##############ECHO OF PROBLEM################# ##############temp/sing1_backpostode.ode################# diff ( y , x , 1 ) = m1 * 2.0 * x / ( x * x + 0.000001 ) / ( x * x + 0.000001 ) ; ! /* BEGIN FIRST INPUT BLOCK */ Digits=32; max_terms=20; ! /* END FIRST INPUT BLOCK */ /* BEGIN SECOND INPUT BLOCK */ x_start=c(-1.5); x_end=c(-2.0); array_y_init[0 + 1] = exact_soln_y(x_start); glob_look_poles=true; glob_max_h=c(0.1); 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=(20.0); glob_subiter_method=3; glob_max_iter=1000; glob_upper_ratio_limit=c(1.11); glob_lower_ratio_limit=c(0.99); /* 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] = -1.5 y[1] (closed_form) = 0.444444246913668 y[1] (numeric) = 0.444444246913668 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 14 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1.5 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.406 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.5 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.5 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.45 y[1] (closed_form) = 0.4756240306187726 y[1] (numeric) = 0.4756240306187726 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 14 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1.45 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.359 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.45 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.45 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.4 y[1] (closed_form) = 0.5102038213245811 y[1] (numeric) = 0.510203821324581 absolute error = 1.110223024625157e-16 relative error = 2.176038238488331e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1.4 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.313 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.4 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.4 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.35 y[1] (closed_form) = 0.5486965439250789 y[1] (numeric) = 0.5486965439250788 absolute error = 1.110223024625157e-16 relative error = 2.023382572602372e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1.35 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.266 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.35 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.35 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.3 y[1] (closed_form) = 0.5917156262037717 y[1] (numeric) = 0.5917156262037715 absolute error = 2.220446049250313e-16 relative error = 3.752556043679077e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1.3 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.219 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.3 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.3 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.25 y[1] (closed_form) = 0.6399995904002623 y[1] (numeric) = 0.6399995904002622 absolute error = 1.110223024625157e-16 relative error = 1.734724586199831e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1.25 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.172 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.25 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.25 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.2 y[1] (closed_form) = 0.6944439621916934 y[1] (numeric) = 0.694443962191693 absolute error = 3.33066907387547e-16 relative error = 4.796166797049747e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1.2 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.125 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.2 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.2 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.15 y[1] (closed_form) = 0.7561430955439735 y[1] (numeric) = 0.7561430955439733 absolute error = 2.220446049250313e-16 relative error = 2.936542120579586e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1.15 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.078 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.15 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.15 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.1 y[1] (closed_form) = 0.8264455979788452 y[1] (numeric) = 0.8264455979788449 absolute error = 3.33066907387547e-16 relative error = 4.030112910058389e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1.1 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.031 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.1 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.1 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.05 y[1] (closed_form) = 0.9070286557563221 y[1] (numeric) = 0.9070286557563216 absolute error = 4.440892098500626e-16 relative error = 4.896087979489035e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1.05 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.9844 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.05 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.05 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.9999999999999996 y[1] (closed_form) = 0.9999990000010009 y[1] (numeric) = 0.9999990000010005 absolute error = 4.440892098500626e-16 relative error = 4.44089653939272e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 1 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.9375 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.9499999999999995 y[1] (closed_form) = 1.108032013260928 y[1] (numeric) = 1.108032013260928 absolute error = 4.440892098500626e-16 relative error = 4.007909559788909e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 0.95 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.8906 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.95 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.95 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.8999999999999995 y[1] (closed_form) = 1.234566377078548 y[1] (numeric) = 1.234566377078548 absolute error = 4.440892098500626e-16 relative error = 3.597127040677601e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 0.9 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.8438 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.9 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.9 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.8499999999999994 y[1] (closed_form) = 1.384081129299477 y[1] (numeric) = 1.384081129299476 absolute error = 6.661338147750939e-16 relative error = 4.812823473088195e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 0.85 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.7969 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.85 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.85 