##############ECHO OF PROBLEM################# ##############temp/sing1postode.ode################# diff ( y , x , 1 ) = neg ( 2.0 ) * x / ( x * x + 0.000001 ) / ( x * x + 0.000001 ) ; ! /* BEGIN FIRST INPUT BLOCK */ Digits=32; max_terms=30; ! /* END FIRST INPUT BLOCK */ /* BEGIN SECOND INPUT BLOCK */ x_start=c(-2.0); x_end=c(-1.5); array_y_init[0 + 1] = exact_soln_y(x_start); glob_look_poles=true; glob_max_h=c(0.001); glob_min_h=c(0.001); glob_type_given_pole=2; array_given_rad_poles[1][1]=c(0.0); array_given_rad_poles[1][2]=c(0.001); array_given_ord_poles[1][1]=c(1.0); array_given_ord_poles[1][2]=c(0.0); /* END SECOND INPUT BLOCK */ /* BEGIN OVERRIDE BLOCK */ glob_desired_digits_correct=8; glob_max_minutes=(3.0); glob_subiter_method=3; glob_max_iter=100000; glob_upper_ratio_limit=c(1.000001); glob_lower_ratio_limit=c(0.999999); glob_look_poles=false; glob_h=c(0.001); glob_display_interval=c(0.01); /* END OVERRIDE BLOCK */ ! /* BEGIN USER DEF BLOCK */ double exact_soln_y (double x) { return(c(1.0) / (c(x) * c(x) + c(0.000001))) ; } /* END USER DEF BLOCK */ #######END OF ECHO OF PROBLEM################# START of Soultion TOP MAIN SOLVE Loop x[1] = -2 y[1] (closed_form) = 0.2499999375000156 y[1] (numeric) = 0.2499999375000156 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.990000000000001 y[1] (closed_form) = 0.2525188120201982 y[1] (numeric) = 0.2525188120201985 absolute error = 2.775557561562891e-16 relative error = 1.099148827510278e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.980000000000002 y[1] (closed_form) = 0.255075947588014 y[1] (numeric) = 0.2550759475880146 absolute error = 6.106226635438361e-16 relative error = 2.393885700779924e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.970000000000003 y[1] (closed_form) = 0.2576721230456527 y[1] (numeric) = 0.2576721230456535 absolute error = 7.771561172376096e-16 relative error = 3.016065952543566e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.960000000000004 y[1] (closed_form) = 0.2603081371542738 y[1] (numeric) = 0.2603081371542749 absolute error = 1.054711873393899e-15 relative error = 4.051782187541893e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.950000000000006 y[1] (closed_form) = 0.2629848092084642 y[1] (numeric) = 0.2629848092084656 absolute error = 1.387778780781446e-15 relative error = 5.277030201700259e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.940000000000007 y[1] (closed_form) = 0.2657029796729231 y[1] (numeric) = 0.2657029796729248 absolute error = 1.665334536937735e-15 relative error = 6.267654728553439e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.930000000000008 y[1] (closed_form) = 0.2684635108422994 y[1] (numeric) = 0.2684635108423014 absolute error = 1.942890293094024e-15 relative error = 7.237073995636282e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.920000000000009 y[1] (closed_form) = 0.2712672875251474 y[1] (numeric) = 0.2712672875251496 absolute error = 2.220446049250313e-15 relative error = 8.185454536402478e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.91000000000001 y[1] (closed_form) = 0.2741152177530144 y[1] (numeric) = 0.2741152177530169 absolute error = 2.498001805406602e-15 relative error = 9.112962884305726e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.900000000000011 y[1] (closed_form) = 0.2770082335157216 y[1] (numeric) = 0.2770082335157243 absolute error = 2.775557561562891e-15 relative error = 1.001976557279972e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.890000000000012 y[1] (closed_form) = 0.2799472915239483 y[1] (numeric) = 0.2799472915239514 absolute error = 3.108624468950438e-15 relative error = 1.110432057416247e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.880000000000013 y[1] (closed_form) = 0.2829333740002863 y[1] (numeric) = 0.2829333740002898 absolute error = 3.441691376337985e-15 relative error = 1.216431744222052e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.870000000000014 y[1] (closed_form) = 0.2859674894999844 y[1] (numeric) = 0.2859674894999882 absolute error = 3.83026943495679e-15 relative error = 1.339407301737004e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.860000000000015 