##############ECHO OF PROBLEM################# ##############temp/h3sinpostode.ode################# diff ( y , x , 3 ) = sin ( x ) ; ! /* BEGIN FIRST INPUT BLOCK */ Digits=32; max_terms=40; ! /* END FIRST INPUT BLOCK */ /* BEGIN SECOND INPUT BLOCK */ x_start=c(0.1); x_end=c(1.5); array_y_init[0 + 1] = exact_soln_y(x_start); array_y_init[1 + 1] = exact_soln_yp(x_start); array_y_init[2 + 1] = exact_soln_ypp(x_start); glob_look_poles=true; glob_type_given_pole=3; /* 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) + cos(c(x))); } double exact_soln_yp (double x) { return(neg(sin(c(x)))); } double exact_soln_ypp (double x) { return(neg(cos(c(x)))); } /* END USER DEF BLOCK */ #######END OF ECHO OF PROBLEM################# START of Soultion TOP MAIN SOLVE Loop x[1] = 0.1 y[1] (closed_form) = 1.995004165278026 y[1] (numeric) = 1.995004165278026 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 14 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.11 y[1] (closed_form) = 1.993956097956697 y[1] (numeric) = 1.993956097956697 absolute error = 2.220446049250313e-16 relative error = 1.113588233725763e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.12 y[1] (closed_form) = 1.992808635853866 y[1] (numeric) = 1.992808635853866 absolute error = 2.220446049250313e-16 relative error = 1.114229439445855e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.13 y[1] (closed_form) = 1.991561893714788 y[1] (numeric) = 1.991561893714788 absolute error = 2.220446049250313e-16 relative error = 1.114926960722569e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.14 y[1] (closed_form) = 1.990215996212637 y[1] (numeric) = 1.990215996212637 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] = 0.15 y[1] (closed_form) = 1.988771077936042 y[1] (numeric) = 1.988771077936042 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] = 0.1600000000000001 y[1] (closed_form) = 1.987227283375627 y[1] (numeric) = 1.987227283375627 absolute error = 2.220446049250313e-16 relative error = 1.117358878788402e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1700000000000001 y[1] (closed_form) = 1.985584766909561 y[1] (numeric) = 1.985584766909561 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] = 0.1800000000000001 y[1] (closed_form) = 1.983843692788121 y[1] (numeric) = 1.983843692788121 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] = 0.1900000000000001 y[1] (closed_form) = 1.98200423511727 y[1] (numeric) = 1.98200423511727 absolute error = 4.440892098500626e-16 relative error = 2.240606765523823e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2000000000000001 y[1] (closed_form) = 1.980066577841242 y[1] (numeric) = 1.980066577841241 absolute error = 6.661338147750939e-16 relative error = 3.364199074060142e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2100000000000001 y[1] (closed_form) = 1.978030914724148 y[1] (numeric) = 1.978030914724147 absolute error = 8.881784197001252e-16 relative error = 4.49021505725045e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2200000000000001 y[1] (closed_form) = 1.975897449330605 y[1] (numeric) = 1.975897449330604 absolute error = 1.110223024625157e-15 relative error = 5.61882918064031e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2300000000000001 y[1] (closed_form) = 1.973666395005375 y[1] (numeric) = 1.973666395005373 absolute error = 1.332267629550188e-15 relative error = 6.750216920760612e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2400000000000001 y[1] (closed_form) = 1.971337974852029 y[1] (numeric) = 1.971337974852028 absolute error = 1.554312234475219e-15 relative error = 7.884554826738359e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2500000000000001 y[1] (closed_form) = 1.968912421710645 y[1] (numeric) = 1.968912421710643 absolute error = 1.332267629550188e-15 relative error = 6.766515436947051e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2600000000000001 y[1] (closed_form) = 1.966389978134513 y[1] (numeric) = 1.966389978134512 absolute error = 1.332267629550188e-15 relative error = 6.775195380186445e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2700000000000001 y[1] (closed_form) = 1.963770896365891 y[1] (numeric) = 1.96377089636589 absolute error = 1.110223024625157e-15 relative error = 5.653526216727775e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2800000000000001 y[1] (closed_form) = 1.961055438310771 y[1] (numeric) = 1.96105543831077 absolute