##############ECHO OF PROBLEM################# ##############temp/sing2postode.ode################# diff ( y , x , 1 ) = 1.0 / ( x * x + 1.0 ) ; ! // BEGIN FIRST INPUT BLOCK Digits=16; max_terms=20; ! // 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(1.0); 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(arctan(c(x))); } // END USER DEF BLOCK #######END OF ECHO OF PROBLEM################# START of Soultion TOP MAIN SOLVE Loop x[1] = -2 y[1] (closed_form) = -1.10714871779409 y[1] (numeric) = -1.10714871779409 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] = -1.990000000000001 y[1] (closed_form) = -1.105140688364495 y[1] (numeric) = -1.105140688364494 absolute error = 2.220446049250313e-16 relative error = 2.009197627621843e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.980000000000002 y[1] (closed_form) = -1.103116481582997 y[1] (numeric) = -1.103116481582997 absolute error = 4.440892098500626e-16 relative error = 4.025768966961534e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.970000000000003 y[1] (closed_form) = -1.101075917953987 y[1] (numeric) = -1.101075917953986 absolute error = 6.661338147750939e-16 relative error = 6.049844555795026e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.960000000000004 y[1] (closed_form) = -1.099018815614148 y[1] (numeric) = -1.099018815614146 absolute error = 1.110223024625157e-15 relative error = 1.010194738117152e-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) = -1.096944990300137 y[1] (numeric) = -1.096944990300136 absolute error = 1.332267629550188e-15 relative error = 1.214525469673428e-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) = -1.094854255315939 y[1] (numeric) = -1.094854255315938 absolute error = 1.77635683940025e-15 relative error = 1.62245963860062e-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) = -1.092746421499887 y[1] (numeric) = -1.092746421499885 absolute error = 1.998401444325282e-15 relative error = 1.828787909991329e-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) = -1.090621297191372 y[1] (numeric) = -1.090621297191369 absolute error = 2.220446049250313e-15 relative error = 2.035945983237746e-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) = -1.088478688197239 y[1] (numeric) = -1.088478688197237 absolute error = 2.220446049250313e-15 relative error = 2.039953628240404e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.900000000000011 y[1] (closed_form) = -1.086318397757876 y[1] (numeric) = -1.086318397757874 absolute error = 2.220446049250313e-15 relative error = 2.044010350771227e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.890000000000012 y[1] (closed_form) = -1.084140226513 y[1] (numeric) = -1.084140226512997 absolute error = 2.442490654175344e-15 relative error = 2.252928721251593e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.880000000000013 y[1] (closed_form) = -1.081943972467152 y[1] (numeric) = -1.081943972467149 absolute error = 2.886579864025407e-15 relative error = 2.66795687898991e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.870000000000014 y[1] (closed_form) = -1.07972943095491 y[1] (numeric) = -1.079729430954907 absolute error = 3.108624468950438e-15 relative error = 2.879077276055335e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.860000000000015 y[1] (closed_form) = -1.077496394605821 y[1] (numeric) = -1.077496394605817 absolute error = 3.774758283725532e-15 relative error = 3.503267669964173e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.850000000000017 y[1] (closed_form) = -1.075244653309072 y[1] (numeric) = -1.075244653309068 absolute error = 4.218847493575595e-15 relative error = 3.923616342189721e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.840000000000018 y[1] (closed_form) = -1.0729739941779 y[1] (numeric) = -1.072973994177895 absolute error = 4.662936703425657e-15 relative error = 4.345805889730204e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.830000000000019 y[1] (closed_form) = -1.070684201513761 y[1] (numeric) = -1.070684201513756 absolute error = 4.884981308350689e-15 relative error = 4.562485653047066e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.82000000000002 y[1] (closed_form) = -1.068375056770271 y[1] (numeric) = -1.068375056770265 absolute error = 5.773159728050814e-15 relative error = 5.403682622002845e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.810000000000021 y[1] (closed_form) = -1.066046338516914 y[1] (numeric) = -1.066046338516908 absolute error = 5.773159728050814e-15 relative error = 5.415486662692774e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.800000000000022 y[1] (closed_form) = -1.063697822402565 y[1] (numeric) = -1.063697822402559 absolute error = 5.773159728050814e-15 relative error = 5.427443402122446e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.790000000000023 y[1] (closed_form) = -1.061329281118805 y[1] (numeric) = -1.0613292811188 absolute error = 5.551115123125783e-15 relative error = 5.230342007782964e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.780000000000024 y[1] (closed_form) = -1.058940484363078 y[1] (numeric) = -1.058940484363072 absolute error = 5.995204332975845e-15 relative error = 5.66151206937922e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.770000000000025 y[1] (closed_form) = -1.056531198801681 y[1] (numeric) = -1.056531198801675 absolute error = 6.439293542825908e-15 relative error = 6.094750017916519e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.760000000000026 y[1] (closed_form) = -1.054101188032626 y[1] (numeric) = -1.054101188032619 absolute error = 6.661338147750939e-15 relative error = 6.319448477411983e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.750000000000028 