1*> \brief \b CLA_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 CLA_PORFSX_EXTENDED + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cla_porfsx_extended.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cla_porfsx_extended.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cla_porfsx_extended.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE CLA_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* COMPLEX 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*> CLA_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 CPORFSX 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 COMPLEX 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 COMPLEX 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 CPOTRF. 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 COMPLEX 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 COMPLEX array, dimension 157*> (LDY,NRHS) 158*> On entry, the solution matrix X, as computed by CPOTRS. 159*> On exit, the improved solution matrix Y. 160*> \endverbatim 161*> 162*> \param[in] LDY 163*> \verbatim 164*> LDY is INTEGER 165*> The leading dimension of the array Y. LDY >= max(1,N). 166*> \endverbatim 167*> 168*> \param[out] BERR_OUT 169*> \verbatim 170*> BERR_OUT is REAL array, dimension (NRHS) 171*> On exit, BERR_OUT(j) contains the componentwise relative backward 172*> error for right-hand-side j from the formula 173*> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) 174*> where abs(Z) is the componentwise absolute value of the matrix 175*> or vector Z. This is computed by CLA_LIN_BERR. 176*> \endverbatim 177*> 178*> \param[in] N_NORMS 179*> \verbatim 180*> N_NORMS is INTEGER 181*> Determines which error bounds to return (see ERR_BNDS_NORM 182*> and ERR_BNDS_COMP). 183*> If N_NORMS >= 1 return normwise error bounds. 184*> If N_NORMS >= 2 return componentwise error bounds. 185*> \endverbatim 186*> 187*> \param[in,out] ERR_BNDS_NORM 188*> \verbatim 189*> ERR_BNDS_NORM is REAL array, dimension 190*> (NRHS, N_ERR_BNDS) 191*> For each right-hand side, this array contains information about 192*> various error bounds and condition numbers corresponding to the 193*> normwise relative error, which is defined as follows: 194*> 195*> Normwise relative error in the ith solution vector: 196*> max_j (abs(XTRUE(j,i) - X(j,i))) 197*> ------------------------------ 198*> max_j abs(X(j,i)) 199*> 200*> The array is indexed by the type of error information as described 201*> below. There currently are up to three pieces of information 202*> returned. 203*> 204*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 205*> right-hand side. 206*> 207*> The second index in ERR_BNDS_NORM(:,err) contains the following 208*> three fields: 209*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 210*> reciprocal condition number is less than the threshold 211*> sqrt(n) * slamch('Epsilon'). 212*> 213*> err = 2 "Guaranteed" error bound: The estimated forward error, 214*> almost certainly within a factor of 10 of the true error 215*> so long as the next entry is greater than the threshold 216*> sqrt(n) * slamch('Epsilon'). This error bound should only 217*> be trusted if the previous boolean is true. 218*> 219*> err = 3 Reciprocal condition number: Estimated normwise 220*> reciprocal condition number. Compared with the threshold 221*> sqrt(n) * slamch('Epsilon') to determine if the error 222*> estimate is "guaranteed". These reciprocal condition 223*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 224*> appropriately scaled matrix Z. 225*> Let Z = S*A, where S scales