error = 1.110223024625157e-15 relative error = 5.661354610053702e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2900000000000001 y[1] (closed_form) = 1.958243875512697 y[1] (numeric) = 1.958243875512696 absolute error = 1.332267629550188e-15 relative error = 6.80337952902511e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3000000000000002 y[1] (closed_form) = 1.955336489125606 y[1] (numeric) = 1.955336489125604 absolute error = 1.554312234475219e-15 relative error = 7.94907803909639e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3100000000000002 y[1] (closed_form) = 1.952333569885713 y[1] (numeric) = 1.952333569885712 absolute error = 1.332267629550188e-15 relative error = 6.823975421516604e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3200000000000002 y[1] (closed_form) = 1.949235418082441 y[1] (numeric) = 1.94923541808244 absolute error = 1.110223024625157e-15 relative error = 5.695684648072616e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3300000000000002 y[1] (closed_form) = 1.946042343528387 y[1] (numeric) = 1.946042343528386 absolute error = 8.881784197001252e-16 relative error = 4.564024121334179e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3400000000000002 y[1] (closed_form) = 1.942754665528346 y[1] (numeric) = 1.942754665528345 absolute error = 1.110223024625157e-15 relative error = 5.714684639932255e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3500000000000002 y[1] (closed_form) = 1.939372712847379 y[1] (numeric) = 1.939372712847378 absolute error = 1.110223024625157e-15 relative error = 5.724650126664573e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3600000000000002 y[1] (closed_form) = 1.935896823677935 y[1] (numeric) = 1.935896823677934 absolute error = 8.881784197001252e-16 relative error = 4.587942956653598e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3700000000000002 y[1] (closed_form) = 1.932327345606034 y[1] (numeric) = 1.932327345606034 absolute error = 4.440892098500626e-16 relative error = 2.298209000974332e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3800000000000002 y[1] (closed_form) = 1.92866463557651 y[1] (numeric) = 1.92866463557651 absolute error = 4.440892098500626e-16 relative error = 2.302573509454727e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3900000000000002 y[1] (closed_form) = 1.924909059857313 y[1] (numeric) = 1.924909059857312 absolute error = 6.661338147750939e-16 relative error = 3.460598885770075e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4000000000000002 y[1] (closed_form) = 1.921060994002885 y[1] (numeric) = 1.921060994002885 absolute error = 4.440892098500626e-16 relative error = 2.311687193880923e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4100000000000003 y[1] (closed_form) = 1.917120822816605 y[1] (numeric) = 1.917120822816605 absolute error = 2.220446049250313e-16 relative error = 1.158219149687221e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4200000000000003 y[1] (closed_form) = 1.913088940312308 y[1] (numeric) = 1.913088940312308 absolute error = 2.220446049250313e-16 relative error = 1.16066012534045e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4300000000000003 y[1] (closed_form) = 1.908965749674885 y[1] (numeric) = 1.908965749674885 absolute error = 2.220446049250313e-16 relative error = 1.163167044578184e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4400000000000003 y[1] (closed_form) = 1.904751663219963 y[1] (numeric) = 1.904751663219963 absolute error = 2.220446049250313e-16 relative error = 1.16574044382073e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4500000000000003 y[1] (closed_form) = 1.900447102352677 y[1] (numeric) = 1.900447102352677 absolute error = 2.220446049250313e-16 relative error = 1.168380875480033e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4600000000000003 y[1] (closed_form) = 1.896052497525525 y[1] (numeric) = 1.896052497525524 absolute error = 6.661338147750939e-16 relative error = 3.513266724652629e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4700000000000003 y[1] (closed_form) = 1.891568288195329 y[1] (numeric) = 1.891568288195328 absolute error = 6.661338147750939e-16 relative error = 3.521595381632381e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4800000000000003 y[1] (closed_form) = 1.886994922779284 y[1] (numeric) = 1.886994922779283 absolute error = 6.661338147750939e-16 relative error = 