y[1] (closed_form) = -1.05165021254838 y[1] (numeric) = -1.051650212548374 absolute error = 6.661338147750939e-15 relative error = 6.33417658102217e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.740000000000029 y[1] (closed_form) = -1.049178029698517 y[1] (numeric) = -1.049178029698511 absolute error = 6.439293542825908e-15 relative error = 6.137465101777101e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.73000000000003 y[1] (closed_form) = -1.046684393652288 y[1] (numeric) = -1.046684393652282 absolute error = 6.661338147750939e-15 relative error = 6.364228021502207e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.720000000000031 y[1] (closed_form) = -1.044169055361154 y[1] (numeric) = -1.044169055361147 absolute error = 7.105427357601002e-15 relative error = 6.80486298757762e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.710000000000032 y[1] (closed_form) = -1.041631762521289 y[1] (numeric) = -1.041631762521282 absolute error = 7.327471962526033e-15 relative error = 7.034608799552875e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.700000000000033 y[1] (closed_form) = -1.039072259536099 y[1] (numeric) = -1.039072259536092 absolute error = 7.549516567451064e-15 relative error = 7.265631911703229e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.690000000000034 y[1] (closed_form) = -1.036490287478765 y[1] (numeric) = -1.036490287478757 absolute error = 7.993605777301127e-15 relative error = 7.712185896835906e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.680000000000035 y[1] (closed_form) = -1.03388558405486 y[1] (numeric) = -1.033885584054852 absolute error = 7.993605777301127e-15 relative error = 7.731615471366287e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.670000000000036 y[1] (closed_form) = -1.031257883565068 y[1] (numeric) = -1.031257883565059 absolute error = 8.659739592076221e-15 relative error = 8.397259046533952e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.660000000000037 y[1] (closed_form) = -1.028606916868028 y[1] (numeric) = -1.028606916868019 absolute error = 8.659739592076221e-15 relative error = 8.418900796860266e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.650000000000039 y[1] (closed_form) = -1.025932411343363 y[1] (numeric) = -1.025932411343354 absolute error = 9.103828801926284e-15 relative error = 8.873712050880296e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.64000000000004 y[1] (closed_form) = -1.02323409085491 y[1] (numeric) = -1.0232340908549 absolute error = 9.325873406851315e-15 relative error = 9.114115225636753e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.630000000000041 y[1] (closed_form) = -1.020511675714205 y[1] (numeric) = -1.020511675714195 absolute error = 9.769962616701378e-15 relative error = 9.573592198113632e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.620000000000042 y[1] (closed_form) = -1.017764882644268 y[1] (numeric) = -1.017764882644258 absolute error = 9.992007221626409e-15 relative error = 9.817598732298613e-13 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 15 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.610000000000043 y[1] (closed_form) = -1.014993424743729 y[1] (numeric) = -1.014993424743719 absolute error = 1.043609643147647e-14 relative error = 1.028193501264447e-12 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.600000000000044 y[1] (closed_form) = -1.012197011451347 y[1] (numeric) = -1.012197011451336 absolute error = 1.043609643147647e-14 relative error = 1.031034108321718e-12 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.590000000000045 y[1] (closed_form) = -1.009375348510965 y[1] (numeric) = -1.009375348510954 absolute error = 1.06581410364015e-14 relative error = 1.0559145368593e-12 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.580000000000046 y[1] (closed_form) = -1.006528137936978 y[1] (numeric) = -1.006528137936967 absolute error = 1.110223024625157e-14 relative error = 1.103022342624932e-12 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 14 h = 0.001 TOP MAIN SOLVE Loop x[1] = -1.570000000000047 y[1] (closed_form) = -1.003655077980341 y[1] (numeric) = -1.003655077980329 absolute error = 1.13242748511766e-14 relative error = 1.128303448029624e-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) = -1.0007558630952 y[1] (numeric) = -1.000755863095188 absolute error = 1.221245327087672e-14 relative error = 1.220322930020643e-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) = -0.997830183906205 y[1] (numeric) = -0.9978301839061924 absolute error = 1.265654248072678e-14 relative error = 1.268406456816152e-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) = -0.9948777271765586 y[1] (numeric) = -0.9948777271765455 absolute error = 1.310063169057685e-14 relative error = 1.316808220016761e-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) = -0.9918981757768925 y[1] (numeric) = -0.9918981757768793 absolute error = 1.321165399303936e-14 relative error = 1.331956678183372e-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) = -0.988891208655027 y[1] (numeric) = -0.9888912086550136 absolute error = 1.343369859796439e-14 relative error = 1.358460716445778e-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) = -0.9858565008067001 y[1] (numeric) = -0.9858565008066863 absolute error = 1.376676550535194e-14 relative error = 1.396426913459207e-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) = -0.982793723247346 y[1] (numeric) = -0.9827937232473321 absolute error = 1.398881011027697e-14 relative error = 1.423371942593931e-12 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 14 h = 0.001 Finished! diff ( y , x , 1 ) = 1.0 / ( x * x + 1.0 ) ; 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 %