##############ECHO OF PROBLEM################# ##############temp/nonlinear2postode.ode################# diff ( y , x , 1 ) = y * y; ! #BEGIN FIRST INPUT BLOCK # Digits:=32; ELIMINATED in preodein.rb 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 # ELIMINATED in preodein.rb ## 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 # ELIMINATED in preodein.rb # ELIMINATED in preodein.rb # ELIMINATED in preodein.rb #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 def exact_soln_y (x) return(2.0/(1.0 - 2.0*x)) end #END USER DEF BLOCK #######END OF ECHO OF PROBLEM################# START of Optimize min_size 0.0 min_size 1.0 $glob_desired_digits_correct 10 desired_abs_gbl_error 1.0e-10 range 0.2 estimated_steps 200000.00000000003 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 1.3421786221786213e-148 estimated_step_error 1.3421786221786213e-148 best_h 2.0e-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 9.007217269175524e-141 estimated_step_error 9.007217269175524e-141 best_h 4.0e-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 6.044653276686248e-133 estimated_step_error 6.044653276686248e-133 best_h 8.0e-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 4.0565143728453123e-125 estimated_step_error 4.0565143728453123e-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 2.7223024922073703e-117 estimated_step_error 2.7223024922073703e-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 8.228576044322444e-94 estimated_step_error 8.228576044322444e-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 5.522810916934549e-86 estimated_step_error 5.522810916934549e-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 3.7072449645186265e-78 estimated_step_error 3.7072449645186265e-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 2.4891650844253993e-70 estimated_step_error 2.4891650844253993e-70 best_h 0.001 START of Soultion TOP MAIN SOLVE Loop x[1] 0.0 y[1] (analytic) 2.0 y[1] (numeric) 2.0 absolute error 0.0 relative error 0.0% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.5 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.4999999999999998 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.5000000000000117 Order of pole (three term test) 651.0000000000155 Radius of convergence (six term test) for eq 1 0.42836993358544667 Order of pole (six term test) -21.1 TOP MAIN SOLVE Loop x[1] 0.010000000000000002 y[1] (analytic) 2.0408163265306123 y[1] (numeric) 2.0408163265306127 absolute error 4.440892098500626e-16 relative error 2.1760371282653068e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.49 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.48999999999999977 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.48999999999999916 Order of pole (three term test) 650.999999999999 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] 0.02000000000000001 y[1] (analytic) 2.0833333333333335 y[1] (numeric) 2.0833333333333335 absolute error 0.0 relative error 0.0% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.48 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.4799999999999997 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.4800000000000033 Order of pole (three term test) 651.0000000000049 Radius of convergence (six term test) for eq 1 0.12175228559505176 Order of pole (six term test) -12.676190476190476 TOP MAIN SOLVE Loop x[1] 0.03000000000000002 y[1] (analytic) 2.127659574468085 y[1] (numeric) 2.1276595744680855 absolute error 4.440892098500626e-16 relative error 2.0872192862952943e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.47 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.4699999999999998 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.47000000000000086 Order of pole (three term test) 651.0000000000014 NO COMPLEX POLE (six term test) for Equation 1 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 4.0856207306205754e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.45999999999999996 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.4599999999999994 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.4599999999999821 Order of pole (three term test) 650.9999999999746 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] 0.05000000000000004 y[1] (analytic) 2.2222222222222223 y[1] (numeric) 2.2222222222222237 absolute error 1.3322676295501878e-15 relative error 5.995204332975845e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.44999999999999996 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.44999999999999957 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.4499999999999929 Order of pole (three term test) 650.99999999999 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] 0.060000000000000046 y[1] (analytic) 2.272727272727273 y[1] (numeric) 2.2727272727272743 absolute error 1.3322676295501878e-15 relative error 5.861977570020827e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.43999999999999995 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.43999999999999956 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.4399999999999918 Order of pole (three term test) 650.9999999999881 Radius of convergence (six term test) for eq 1 0.03016988933062603 Order of pole (six term test) -11.522222222222222 TOP MAIN SOLVE Loop x[1] 0.07000000000000005 y[1] (analytic) 2.3255813953488373 y[1] (numeric) 2.325581395348839 absolute error 1.7763568394002505e-15 relative error 7.638334409421077e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.42999999999999994 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.4299999999999995 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.4300000000000039 Order of pole (three term test) 651.0000000000068 Radius of convergence (six term test) for eq 1 0.29584633280062633 Order of pole (six term test) -16.263157894736842 TOP MAIN SOLVE Loop x[1] 0.08000000000000006 y[1] (analytic) 2.3809523809523814 y[1] (numeric) 2.380952380952383 absolute error 1.7763568394002505e-15 relative error 7.460698725481051e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.41999999999999993 