##############ECHO OF PROBLEM################# ##############temp/nonlinear2postode.ode################# diff ( y , x , 1 ) = y * y; ! /* BEGIN FIRST INPUT BLOCK */ Digits=32; max_terms=30; ! /* END FIRST INPUT BLOCK */ /* BEGIN SECOND INPUT BLOCK */ /* # problem from Boyce DePrima - */ /* # _Elementary Differential Equations and Boundary Value Problems_ */ /* # page 23 */ /* # Singularity at x = 0.5 (which depends on init condition) */ /* # */ x_start=0.0; x_end=0.2; array_y_init[0 + 1] = exact_soln_y(x_start); glob_look_poles=true; glob_max_iter=1000000; glob_max_h=0.001; /* # Not Given = 0 */ /* # No Pole = 3 */ /* # Pole = 4 */ glob_type_given_pole=4; /* # Real Part */ array_given_rad_poles[1][1]=0.5; /* # Imag Part */ array_given_rad_poles[1][2]=0.0; /* # Order */ array_given_ord_poles[1][1]=0.0; /* # Not Used */ array_given_ord_poles[1][2]=0.0; /* END SECOND INPUT BLOCK */ /* BEGIN OVERRIDE BLOCK */ glob_desired_digits_correct=10; glob_display_interval=0.01; glob_look_poles=true; glob_max_iter=10000000; glob_max_minutes=3; glob_subiter_method=3; /* END OVERRIDE BLOCK */ ! /* BEGIN USER DEF BLOCK */ double exact_soln_y (double x) { return(2.0/(1.0 - 2.0*x)); } /* END USER DEF BLOCK */ #######END OF ECHO OF PROBLEM################# START of Optimize min_size = 0 min_size = 1 glob_desired_digits_correct = 10 desired_abs_gbl_error = 1e-10 range = 0.2 estimated_steps = 200000 step_error = 4.999999999999999e-16 est_needed_step_err = 4.999999999999999e-16 opt_iter = 1 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 1.342178622178622e-148 estimated_step_error = 1.342178622178622e-148 best_h = 2e-06 opt_iter = 2 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 9.007217269175527e-141 estimated_step_error = 9.007217269175527e-141 best_h = 4e-06 opt_iter = 3 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 6.04465327668625e-133 estimated_step_error = 6.04465327668625e-133 best_h = 8e-06 opt_iter = 4 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 4.056514372845313e-125 estimated_step_error = 4.056514372845313e-125 best_h = 1.6e-05 opt_iter = 5 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 2.722302492207371e-117 estimated_step_error = 2.722302492207371e-117 best_h = 3.2e-05 opt_iter = 6 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 1.826935508599843e-109 estimated_step_error = 1.826935508599843e-109 best_h = 6.4e-05 opt_iter = 7 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 1.226074901486084e-101 estimated_step_error = 1.226074901486084e-101 best_h = 0.000128 opt_iter = 8 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 8.228576044322446e-94 estimated_step_error = 8.228576044322446e-94 best_h = 0.000256 opt_iter = 9 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 5.522810916934551e-86 estimated_step_error = 5.522810916934551e-86 best_h = 0.000512 opt_iter = 10 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 3.707244964518627e-78 estimated_step_error = 3.707244964518627e-78 best_h = 0.001024 opt_iter = 11 hn_div_ho = 0.5 hn_div_ho_2 = 0.25 hn_div_ho_3 = 0.125 max_estimated_step_error = 2.4891650844254e-70 estimated_step_error = 2.4891650844254e-70 best_h = 0.001 START of Soultion TOP MAIN SOLVE Loop x[1] = 0 y[1] (analytic) = 2 y[1] (numeric) = 2 absolute error = 0 relative error = 0 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.5 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.5 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.5 Order of pole (three term test) = 651 Radius of convergence (six term test) for eq 1 = 0.2073 Order of pole (six term test) = -14.62 TOP MAIN SOLVE Loop x[1] = 0.01 y[1] (analytic) = 2.040816326530612 y[1] (numeric) = 2.040816326530613 absolute error = 4.440892098500626e-16 relative error = 2.176037128265307e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.49 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.49 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.49 Order of pole (three term test) = 651 Radius of convergence (six term test) for eq 1 = 0.6194 Order of pole (six term test) = -27.38 TOP MAIN SOLVE Loop x[1] = 0.02000000000000001 y[1] (analytic) = 2.083333333333333 y[1] (numeric) = 2.083333333333333 absolute error = 0 relative error = 0 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.48 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.48 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.48 Order of pole (three term test) = 651 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] = 0.03000000000000002 y[1] (analytic) = 2.127659574468085 y[1] (numeric) = 2.127659574468086 absolute error = 4.440892098500626e-16 relative error = 2.087219286295294e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.47 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.47 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.47 Order of pole (three term test) = 651 Radius of convergence (six term test) for eq 1 = 0.04124 Order of pole (six term test) = -11.76 TOP MAIN SOLVE Loop x[1] = 0.04000000000000003 y[1] (analytic) = 2.173913043478261 y[1] (numeric) = 2.173913043478262 absolute error = 8.881784197001252e-16 relative error = 4.085620730620575e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.46 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.46 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.46 Order of pole (three term test) = 651 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] = 0.05000000000000004 y[1] (analytic) = 2.222222222222222 y[1] (numeric) = 2.222222222222224 absolute error = 1.332267629550188e-15 relative error = 5.995204332975845e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.45 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.45 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.45 Order of pole (three term test) = 651 Radius of convergence (six term test) for eq 1 = 0.244 Order of pole (six term test) = -16.22 TOP MAIN SOLVE Loop x[1] = 0.06000000000000005 y[1] (analytic) = 2.272727272727273 y[1] (numeric) = 2.272727272727274 absolute error = 1.332267629550188e-15 relative error = 5.861977570020827e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.44 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.44 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.44 Order