each row by a power of the 226*> radix so all absolute row 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*> \endverbatim 233*> 234*> \param[in,out] ERR_BNDS_COMP 235*> \verbatim 236*> ERR_BNDS_COMP is REAL array, dimension 237*> (NRHS, N_ERR_BNDS) 238*> For each right-hand side, this array contains information about 239*> various error bounds and condition numbers corresponding to the 240*> componentwise relative error, which is defined as follows: 241*> 242*> Componentwise relative error in the ith solution vector: 243*> abs(XTRUE(j,i) - X(j,i)) 244*> max_j ---------------------- 245*> abs(X(j,i)) 246*> 247*> The array is indexed by the right-hand side i (on which the 248*> componentwise relative error depends), and the type of error 249*> information as described below. There currently are up to three 250*> pieces of information returned for each right-hand side. If 251*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 252*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most 253*> the first (:,N_ERR_BNDS) entries are returned. 254*> 255*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 256*> right-hand side. 257*> 258*> The second index in ERR_BNDS_COMP(:,err) contains the following 259*> three fields: 260*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 261*> reciprocal condition number is less than the threshold 262*> sqrt(n) * slamch('Epsilon'). 263*> 264*> err = 2 "Guaranteed" error bound: The estimated forward error, 265*> almost certainly within a factor of 10 of the true error 266*> so long as the next entry is greater than the threshold 267*> sqrt(n) * slamch('Epsilon'). This error bound should only 268*> be trusted if the previous boolean is true. 269*> 270*> err = 3 Reciprocal condition number: Estimated componentwise 271*> reciprocal condition number. Compared with the threshold 272*> sqrt(n) * slamch('Epsilon') to determine if the error 273*> estimate is "guaranteed". These reciprocal condition 274*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 275*> appropriately scaled matrix Z. 276*> Let Z = S*(A*diag(x)), where x is the solution for the 277*> current right-hand side and S scales each row of 278*> A*diag(x) by a power of the radix so all absolute row 279*> sums of Z are approximately 1. 280*> 281*> This subroutine is only responsible for setting the second field 282*> above. 283*> See Lapack Working Note 165 for further details and extra 284*> cautions. 285*> \endverbatim 286*> 287*> \param[in] RES 288*> \verbatim 289*> RES is COMPLEX array, dimension (N) 290*> Workspace to hold the intermediate residual. 291*> \endverbatim 292*> 293*> \param[in] AYB 294*> \verbatim 295*> AYB is REAL array, dimension (N) 296*> Workspace. 297*> \endverbatim 298*> 299*> \param[in] DY 300*> \verbatim 301*> DY is COMPLEX array, dimension (N) 302*> Workspace to hold the intermediate solution. 303*> \endverbatim 304*> 305*> \param[in] Y_TAIL 306*> \verbatim 307*> Y_TAIL is COMPLEX array, dimension (N) 308*> Workspace to hold the trailing bits of the intermediate solution. 309*> \endverbatim 310*> 311*> \param[in] RCOND 312*> \verbatim 313*> RCOND is REAL 314*> Reciprocal scaled condition number. This is an estimate of the 315*> reciprocal Skeel condition number of the matrix A after 316*> equilibration (if done). If this is less than the machine 317*> precision (in particular, if it is zero), the matrix is singular 318*> to working precision. Note that the error may still be small even 319*> if this number is very small and the matrix appears ill- 320*> conditioned. 