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.4199999999999995 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.41999999999999904 Order of pole (three term test) 650.9999999999991 Radius of convergence (six term test) for eq 1 0.256 Order of pole (six term test) -22.0 TOP MAIN SOLVE Loop x[1] 0.09000000000000007 y[1] (analytic) 2.439024390243903 y[1] (numeric) 2.4390243902439037 absolute error 8.881784197001252e-16 relative error 3.641531520770513e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.4099999999999999 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.4099999999999995 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.4100000000000043 Order of pole (three term test) 651.0000000000074 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] 0.10000000000000007 y[1] (analytic) 2.5000000000000004 y[1] (numeric) 2.5000000000000018 absolute error 1.3322676295501878e-15 relative error 5.32907051820075e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.3999999999999999 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.39999999999999963 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.4000000000000152 Order of pole (three term test) 651.0000000000257 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] 0.11000000000000008 y[1] (analytic) 2.5641025641025648 y[1] (numeric) 2.5641025641025657 absolute error 8.881784197001252e-16 relative error 3.463895836830488e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.3899999999999999 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.3899999999999997 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.39000000000000773 Order of pole (three term test) 651.0000000000136 Radius of convergence (six term test) for eq 1 0.4147674046981031 Order of pole (six term test) -25.5 TOP MAIN SOLVE Loop x[1] 0.12000000000000009 y[1] (analytic) 2.6315789473684217 y[1] (numeric) 2.6315789473684226 absolute error 8.881784197001252e-16 relative error 3.375077994860475e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.3799999999999999 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.37999999999999967 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.38000000000001083 Order of pole (three term test) 651.0000000000192 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] 0.1300000000000001 y[1] (analytic) 2.7027027027027035 y[1] (numeric) 2.7027027027027044 absolute error 8.881784197001252e-16 relative error 3.286260152890462e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.3699999999999999 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.3699999999999997 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.37 Order of pole (three term test) 651.0000000000002 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] 0.1400000000000001 y[1] (analytic) 2.7777777777777786 y[1] (numeric) 2.7777777777777803 absolute error 1.7763568394002505e-15 relative error 6.3948846218409e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.3599999999999999 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.35999999999999954 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.3599999999999959 Order of pole (three term test) 650.9999999999928 Radius of convergence (six term test) for eq 1 0.0905096679918781 Order of pole (six term test) -13.5 TOP MAIN SOLVE Loop x[1] 0.1500000000000001 y[1] (analytic) 2.857142857142858 y[1] (numeric) 2.85714285714286 absolute error 1.7763568394002505e-15 relative error 6.217248937900874e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.34999999999999987 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.3499999999999996 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.34999999999999754 Order of pole (three term test) 650.9999999999958 Radius of convergence (six term test) for eq 1 0.06618492611195123 Order of pole (six term test) -12.340277777777779 TOP MAIN SOLVE Loop x[1] 0.16000000000000011 y[1] (analytic) 2.9411764705882364 y[1] (numeric) 2.941176470588238 absolute error 1.7763568394002505e-15 relative error 6.039613253960849e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.33999999999999986 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.3399999999999996 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.3399999999999964 Order of pole (three term test) 650.9999999999936 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] 0.17000000000000012 y[1] (analytic) 3.0303030303030316 y[1] (numeric) 3.0303030303030334 absolute error 1.7763568394002505e-15 relative error 5.861977570020824e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.32999999999999985 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.32999999999999957 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.32999999999999335 Order of pole (three term test) 650.9999999999874 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] 0.18000000000000013 y[1] (analytic) 3.1250000000000018 y[1] (numeric) 3.125000000000003 absolute error 1.3322676295501878e-15 relative error 4.2632564145605986e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.31999999999999984 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.31999999999999956 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.31999999999999545 Order of pole (three term test) 650.9999999999914 NO COMPLEX POLE (six term test) for Equation 1 TOP MAIN SOLVE Loop x[1] 0.19000000000000014 y[1] (analytic) 3.225806451612905 y[1] (numeric) 3.225806451612906 absolute error 1.3322676295501878e-15 relative error 4.1300296516055804e-14% Correct digits 16 h 0.001 Radius of convergence (given) for eq 1 0.30999999999999983 Order of pole (given) 0.0 Radius of convergence (ratio test) for eq 1 0.30999999999999966 Order of pole (ratio test) Not computed Radius of convergence (three term test) for eq 1 0.3100000000000059 Order of pole (three term test) 651.0000000000132 NO COMPLEX POLE (six term test) for Equation 1 Finished! diff ( y , x , 1 ) = y * y; Iterations 200 Total Elapsed Time 0.0 Seconds Elapsed Time(since restart) 0.0 Seconds Time to Timeout 3 Minutes 0.0 Seconds Percent Done 100.50000000000007%