3.530130403286779e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4900000000000003 y[1] (closed_form) = 1.882332858610121 y[1] (numeric) = 1.882332858610121 absolute error = 6.661338147750939e-16 relative error = 3.538873646752118e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.5000000000000003 y[1] (closed_form) = 1.877582561890373 y[1] (numeric) = 1.877582561890372 absolute error = 6.661338147750939e-16 relative error = 3.547827021275818e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.5100000000000003 y[1] (closed_form) = 1.872744507645751 y[1] (numeric) = 1.87274450764575 absolute error = 8.881784197001252e-16 relative error = 4.74265665217016e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.5200000000000004 y[1] (closed_form) = 1.86781917967765 y[1] (numeric) = 1.867819179677649 absolute error = 6.661338147750939e-16 relative error = 3.566372066540488e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.5300000000000004 y[1] (closed_form) = 1.862807070514761 y[1] (numeric) = 1.86280707051476 absolute error = 1.110223024625157e-15 relative error = 5.959946374470019e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.5400000000000004 y[1] (closed_form) = 1.857708681363824 y[1] (numeric) = 1.857708681363823 absolute error = 8.881784197001252e-16 relative error = 4.781042520876175e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.5500000000000004 y[1] (closed_form) = 1.852524522059506 y[1] (numeric) = 1.852524522059505 absolute error = 8.881784197001252e-16 relative error = 4.794421931390745e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.5600000000000004 y[1] (closed_form) = 1.847255111013416 y[1] (numeric) = 1.847255111013415 absolute error = 4.440892098500626e-16 relative error = 2.40404916030484e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.5700000000000004 y[1] (closed_form) = 1.841900975162269 y[1] (numeric) = 1.841900975162268 absolute error = 6.661338147750939e-16 relative error = 3.616556067659438e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.5800000000000004 y[1] (closed_form) = 1.836462649915187 y[1] (numeric) = 1.836462649915186 absolute error = 4.440892098500626e-16 relative error = 2.418177194458989e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.5900000000000004 y[1] (closed_form) = 1.830940679100163 y[1] (numeric) = 1.830940679100163 absolute error = 6.661338147750939e-16 relative error = 3.638205335535355e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.6000000000000004 y[1] (closed_form) = 1.825335614909678 y[1] (numeric) = 1.825335614909677 absolute error = 8.881784197001252e-16 relative error = 4.865836246470622e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.6100000000000004 y[1] (closed_form) = 1.819648017845479 y[1] (numeric) = 1.819648017845479 absolute error = 6.661338147750939e-16 relative error = 3.660783889204118e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.6200000000000004 y[1] (closed_form) = 1.813878456662534 y[1] (numeric) = 1.813878456662533 absolute error = 6.661338147750939e-16 relative error = 3.672428063348602e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.6300000000000004 y[1] (closed_form) = 1.808027508312152 y[1] (numeric) = 1.808027508312151 absolute error = 4.440892098500626e-16 relative error = 2.456208259047084e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.6400000000000005 y[1] (closed_form) = 1.802095757884292 y[1] (numeric) = 1.802095757884292 absolute error = 2.220446049250313e-16 relative error = 1.232146538015924e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.6500000000000005 y[1] (closed_form) = 1.796083798549056 y[1] (numeric) = 1.796083798549055 absolute error = 2.220446049250313e-16 relative error = 1.236270852754239e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.6600000000000005 y[1] (closed_form) = 1.789992231497365 y[1] (numeric) = 1.789992231497365 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.6700000000000005 y[1] (closed_form) = 1.783821665880849 y[1] (numeric) = 1.783821665880849 absolute error = 2.220446049250313e-16 relative error = 1.244769077380759e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.6800000000000005 y[1] (closed_form) = 1.777572718750928 y[1] (numeric) = 1.777572718750928 absolute error = 2.220446049250313e-16 relative error = 1.249144986209389e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.6900000000000005 y[1] (closed_form) = 1.771246014997106 y[1] (numeric) = 