321*> \endverbatim 322*> 323*> \param[in] ITHRESH 324*> \verbatim 325*> ITHRESH is INTEGER 326*> The maximum number of residual computations allowed for 327*> refinement. The default is 10. For 'aggressive' set to 100 to 328*> permit convergence using approximate factorizations or 329*> factorizations other than LU. If the factorization uses a 330*> technique other than Gaussian elimination, the guarantees in 331*> ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. 332*> \endverbatim 333*> 334*> \param[in] RTHRESH 335*> \verbatim 336*> RTHRESH is REAL 337*> Determines when to stop refinement if the error estimate stops 338*> decreasing. Refinement will stop when the next solution no longer 339*> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is 340*> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The 341*> default value is 0.5. For 'aggressive' set to 0.9 to permit 342*> convergence on extremely ill-conditioned matrices. See LAWN 165 343*> for more details. 344*> \endverbatim 345*> 346*> \param[in] DZ_UB 347*> \verbatim 348*> DZ_UB is REAL 349*> Determines when to start considering componentwise convergence. 350*> Componentwise convergence is only considered after each component 351*> of the solution Y is stable, which we definte as the relative 352*> change in each component being less than DZ_UB. The default value 353*> is 0.25, requiring the first bit to be stable. See LAWN 165 for 354*> more details. 355*> \endverbatim 356*> 357*> \param[in] IGNORE_CWISE 358*> \verbatim 359*> IGNORE_CWISE is LOGICAL 360*> If .TRUE. then ignore componentwise convergence. Default value 361*> is .FALSE.. 362*> \endverbatim 363*> 364*> \param[out] INFO 365*> \verbatim 366*> INFO is INTEGER 367*> = 0: Successful exit. 368*> < 0: if INFO = -i, the ith argument to CPOTRS had an illegal 369*> value 370*> \endverbatim 371* 372* Authors: 373* ======== 374* 375*> \author Univ. of Tennessee 376*> \author Univ. of California Berkeley 377*> \author Univ. of Colorado Denver 378*> \author NAG Ltd. 379* 380*> \date September 2012 381* 382*> \ingroup complexPOcomputational 383* 384* ===================================================================== 385 SUBROUTINE CLA_PORFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA, 386 $ AF, LDAF, COLEQU, C, B, LDB, Y, 387 $ LDY, BERR_OUT, N_NORMS, 388 $ ERR_BNDS_NORM, ERR_BNDS_COMP, RES, 389 $ AYB, DY, Y_TAIL, RCOND, ITHRESH, 390 $ RTHRESH, DZ_UB, IGNORE_CWISE, 391 $ INFO ) 392* 393* -- LAPACK computational routine (version 3.4.2) -- 394* -- LAPACK is a software package provided by Univ. of Tennessee, -- 395* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 396* September 2012 397* 398* .. Scalar Arguments .. 399 INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, 400 $ N_NORMS, ITHRESH 401 CHARACTER UPLO 402 LOGICAL COLEQU, IGNORE_CWISE 403 REAL RTHRESH, DZ_UB 404* .. 405* .. Array Arguments .. 406 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 407 $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) 408 REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ), 409 $ ERR_BNDS_NORM( NRHS, * ), 410 $ ERR_BNDS_COMP( NRHS, * ) 411* .. 412* 413* ===================================================================== 414* 415* .. Local Scalars .. 416 INTEGER UPLO2, CNT, I, J, X_STATE, Z_STATE, 417 $ Y_PREC_STATE 418 REAL YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT, 419 $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX, 420 $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z, 421 $ EPS, HUGEVAL, INCR_THRESH 422 LOGICAL INCR_PREC 423 COMPLEX ZDUM 424* .. 