / (1 - DXRATMAX) 675 END IF 676 IF ( N_NORMS .GE. 2 ) THEN 677 ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 678 $ FINAL_DZ_Z / (1 - DZRATMAX) 679 END IF 680* 681* Compute componentwise relative backward error from formula 682* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) 683* where abs(Z) is the componentwise absolute value of the matrix 684* or vector Z. 685* 686* Compute residual RES = B_s - op(A_s) * Y, 687* op(A) = A, A**T, or A**H depending on TRANS (and type). 688 CALL SCOPY( N, B( 1, J ), 1, RES, 1 ) 689 CALL SSYMV( UPLO, N, -1.0, A, LDA, Y(1,J), 1, 1.0, RES, 1 ) 690 691 DO I = 1, N 692 AYB( I ) = ABS( B( I, J ) ) 693 END DO 694* 695* Compute abs(op(A_s))*abs(Y) + abs(B_s). 696* 697 CALL SLA_SYAMV( UPLO2, N, 1.0, 698 $ A, LDA, Y(1, J), 1, 1.0, AYB, 1 ) 699 700 CALL SLA_LIN_BERR( N, N, 1, RES, AYB, BERR_OUT( J ) ) 701* 702* End of loop for each RHS. 703* 704 END DO 705* 706 RETURN 707 END 708