1.771246014997107 absolute error = 4.440892098500626e-16 relative error = 2.507213600425733e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.7000000000000005 y[1] (closed_form) = 1.764842187284488 y[1] (numeric) = 1.764842187284489 absolute error = 4.440892098500626e-16 relative error = 2.516311163965147e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.7100000000000005 y[1] (closed_form) = 1.758361875990508 y[1] (numeric) = 1.758361875990508 absolute error = 4.440892098500626e-16 relative error = 2.525584840719442e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.7200000000000005 y[1] (closed_form) = 1.751805729140895 y[1] (numeric) = 1.751805729140895 absolute error = 6.661338147750939e-16 relative error = 3.802555293056231e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.7300000000000005 y[1] (closed_form) = 1.74517440234487 y[1] (numeric) = 1.745174402344871 absolute error = 6.661338147750939e-16 relative error = 3.817004271206683e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.7400000000000005 y[1] (closed_form) = 1.738468558729588 y[1] (numeric) = 1.738468558729588 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.7500000000000006 y[1] (closed_form) = 1.731688868873821 y[1] (numeric) = 1.731688868873821 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.7600000000000006 y[1] (closed_form) = 1.724836010740905 y[1] (numeric) = 1.724836010740905 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.7700000000000006 y[1] (closed_form) = 1.717910669610943 y[1] (numeric) = 1.717910669610943 absolute error = 2.220446049250313e-16 relative error = 1.292527072873457e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.7800000000000006 y[1] (closed_form) = 1.710913538012277 y[1] (numeric) = 1.710913538012277 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.7900000000000006 y[1] (closed_form) = 1.703845315652236 y[1] (numeric) = 1.703845315652236 absolute error = 2.220446049250313e-16 relative error = 1.303196967971427e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.8000000000000006 y[1] (closed_form) = 1.696706709347165 y[1] (numeric) = 1.696706709347165 absolute error = 2.220446049250313e-16 relative error = 1.308679948642783e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.8100000000000006 y[1] (closed_form) = 1.689498432951747 y[1] (numeric) = 1.689498432951747 absolute error = 2.220446049250313e-16 relative error = 1.314263455912735e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.8200000000000006 y[1] (closed_form) = 1.682221207287613 y[1] (numeric) = 1.682221207287614 absolute error = 6.661338147750939e-16 relative error = 3.959846730556664e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.8300000000000006 y[1] (closed_form) = 1.674875760071266 y[1] (numeric) = 1.674875760071268 absolute error = 1.110223024625157e-15 relative error = 6.62868883228638e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.8400000000000006 y[1] (closed_form) = 1.667462825841308 y[1] (numeric) = 1.667462825841309 absolute error = 8.881784197001252e-16 relative error = 5.326526060645463e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.8500000000000006 y[1] (closed_form) = 1.659983145884982 y[1] (numeric) = 1.659983145884983 absolute error = 1.554312234475219e-15 relative error = 9.363421781289071e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.8600000000000007 y[1] (closed_form) = 1.652437468164051 y[1] (numeric) = 1.652437468164053 absolute error = 1.554312234475219e-15 relative error = 9.406178838356559e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.8700000000000007 y[1] (closed_form) = 1.644826547240001 y[1] (numeric) = 1.644826547240002 absolute error = 1.77635683940025e-15 relative error = 1.079966056226996e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.8800000000000007 y[1] (closed_form) = 1.63715114419858 y[1] (numeric) = 1.637151144198582 absolute error = 2.220446049250313e-15 relative error = 1.356286532931734e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.8900000000000007 y[1] (closed_form) = 1.629412026573696 y[1] (numeric) = 1.629412026573699 absolute error = 2.220446049250313e-15 relative error = 1.362728403275281e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.9000000000000007 y[1] (closed_form) = 1.621609968270664 y[1] (numeric) = 1.621609968270666 absolute error = 2.220446049250313e-15 relative error = 1.369284903704845e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.9100000000000007 