425* .. Parameters .. 426 INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE, 427 $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL, 428 $ EXTRA_Y 429 PARAMETER ( UNSTABLE_STATE = 0, WORKING_STATE = 1, 430 $ CONV_STATE = 2, NOPROG_STATE = 3 ) 431 PARAMETER ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1, 432 $ EXTRA_Y = 2 ) 433 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 434 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 435 INTEGER CMP_ERR_I, PIV_GROWTH_I 436 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 437 $ BERR_I = 3 ) 438 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 439 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 440 $ PIV_GROWTH_I = 9 ) 441 INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I, 442 $ LA_LINRX_CWISE_I 443 PARAMETER ( LA_LINRX_ITREF_I = 1, 444 $ LA_LINRX_ITHRESH_I = 2 ) 445 PARAMETER ( LA_LINRX_CWISE_I = 3 ) 446 INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I, 447 $ LA_LINRX_RCOND_I 448 PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 ) 449 PARAMETER ( LA_LINRX_RCOND_I = 3 ) 450* .. 451* .. External Functions .. 452 LOGICAL LSAME 453 EXTERNAL ILAUPLO 454 INTEGER ILAUPLO 455* .. 456* .. External Subroutines .. 457 EXTERNAL CAXPY, CCOPY, CPOTRS, CHEMV, BLAS_CHEMV_X, 458 $ BLAS_CHEMV2_X, CLA_HEAMV, CLA_WWADDW, 459 $ CLA_LIN_BERR, SLAMCH 460 REAL SLAMCH 461* .. 462* .. Intrinsic Functions .. 463 INTRINSIC ABS, REAL, AIMAG, MAX, MIN 464* .. 465* .. Statement Functions .. 466 REAL CABS1 467* .. 468* .. Statement Function Definitions .. 469 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) ) 470* .. 471* .. Executable Statements .. 472* 473 IF (INFO.NE.0) RETURN 474 EPS = SLAMCH( 'Epsilon' ) 475 HUGEVAL = SLAMCH( 'Overflow' ) 476* Force HUGEVAL to Inf 477 HUGEVAL = HUGEVAL * HUGEVAL 478* Using HUGEVAL may lead to spurious underflows. 479 INCR_THRESH = REAL(N) * EPS 480 481 IF (LSAME (UPLO, 'L')) THEN 482 UPLO2 = ILAUPLO( 'L' ) 483 ELSE 484 UPLO2 = ILAUPLO( 'U' ) 485 ENDIF 486 487 DO J = 1, NRHS 488 Y_PREC_STATE = EXTRA_RESIDUAL 489 IF (Y_PREC_STATE .EQ. EXTRA_Y) THEN 490 DO I = 1, N 491 Y_TAIL( I ) = 0.0 492 END DO 493 END IF 494 495 DXRAT = 0.0 496 DXRATMAX = 0.0 497 DZRAT = 0.0 498 DZRATMAX = 0.0 499 FINAL_DX_X = HUGEVAL 500 FINAL_DZ_Z = HUGEVAL 501 PREVNORMDX = HUGEVAL 502 PREV_DZ_Z = HUGEVAL 503 DZ_Z = HUGEVAL 504 DX_X = HUGEVAL 505 506 X_STATE = WORKING_STATE 507 Z_STATE = UNSTABLE_STATE 508 INCR_PREC = .FALSE. 509 510 DO CNT = 1, ITHRESH 511* 512* Compute residual RES = B_s - op(A_s) * Y, 513* op(A) = A, A**T, or A**H depending on TRANS (and type). 514* 515 CALL CCOPY( N, B( 1, J ), 1, RES, 1 ) 516 IF (Y_PREC_STATE .EQ. BASE_RESIDUAL) THEN 517 CALL CHEMV(UPLO, N, CMPLX(-1.0), A, LDA, Y(1,J), 1, 518 $ CMPLX(1.0), RES, 1) 519 ELSE IF (Y_PREC_STATE .EQ. EXTRA_RESIDUAL) THEN 520 CALL BLAS_CHEMV_X(UPLO2, N, CMPLX(-1.0), A, LDA, 521 $ Y( 1, J ), 1, CMPLX(1.0), RES, 1, PREC_TYPE) 522 ELSE 523 CALL BLAS_CHEMV2_X(UPLO2, N, CMPLX(-1.0), A, LDA, 524 $ Y(1, J), Y_TAIL, 1, CMPLX(1.0), RES, 1, PREC_TYPE) 525 END IF 526 527! XXX: RES is no longer needed. 