of pole (three term test) = 651 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] = 0.07000000000000005 y[1] (analytic) = 2.325581395348837 y[1] (numeric) = 2.325581395348839 absolute error = 1.77635683940025e-15 relative error = 7.638334409421077e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.43 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.43 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.43 Order of pole (three term test) = 651 Radius of convergence (six term test) for eq 1 = 0.1639 Order of pole (six term test) = -11.93 TOP MAIN SOLVE Loop x[1] = 0.08000000000000006 y[1] (analytic) = 2.380952380952381 y[1] (numeric) = 2.380952380952383 absolute error = 1.77635683940025e-15 relative error = 7.460698725481051e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.42 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.42 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.42 Order of pole (three term test) = 651 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] = 0.09000000000000007 y[1] (analytic) = 2.439024390243903 y[1] (numeric) = 2.439024390243904 absolute error = 8.881784197001252e-16 relative error = 3.641531520770513e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.41 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.41 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.41 Order of pole (three term test) = 651 Radius of convergence (six term test) for eq 1 = 0.1501 Order of pole (six term test) = -12.39 TOP MAIN SOLVE Loop x[1] = 0.1000000000000001 y[1] (analytic) = 2.5 y[1] (numeric) = 2.500000000000002 absolute error = 1.332267629550188e-15 relative error = 5.32907051820075e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.4 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.4 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.4 Order of pole (three term test) = 651 Radius of convergence (six term test) for eq 1 = 0.09295 Order of pole (six term test) = -12.98 TOP MAIN SOLVE Loop x[1] = 0.1100000000000001 y[1] (analytic) = 2.564102564102565 y[1] (numeric) = 2.564102564102566 absolute error = 8.881784197001252e-16 relative error = 3.463895836830488e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.39 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.39 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.39 Order of pole (three term test) = 651 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] = 0.1200000000000001 y[1] (analytic) = 2.631578947368422 y[1] (numeric) = 2.631578947368423 absolute error = 8.881784197001252e-16 relative error = 3.375077994860475e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.38 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.38 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.38 Order of pole (three term test) = 651 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] = 0.1300000000000001 y[1] (analytic) = 2.702702702702704 y[1] (numeric) = 2.702702702702704 absolute error = 8.881784197001252e-16 relative error = 3.286260152890462e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.37 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.37 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.37 Order of pole (three term test) = 651 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] = 0.1400000000000001 y[1] (analytic) = 2.777777777777779 y[1] (numeric) = 2.77777777777778 absolute error = 1.77635683940025e-15 relative error = 6.3948846218409e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.36 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.36 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.36 Order of pole (three term test) = 651 Radius of convergence (six term test) for eq 1 = 0.05972 Order of pole (six term test) = -11.72 TOP MAIN SOLVE Loop x[1] = 0.1500000000000001 y[1] (analytic) = 2.857142857142858 y[1] (numeric) = 2.85714285714286 absolute error = 1.77635683940025e-15 relative error = 6.217248937900874e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.35 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.35 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.35 Order of pole (three term test) = 651 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] = 0.1600000000000001 y[1] (analytic) = 2.941176470588236 y[1] (numeric) = 2.941176470588238 absolute error = 1.77635683940025e-15 relative error = 6.039613253960849e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.34 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.34 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.34 Order of pole (three term test) = 651 Radius of convergence (six term test) for eq 1 = 0.06667 Order of pole (six term test) = -11.98 TOP MAIN SOLVE Loop x[1] = 0.1700000000000001 y[1] (analytic) = 3.030303030303032 y[1] (numeric) = 3.030303030303033 absolute error = 1.77635683940025e-15 relative error = 5.861977570020824e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.33 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.33 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.33 Order of pole (three term test) = 651 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] = 0.1800000000000001 y[1] (analytic) = 3.125000000000001 y[1] (numeric) = 3.125000000000003 absolute error = 1.77635683940025e-15 relative error = 5.684341886080799e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.32 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.32 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.32 Order of pole (three term test) = 651 Radius of convergence (six term test) for eq 1 = 0.127 Order of pole (six term test) = -13.03 TOP MAIN SOLVE Loop x[1] = 0.1900000000000001 y[1] (analytic) = 3.225806451612905 y[1] (numeric) = 3.225806451612906 absolute error = 1.332267629550188e-15 relative error = 4.13002965160558e-14 % Correct digits = 16 h = 0.001 Radius of convergence (given) for eq 1 = 0.31 Order of pole (given) = 0 Radius of convergence (ratio test) for eq 1 = 0.31 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 = 0.31 Order of pole (three term test) = 651 NO COMPLEX POLE (six term test) for Equation 1 Finished! diff ( y , x , 1 ) = y * y; Iterations = 200 Total Elapsed Time = 0 Seconds Elapsed Time(since restart) = 0 Seconds Time to Timeout = 3 Minutes 0 Seconds Percent Done = 100.5 %