y[1] (closed_form) = 1.613745749488811 y[1] (numeric) = 1.613745749488813 absolute error = 2.220446049250313e-15 relative error = 1.375957798775729e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.9200000000000007 y[1] (closed_form) = 1.605820156643462 y[1] (numeric) = 1.605820156643465 absolute error = 2.664535259100376e-15 relative error = 1.659298675556467e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.9300000000000007 y[1] (closed_form) = 1.597833982287298 y[1] (numeric) = 1.5978339822873 absolute error = 2.220446049250313e-15 relative error = 1.38966004845619e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.9400000000000007 y[1] (closed_form) = 1.589788025031098 y[1] (numeric) = 1.5897880250311 absolute error = 2.664535259100376e-15 relative error = 1.676031783575836e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.9500000000000007 y[1] (closed_form) = 1.581683089463883 y[1] (numeric) = 1.581683089463885 absolute error = 2.442490654175344e-15 relative error = 1.544235169766679e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.9600000000000007 y[1] (closed_form) = 1.573519986072456 y[1] (numeric) = 1.573519986072458 absolute error = 2.442490654175344e-15 relative error = 1.552246349454931e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.9700000000000008 y[1] (closed_form) = 1.565299531160354 y[1] (numeric) = 1.565299531160357 absolute error = 2.886579864025407e-15 relative error = 1.844107026522643e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.9800000000000008 y[1] (closed_form) = 1.557022546766217 y[1] (numeric) = 1.55702254676622 absolute error = 2.886579864025407e-15 relative error = 1.853910124821603e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.9900000000000008 y[1] (closed_form) = 1.548689860581587 y[1] (numeric) = 1.54868986058159 absolute error = 3.108624468950438e-15 relative error = 2.00726081320313e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.000000000000001 y[1] (closed_form) = 1.540302305868139 y[1] (numeric) = 1.540302305868142 absolute error = 2.664535259100376e-15 relative error = 1.729878121294249e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.01 y[1] (closed_form) = 1.531860721374356 y[1] (numeric) = 1.531860721374358 absolute error = 1.77635683940025e-15 relative error = 1.159607276702374e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.019999999999998 y[1] (closed_form) = 1.523365951251651 y[1] (numeric) = 1.523365951251652 absolute error = 1.110223024625157e-15 relative error = 7.287960084134468e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.029999999999997 y[1] (closed_form) = 1.514818844969958 y[1] (numeric) = 1.514818844969958 absolute error = 2.220446049250313e-16 relative error = 1.46581623051722e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.039999999999996 y[1] (closed_form) = 1.506220257232782 y[1] (numeric) = 1.506220257232781 absolute error = 6.661338147750939e-16 relative error = 4.422552489095524e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.049999999999995 y[1] (closed_form) = 1.497571047891731 y[1] (numeric) = 1.49757104789173 absolute error = 1.554312234475219e-15 relative error = 1.037888811127437e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.059999999999994 y[1] (closed_form) = 1.488872081860533 y[1] (numeric) = 1.48887208186053 absolute error = 2.442490654175344e-15 relative error = 1.640497316010617e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.069999999999993 y[1] (closed_form) = 1.48012422902854 y[1] (numeric) = 1.480124229028537 absolute error = 3.33066907387547e-15 relative error = 2.250263193152043e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.079999999999992 y[1] (closed_form) = 1.471328364173747 y[1] (numeric) = 1.471328364173743 absolute error = 4.218847493575595e-15 relative error = 2.867373182154868e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.089999999999991 y[1] (closed_form) = 1.462485366875309 y[1] (numeric) = 1.462485366875304 absolute error = 5.329070518200751e-15 relative error = 3.643845360030263e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.09999999999999 y[1] (closed_form) = 1.453596121425587 y[1] (numeric) = 1.45359612142558 absolute error = 6.439293542825908e-15 relative error = 4.429905561739319e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.109999999999989 y[1] (closed_form) = 1.444661516741717 