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.7999999999999994 y[1] (closed_form) = 1.562497558597567 y[1] (numeric) = 1.562497558597566 absolute error = 8.881784197001252e-16 relative error = 5.68435076786499e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 0.8 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.75 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.8 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.8 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.7499999999999993 y[1] (closed_form) = 1.777774617289572 y[1] (numeric) = 1.777774617289571 absolute error = 8.881784197001252e-16 relative error = 4.996012492597393e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 0.75 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.7031 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.75 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.75 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.6999999999999993 y[1] (closed_form) = 2.040812161607838 y[1] (numeric) = 2.040812161607836 absolute error = 1.332267629550188e-15 relative error = 6.528124707472203e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 0.7 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.6563 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.7 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.7 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.6499999999999992 y[1] (closed_form) = 2.366858303293962 y[1] (numeric) = 2.366858303293961 absolute error = 1.332267629550188e-15 relative error = 5.628844057525826e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 0.65 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.6094 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.65 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.65 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.5999999999999992 y[1] (closed_form) = 2.777770061749836 y[1] (numeric) = 2.777770061749834 absolute error = 2.220446049250313e-15 relative error = 7.993627981761598e-14 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 0.6 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.5625 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.6 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.6 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.5499999999999992 y[1] (closed_form) = 3.305774195787793 y[1] (numeric) = 3.30577419578779 absolute error = 2.664535259100376e-15 relative error = 8.060245804131202e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 0.55 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.5156 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.55 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.55 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.4999999999999992 y[1] (closed_form) = 3.999984000064014 y[1] (numeric) = 3.999984000064011 absolute error = 2.220446049250313e-15 relative error = 5.551137327586256e-14 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 16 h = 0.05 Radius of convergence (given) for eq 1 = 0.5 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.4688 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.5 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.5 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.4499999999999992 y[1] (closed_form) = 4.938247218532272 y[1] (numeric) = 4.938247218532271 absolute error = 8.881784197001252e-16 relative error = 1.798570181676944e-14 % Desired digits = 8 Estimated correct digits = 11 Correct digits = 16 h = 0.05 SETTING H FOR POLE Radius of convergence (given) for eq 1 = 0.45 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.4219 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.45 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.45 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.4950001107652434 y[1] (closed_form) = 4.081197719693583 y[1] (numeric) = 4.081197719693582 absolute error = 1.77635683940025e-15 relative error = 4.352538057219192e-14 % Desired digits = 8 Estimated correct digits = 11 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.495 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.4641 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.495 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.495 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.5400002215304875 y[1] (closed_form) = 3.429340707062155 y[1] (numeric) = 3.429340707062153 absolute error = 2.220446049250313e-15 relative error = 6.474848196557649e-14 % Desired digits = 8 Estimated correct digits = 11 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.54 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.5063 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.54 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.54 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.5850003322957318 y[1] (closed_form) = 2.922042346141157 y[1] (numeric) = 2.922042346141156 absolute error = 1.332267629550188e-15 relative error = 4.559371397576006e-14 % Desired digits = 8 Estimated correct digits = 11 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.585 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.5484 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.585 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.585 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.630000443060976 y[1] (closed_form) = 2.519516437253236 y[1] (numeric) = 2.519516437253235 absolute error = 1.332267629550188e-15 relative error = 5.287790981838639e-14 % Desired digits = 8 Estimated correct digits = 11 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.63 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.5906 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.63 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.63 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.6750005538262203 y[1] (closed_form) = 2.194778961338999 y[1] (numeric) = 2.194778961338998 absolute error = 8.881784197001252e-16 relative error = 4.046778447148327e-14 % Desired digits = 8 Estimated correct digits = 11 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.675 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.6328 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.675 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.675 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.7200006645914645 y[1] (closed_form) = 1.929005063490841 y[1] (numeric) = 1.929005063490841 absolute error = 2.220446049250313e-16 relative error = 1.15108357737126e-14 % Desired digits = 8 Estimated correct digits = 11 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.72 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.675 