y[1] (closed_form) = 0.2890506737626632 y[1] (numeric) = 0.2890506737626674 absolute error = 4.218847493575595e-15 relative error = 1.459552900762186e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.850000000000017 y[1] (closed_form) = 0.2921839905963452 y[1] (numeric) = 0.2921839905963498 absolute error = 4.6074255521944e-15 relative error = 1.576891855981117e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.840000000000018 y[1] (closed_form) = 0.2953685327952056 y[1] (numeric) = 0.2953685327952104 absolute error = 4.829470157119431e-15 relative error = 1.635065899341402e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.830000000000019 y[1] (closed_form) = 0.2986054230925249 y[1] (numeric) = 0.2986054230925301 absolute error = 5.218048215738236e-15 relative error = 1.747472688773435e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.82000000000002 y[1] (closed_form) = 0.3018958151503934 y[1] (numeric) = 0.3018958151503989 absolute error = 5.551115123125783e-15 relative error = 1.838751928495737e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.810000000000021 y[1] (closed_form) = 0.3052408945877971 y[1] (numeric) = 0.305240894587803 absolute error = 5.88418203051333e-15 relative error = 1.92771746343472e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.800000000000022 y[1] (closed_form) = 0.3086418800487949 y[1] (numeric) = 0.3086418800488011 absolute error = 6.217248937900877e-15 relative error = 2.014389277604827e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.790000000000023 y[1] (closed_form) = 0.3121000243125838 y[1] (numeric) = 0.3121000243125904 absolute error = 6.605826996519681e-15 relative error = 2.116573688537626e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.780000000000024 y[1] (closed_form) = 0.3156166154473417 y[1] (numeric) = 0.3156166154473486 absolute error = 6.994405055138486e-15 relative error = 2.216107997110644e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.770000000000025 y[1] (closed_form) = 0.319192978009829 y[1] (numeric) = 0.3191929780098363 absolute error = 7.271960811294775e-15 relative error = 2.278233329766686e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.760000000000026 y[1] (closed_form) = 0.3228304742928382 y[1] (numeric) = 0.3228304742928459 absolute error = 7.771561172376096e-15 relative error = 2.407319565911409e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.750000000000028 y[1] (closed_form) = 0.3265305056226818 y[1] (numeric) = 0.32653050562269 absolute error = 8.215650382226158e-15 relative error = 2.516043751121878e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.740000000000029 y[1] (closed_form) = 0.3302945137090281 y[1] (numeric) = 0.3302945137090367 absolute error = 8.604228440844963e-15 relative error = 2.605017063173151e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.73000000000003 y[1] (closed_form) = 0.3341239820495117 y[1] (numeric) = 0.3341239820495207 absolute error = 8.93729534823251e-15 relative error = 2.674844018502135e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.720000000000031 y[1] (closed_form) = 0.3380204373916734 y[1] (numeric) = 0.3380204373916829 absolute error = 9.436895709313831e-15 relative error = 2.791812170333075e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.710000000000032 y[1] (closed_form) = 0.3419854512549199 y[1] (numeric) = 0.3419854512549297 absolute error = 9.825473767932635e-15 relative error = 2.873067767028666e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.700000000000033 y[1] (closed_form) = 0.3460206415153355 y[1] (numeric) = 0.3460206415153458 absolute error = 1.032507412901396e-14 relative error = 2.983947455792563e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.690000000000034 y[1] (closed_form) = 0.3501276740563304 y[1] (numeric) = 0.3501276740563413 absolute error = 1.082467449009528e-14 relative error = 3.091636363583686e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.680000000000035 y[1] (closed_form) = 0.3543082644882701 y[1] (numeric) = 0.3543082644882813 absolute error = 1.121325254871408e-14 relative error = 3.16482952067445e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.670000000000036 