528 CALL CCOPY( N, RES, 1, DY, 1 ) 529 CALL CPOTRS( UPLO, N, 1, AF, LDAF, DY, N, INFO) 530* 531* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT. 532* 533 NORMX = 0.0 534 NORMY = 0.0 535 NORMDX = 0.0 536 DZ_Z = 0.0 537 YMIN = HUGEVAL 538 539 DO I = 1, N 540 YK = CABS1(Y(I, J)) 541 DYK = CABS1(DY(I)) 542 543 IF (YK .NE. 0.0) THEN 544 DZ_Z = MAX( DZ_Z, DYK / YK ) 545 ELSE IF (DYK .NE. 0.0) THEN 546 DZ_Z = HUGEVAL 547 END IF 548 549 YMIN = MIN( YMIN, YK ) 550 551 NORMY = MAX( NORMY, YK ) 552 553 IF ( COLEQU ) THEN 554 NORMX = MAX(NORMX, YK * C(I)) 555 NORMDX = MAX(NORMDX, DYK * C(I)) 556 ELSE 557 NORMX = NORMY 558 NORMDX = MAX(NORMDX, DYK) 559 END IF 560 END DO 561 562 IF (NORMX .NE. 0.0) THEN 563 DX_X = NORMDX / NORMX 564 ELSE IF (NORMDX .EQ. 0.0) THEN 565 DX_X = 0.0 566 ELSE 567 DX_X = HUGEVAL 568 END IF 569 570 DXRAT = NORMDX / PREVNORMDX 571 DZRAT = DZ_Z / PREV_DZ_Z 572* 573* Check termination criteria. 574* 575 IF (YMIN*RCOND .LT. INCR_THRESH*NORMY 576 $ .AND. Y_PREC_STATE .LT. EXTRA_Y) 577 $ INCR_PREC = .TRUE. 578 579 IF (X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH) 580 $ X_STATE = WORKING_STATE 581 IF (X_STATE .EQ. WORKING_STATE) THEN 582 IF (DX_X .LE. EPS) THEN 583 X_STATE = CONV_STATE 584 ELSE IF (DXRAT .GT. RTHRESH) THEN 585 IF (Y_PREC_STATE .NE. EXTRA_Y) THEN 586 INCR_PREC = .TRUE. 587 ELSE 588 X_STATE = NOPROG_STATE 589 END IF 590 ELSE 591 IF (DXRAT .GT. DXRATMAX) DXRATMAX = DXRAT 592 END IF 593 IF (X_STATE .GT. WORKING_STATE) FINAL_DX_X = DX_X 594 END IF 595 596 IF (Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB) 597 $ Z_STATE = WORKING_STATE 598 IF (Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH) 599 $ Z_STATE = WORKING_STATE 600 IF (Z_STATE .EQ. WORKING_STATE) THEN 601 IF (DZ_Z .LE. EPS) THEN 602 Z_STATE = CONV_STATE 603 ELSE IF (DZ_Z .GT. DZ_UB) THEN 604 Z_STATE = UNSTABLE_STATE 605 DZRATMAX = 0.0 606 FINAL_DZ_Z = HUGEVAL 607 ELSE IF (DZRAT .GT. RTHRESH) THEN 608 IF (Y_PREC_STATE .NE. EXTRA_Y) THEN 609 INCR_PREC = .TRUE. 610 ELSE 611 Z_STATE = NOPROG_STATE 612 END IF 613 ELSE 614 IF (DZRAT .GT. DZRATMAX) DZRATMAX = DZRAT 615 END IF 616 IF (Z_STATE .GT. WORKING_STATE) FINAL_DZ_Z = DZ_Z 617 END IF 618 619 IF ( X_STATE.NE.WORKING_STATE.AND. 620 $ (IGNORE_CWISE.OR.Z_STATE.NE.WORKING_STATE) ) 621 $ GOTO 666 622 623 IF (INCR_PREC) THEN 624 INCR_PREC = .FALSE. 625 Y_PREC_STATE = Y_PREC_STATE + 1 626 DO I = 1, N 627 Y_TAIL( I ) = 0.0 628 END DO 629 END IF 630 631 PREVNORMDX = NORMDX 632 PREV_DZ_Z = DZ_Z 633* 634* Update soluton. 635* 636 IF (Y_PREC_STATE .LT. EXTRA_Y) THEN 637 CALL CAXPY( N, CMPLX(1.0), DY, 1, Y(1,J), 1 ) 638 ELSE 639 CALL CLA_WWADDW(N, Y(1,J), Y_TAIL, DY) 640 END IF 641 642 END DO 643* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT. 644 666 CONTINUE 645* 646* Set final_* when cnt hits ithresh. 647* 648 IF (X_STATE .EQ. WORKING_STATE) FINAL_DX_X = DX_X 649 IF (Z_STATE .EQ. WORKING_STATE) FINAL_DZ_Z = DZ_Z 650* 651* Compute error bounds. 652* 653 IF (N_NORMS .GE. 1) THEN 654 ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 655 $ FINAL_DX_X / (1 - DXRATMAX) 656 END IF 657 IF (N_NORMS .GE. 2) THEN 658 ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 659 $ FINAL_DZ_Z / (1 - DZRATMAX) 660 END IF 661* 662* Compute componentwise relative backward error from formula 663* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) 664* where abs(Z) is the componentwise absolute value of the matrix 665* or vector Z. 666* 667* Compute residual RES = B_s - op(A_s) * Y, 668* op(A) = A, A**T, or A**H depending on TRANS (and type). 669* 670 CALL CCOPY( N, B( 1, J ), 1, RES, 1 ) 671 CALL CHEMV(UPLO, N, CMPLX(-1.0), A, LDA, Y(1,J), 1, CMPLX(1.0), 672 $ RES, 1) 673 674 DO I = 1, N 675 AYB( I ) = CABS1( B( I, J ) ) 676 END DO 677* 678* Compute abs(op(A_s))*abs(Y) + abs(B_s). 679* 680 CALL CLA_HEAMV (UPLO2, N, 1.0, 681 $ A, LDA, Y(1, J), 1, 1.0, AYB, 1) 682 683 CALL CLA_LIN_BERR (N, N, 1, RES, AYB, BERR_OUT(J)) 684* 685* End of loop for each RHS. 686* 687 END DO 688* 689 RETURN 690 END 691