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.72 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.72 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.7650007753567087 y[1] (closed_form) = 1.708738116437381 y[1] (numeric) = 1.708738116437381 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.765 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.7172 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.765 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.765 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.810000886121953 y[1] (closed_form) = 1.524152244931018 y[1] (numeric) = 1.524152244931018 absolute error = 0 relative error = 0 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.81 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.7594 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.81 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.81 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.8550009968871972 y[1] (closed_form) = 1.367937211682473 y[1] (numeric) = 1.367937211682474 absolute error = 4.440892098500626e-16 relative error = 3.246415157490029e-14 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.855 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.8016 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.855 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.855 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.9000011076524415 y[1] (closed_form) = 1.23456333826466 y[1] (numeric) = 1.234563338264661 absolute error = 6.661338147750939e-16 relative error = 5.395703842230015e-14 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.9 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.8438 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.9 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.9 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.9450012184176857 y[1] (closed_form) = 1.119785338104496 y[1] (numeric) = 1.119785338104497 absolute error = 6.661338147750939e-16 relative error = 5.948763500535595e-14 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.945 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.8859 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.945 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.945 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -0.9900013291829299 y[1] (closed_form) = 1.020300269860053 y[1] (numeric) = 1.020300269860053 absolute error = 6.661338147750939e-16 relative error = 6.528801711151785e-14 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 0.99 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.9281 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 0.99 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 0.99 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.035001439948174 y[1] (closed_form) = 0.9335072314337546 y[1] (numeric) = 0.9335072314337554 absolute error = 7.771561172376096e-16 relative error = 8.325121553091683e-14 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 16 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.035 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 0.9703 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.035 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.035 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.080001550713418 y[1] (closed_form) = 0.8573356232694685 y[1] (numeric) = 0.8573356232694694 absolute error = 8.881784197001252e-16 relative error = 1.035975171908799e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.08 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.013 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.08 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.08 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.125001661478663 y[1] (closed_form) = 0.7901204986852614 y[1] (numeric) = 0.7901204986852622 absolute error = 8.881784197001252e-16 relative error = 1.124105020915202e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.125 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.055 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.125 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.125 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.170001772243907 y[1] (closed_form) = 0.7305108043111463 y[1] (numeric) = 0.7305108043111472 absolute error = 8.881784197001252e-16 relative error = 1.215832010229685e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.17 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.097 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.17 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.17 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.215001883009151 y[1] (closed_form) = 0.6774009537874041 y[1] (numeric) = 0.6774009537874051 absolute error = 9.992007221626409e-16 relative error = 1.475050657333782e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.215 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.139 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.215 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.215 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.260001993774395 y[1] (closed_form) = 0.6298791921195833 y[1] (numeric) = 0.6298791921195843 absolute error = 9.992007221626409e-16 relative error = 1.586337086005758e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.26 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.181 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.26 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.26 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.30500210453964 y[1] (closed_form) = 0.5871882018604468 y[1] (numeric) = 0.587188201860448 absolute error = 1.110223024625157e-15 relative error = 1.89074477502703e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.305 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.223 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.305 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.305 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.350002215304884 y[1] (closed_form) = 0.5486947431451165 y[1] (numeric) = 0.5486947431451176 absolute error = 1.110223024625157e-15 relative error = 2.02338921321054e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.35 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.266 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.35 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.35 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.395002326070128 y[1] (closed_form) = 0.5138660352649387 y[1] (numeric) = 0.5138660352649398 absolute error = 1.110223024625157e-15 relative error = 