y[1] (closed_form) = 0.3585641799403982 y[1] (numeric) = 0.3585641799404099 absolute error = 1.17128529097954e-14 relative error = 3.266598719298273e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.660000000000037 y[1] (closed_form) = 0.3628972409285506 y[1] (numeric) = 0.3628972409285628 absolute error = 1.221245327087672e-14 relative error = 3.365264844568269e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.650000000000039 y[1] (closed_form) = 0.3673093233023434 y[1] (numeric) = 0.3673093233023562 absolute error = 1.282307593442056e-14 relative error = 3.491083705453754e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.64000000000004 y[1] (closed_form) = 0.3718023602757255 y[1] (numeric) = 0.3718023602757388 absolute error = 1.332267629550188e-14 relative error = 3.583268348705988e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.630000000000041 y[1] (closed_form) = 0.3763783445449981 y[1] (numeric) = 0.376378344545012 absolute error = 1.393329895904571e-14 relative error = 3.701939593758937e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.620000000000042 y[1] (closed_form) = 0.381039330498635 y[1] (numeric) = 0.3810393304986495 absolute error = 1.443289932012704e-14 relative error = 3.787771540864267e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.610000000000043 y[1] (closed_form) = 0.385787436523479 y[1] (numeric) = 0.385787436523494 absolute error = 1.493249968120836e-14 relative error = 3.870654735616193e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.600000000000044 y[1] (closed_form) = 0.3906248474121475 y[1] (numeric) = 0.3906248474121629 absolute error = 1.543210004228968e-14 relative error = 3.950619154036378e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.590000000000045 y[1] (closed_form) = 0.3955538168767558 y[1] (numeric) = 0.3955538168767718 absolute error = 1.604272270583351e-14 relative error = 4.055762331534271e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.580000000000046 y[1] (closed_form) = 0.4005766701743596 y[1] (numeric) = 0.4005766701743761 absolute error = 1.648681191568357e-14 relative error = 4.11576937531268e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.570000000000047 y[1] (closed_form) = 0.4056958068498246 y[1] (numeric) = 0.4056958068498417 absolute error = 1.709743457922741e-14 relative error = 4.214348359177477e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.560000000000048 y[1] (closed_form) = 0.4109137036021672 y[1] (numeric) = 0.4109137036021849 absolute error = 1.770805724277125e-14 relative error = 4.309434581406803e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.55000000000005 y[1] (closed_form) = 0.416232917280757 y[1] (numeric) = 0.4162329172807753 absolute error = 1.826316875508383e-14 relative error = 4.387728119726045e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.540000000000051 y[1] (closed_form) = 0.4216560880181506 y[1] (numeric) = 0.4216560880181695 absolute error = 1.88182802673964e-14 relative error = 4.462945230044052e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.530000000000052 y[1] (closed_form) = 0.4271859425067238 y[1] (numeric) = 0.4271859425067432 absolute error = 1.937339177970898e-14 relative error = 4.535119219051561e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.520000000000053 y[1] (closed_form) = 0.4328252974266936 y[1] (numeric) = 0.4328252974267136 absolute error = 1.998401444325282e-14 relative error = 4.617108695370897e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.510000000000054 y[1] (closed_form) = 0.4385770630335799 y[1] (numeric) = 0.4385770630336005 absolute error = 2.065014825802791e-14 relative error = 4.708442369328107e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.500000000000055 y[1] (closed_form) = 0.4444442469136354 y[1] (numeric) = 0.4444442469136567 absolute error = 2.131628207280301e-14 relative error = 4.796165598009236e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 Finished! diff ( y , x , 1 ) = neg ( 2.0 ) * x / ( x * x + 0.000001 ) / ( x * x + 0.000001 ) ; Iterations = 501 Total Elapsed Time = 0.0 Seconds Elapsed Time(since restart) = 0.0 Seconds Time to Timeout = 3 Minutes 0.0 Seconds Percent Done = 100.4 %