y[1] (numeric) = 1.44466151674171 absolute error = 7.327471962526033e-15 relative error = 5.072102965026977e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.119999999999987 y[1] (closed_form) = 1.435682446276723 y[1] (numeric) = 1.435682446276715 absolute error = 8.215650382226158e-15 relative error = 5.7224704554496e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.129999999999986 y[1] (closed_form) = 1.42665980793017 y[1] (numeric) = 1.426659807930161 absolute error = 9.103828801926284e-15 relative error = 6.381219090439173e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.139999999999985 y[1] (closed_form) = 1.417594503958371 y[1] (numeric) = 1.417594503958362 absolute error = 9.992007221626409e-15 relative error = 7.048565152958458e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.149999999999984 y[1] (closed_form) = 1.408487440884172 y[1] (numeric) = 1.408487440884161 absolute error = 1.110223024625157e-14 relative error = 7.882377878557575e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.159999999999983 y[1] (closed_form) = 1.399339529406289 y[1] (numeric) = 1.399339529406277 absolute error = 1.199040866595169e-14 relative error = 8.568619991060337e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.169999999999982 y[1] (closed_form) = 1.390151684308247 y[1] (numeric) = 1.390151684308234 absolute error = 1.287858708565182e-14 relative error = 9.264159610079044e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.179999999999981 y[1] (closed_form) = 1.380924824366899 y[1] (numeric) = 1.380924824366886 absolute error = 1.376676550535194e-14 relative error = 9.969236023882379e-13 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.18999999999998 y[1] (closed_form) = 1.371659872260552 y[1] (numeric) = 1.371659872260537 absolute error = 1.509903313490213e-14 relative error = 1.100785511062469e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.199999999999979 y[1] (closed_form) = 1.362357754476693 y[1] (numeric) = 1.362357754476678 absolute error = 1.576516694967722e-14 relative error = 1.157197285211836e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.209999999999978 y[1] (closed_form) = 1.353019401219351 y[1] (numeric) = 1.353019401219334 absolute error = 1.687538997430238e-14 relative error = 1.247239319635339e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.219999999999976 y[1] (closed_form) = 1.343645746316069 y[1] (numeric) = 1.343645746316051 absolute error = 1.798561299892754e-14 relative error = 1.338568074824741e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.229999999999975 y[1] (closed_form) = 1.334237727124526 y[1] (numeric) = 1.334237727124507 absolute error = 1.931788062847772e-14 relative error = 1.447858971137815e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.239999999999974 y[1] (closed_form) = 1.324796284438801 y[1] (numeric) = 1.32479628443878 absolute error = 2.020605904817785e-14 relative error = 1.525220087459513e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.249999999999973 y[1] (closed_form) = 1.315322362395294 y[1] (numeric) = 1.315322362395273 absolute error = 2.131628207280301e-14 relative error = 1.620612762485431e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.259999999999972 y[1] (closed_form) = 1.305816908378316 y[1] (numeric) = 1.305816908378294 absolute error = 2.242650509742816e-14 relative error = 1.717431054348918e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.269999999999971 y[1] (closed_form) = 1.296280872925347 y[1] (numeric) = 1.296280872925323 absolute error = 2.353672812205332e-14 relative error = 1.815712058524589e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.27999999999997 y[1] (closed_form) = 1.286715209631984 y[1] (numeric) = 1.28671520963196 absolute error = 2.442490654175344e-14 relative error = 1.898237182471734e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.289999999999969 y[1] (closed_form) = 1.277120875056588 y[1] (numeric) = 1.277120875056562 absolute error = 2.531308496145357e-14 relative error = 1.982043004373566e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.299999999999968 y[1] (closed_form) = 1.267498828624618 y[1] (numeric) = 1.267498828624592 absolute error = 2.620126338115369e-14 relative error = 2.067162729419251e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.309999999999967 y[1] (closed_form) = 1.257850032532702 y[1] (numeric) = 1.257850032532675 absolute error = 2.731148640577885e-14 