2.160530076779152e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.395 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.308 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.395 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.395 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.440002436835372 y[1] (closed_form) = 0.4822512216750718 y[1] (numeric) = 0.4822512216750729 absolute error = 1.110223024625157e-15 relative error = 2.302167365732866e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.44 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.35 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.44 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.44 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.485002547600617 y[1] (closed_form) = 0.4534667054072661 y[1] (numeric) = 0.4534667054072672 absolute error = 1.165734175856414e-15 relative error = 2.570716134075265e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.485 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.392 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.485 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.485 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.530002658365861 y[1] (closed_form) = 0.427184458045209 y[1] (numeric) = 0.4271844580452102 absolute error = 1.165734175856414e-15 relative error = 2.728877780785378e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.53 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.434 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.53 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.53 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.575002769131105 y[1] (closed_form) = 0.4031226326152078 y[1] (numeric) = 0.4031226326152089 absolute error = 1.165734175856414e-15 relative error = 2.891760674149847e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.575 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.477 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.575 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.575 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.620002879896349 y[1] (closed_form) = 0.3810379757450343 y[1] (numeric) = 0.3810379757450355 absolute error = 1.221245327087672e-15 relative error = 3.20504885293861e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.62 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.519 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.62 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.62 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.665002990661594 y[1] (closed_form) = 0.3607196554771062 y[1] (numeric) = 0.3607196554771074 absolute error = 1.165734175856414e-15 relative error = 3.231690200841855e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.665 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.561 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.665 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.665 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.710003101426838 y[1] (closed_form) = 0.3419842107407708 y[1] (numeric) = 0.341984210740772 absolute error = 1.165734175856414e-15 relative error = 3.408736834169395e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.71 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.603 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.71 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.71 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.755003212192082 y[1] (closed_form) = 0.3246713954341559 y[1] (numeric) = 0.324671395434157 absolute error = 1.165734175856414e-15 relative error = 3.590504714151291e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.755 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.645 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.755 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.755 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.800003322957326 y[1] (closed_form) = 0.3086407404925359 y[1] (numeric) = 0.308640740492537 absolute error = 1.165734175856414e-15 relative error = 3.776993840787543e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.8 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.688 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.8 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.8 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.84500343372257 y[1] (closed_form) = 0.2937686956030877 y[1] (numeric) = 0.2937686956030889 absolute error = 1.221245327087672e-15 relative error = 4.157166319510445e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.845 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.73 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.845 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.845 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.890003544487815 y[1] (closed_form) = 0.2799462415062858 y[1] (numeric) = 0.279946241506287 absolute error = 1.221245327087672e-15 relative error = 4.362428016595647e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.89 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.772 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.89 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.89 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.935003655253059 y[1] (closed_form) = 0.2670768863771515 y[1] (numeric) = 0.2670768863771528 absolute error = 1.27675647831893e-15 relative error = 4.780482862586482e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.935 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.814 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.935 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.935 Order of pole (six term test) = 1 TOP MAIN SOLVE Loop x[1] = -1.980003766018303 y[1] (closed_form) = 0.2550749772671039 y[1] (numeric) = 0.2550749772671052 absolute error = 1.27675647831893e-15 relative error = 5.005416415197653e-13 % Desired digits = 8 Estimated correct digits = 10 Correct digits = 15 h = -0.04500011076524419 Radius of convergence (given) for eq 1 = 1.98 Order of pole (given) = 1 Radius of convergence (ratio test) for eq 1 = 1.856 Order of pole (ratio test) = 1 Radius of convergence (three term test) for eq 1 = 1.98 Order of pole (three term test) = 2 Radius of convergence (six term test) for eq 1 = 1.98 Order of pole (six term test) = 1 Finished! diff ( y , x , 1 ) = m1 * 2.0 * x / ( x * x + 0.000001 ) / ( x * x + 0.000001 ) ; Iterations = 56 Total Elapsed Time = 0.0 Seconds Elapsed Time(since restart) = 0.0 Seconds Expected Time Remaining = 0.0 Seconds Optimized Time Remaining = 0.0 Seconds Expected Total Time = 0.0 Seconds Time to Timeout = 20 Minutes 0.0 Seconds Percent Done = 0 %