relative error = 2.171283197472016e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.319999999999965 y[1] (closed_form) = 1.248175451652406 y[1] (numeric) = 1.248175451652378 absolute error = 2.864375403532904e-14 relative error = 2.294849974609643e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.329999999999964 y[1] (closed_form) = 1.238476053433758 y[1] (numeric) = 1.238476053433728 absolute error = 2.997602166487923e-14 relative error = 2.420395742151711e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.339999999999963 y[1] (closed_form) = 1.228752807808495 y[1] (numeric) = 1.228752807808464 absolute error = 3.130828929442941e-14 relative error = 2.547972960506871e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.349999999999962 y[1] (closed_form) = 1.219006687093078 y[1] (numeric) = 1.219006687093046 absolute error = 3.241851231905457e-14 relative error = 2.659420384014614e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.359999999999961 y[1] (closed_form) = 1.209238665891458 y[1] (numeric) = 1.209238665891424 absolute error = 3.352873534367973e-14 relative error = 2.77271445988391e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.36999999999996 y[1] (closed_form) = 1.199449720997612 y[1] (numeric) = 1.199449720997578 absolute error = 3.441691376337985e-14 relative error = 2.869391952065689e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.379999999999959 y[1] (closed_form) = 1.189640831297875 y[1] (numeric) = 1.189640831297839 absolute error = 3.530509218307998e-14 relative error = 2.967710190693675e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.389999999999958 y[1] (closed_form) = 1.179812977673041 y[1] (numeric) = 1.179812977673005 absolute error = 3.61932706027801e-14 relative error = 3.067712534758222e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.399999999999957 y[1] (closed_form) = 1.169967142900284 y[1] (numeric) = 1.169967142900246 absolute error = 3.752553823233029e-14 relative error = 3.20740103344326e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.409999999999956 y[1] (closed_form) = 1.160104311554875 y[1] (numeric) = 1.160104311554837 absolute error = 3.841371665203042e-14 relative error = 3.31122953939762e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.419999999999954 y[1] (closed_form) = 1.150225469911731 y[1] (numeric) = 1.150225469911691 absolute error = 3.97459842815806e-14 relative error = 3.455495059123559e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.429999999999953 y[1] (closed_form) = 1.140331605846783 y[1] (numeric) = 1.140331605846742 absolute error = 4.063416270128073e-14 relative error = 3.563363717443118e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.439999999999952 y[1] (closed_form) = 1.130423708738193 y[1] (numeric) = 1.130423708738151 absolute error = 4.196643033083092e-14 relative error = 3.712451358409219e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.449999999999951 y[1] (closed_form) = 1.120502769367415 y[1] (numeric) = 1.120502769367372 absolute error = 4.307665335545607e-14 relative error = 3.844404006227953e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.45999999999995 y[1] (closed_form) = 1.110569779820119 y[1] (numeric) = 1.110569779820075 absolute error = 4.39648317751562e-14 relative error = 3.958763562094878e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.469999999999949 y[1] (closed_form) = 1.100625733386982 y[1] (numeric) = 1.100625733386938 absolute error = 4.485301019485632e-14 relative error = 4.075228193768381e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.479999999999948 y[1] (closed_form) = 1.090671624464362 y[1] (numeric) = 1.090671624464315 absolute error = 4.640732242933154e-14 relative error = 4.254930759028647e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.489999999999947 y[1] (closed_form) = 1.080708448454854 y[1] (numeric) = 1.080708448454806 absolute error = 4.75175454539567e-14 relative error = 4.396888496790699e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = 1.499999999999946 y[1] (closed_form) = 1.070737201667757 y[1] (numeric) = 1.070737201667709 absolute error = 4.840572387365683e-14 relative error = 4.520784726472669e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 Finished! diff ( y , x , 3 ) = sin ( x ) ; Iterations = 1401 Total Elapsed Time = 0.0 Seconds Elapsed Time(since restart) = 0.0 Seconds Time to Timeout = 3 Minutes 0.0 Seconds Percent Done = 100.1 %