##############ECHO OF PROBLEM################# ##############temp/arccos_sqrtpostode.ode################# diff ( y , x , 1 ) = arccos ( sqrt ( 0.1 * x + 0.2 ) ) ; ! // BEGIN FIRST INPUT BLOCK Digits=32; max_terms=30; ! // END FIRST INPUT BLOCK // BEGIN SECOND INPUT BLOCK x_start=c(0.0); x_end=c(0.5); array_y_init[0 + 1] = exact_soln_y(x_start); glob_look_poles=true; glob_max_h=c(0.001); glob_type_given_pole=1; array_given_rad_poles[1][1]=c(-2.0); array_given_rad_poles[1][2]=c(0.0); array_given_ord_poles[1][1]=c(0.5); 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(c(10.0) * (c(0.1) * c(x) + c(0.2)) * arccos(sqrt ( c(0.1) * c(x) + c(0.2))) - c(5.0) * sqrt( c(0.1) * c(x) + c(0.2)) * sqrt( c(0.8) - c(0.1) * c(x)) + c(5.0) * arcsin(sqrt( c(0.1) * c(x) + c(0.2)))); } // END USER DEF BLOCK #######END OF ECHO OF PROBLEM################# START of Soultion TOP MAIN SOLVE Loop x[1] = 0 y[1] (closed_form) = 2.532535480592211 y[1] (numeric) = 2.532535480592211 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] = 0.01 y[1] (closed_form) = 2.543600721667676 y[1] (numeric) = 2.54360072168069 absolute error = 1.301447838386594e-11 relative error = 5.1165571203853e-10 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 12 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.02000000000000001 y[1] (closed_form) = 2.554653486058796 y[1] (numeric) = 2.55465348608482 absolute error = 2.602318360800382e-11 relative error = 1.018658058715869e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.03000000000000002 y[1] (closed_form) = 2.565693796891293 y[1] (numeric) = 2.565693796930316 absolute error = 3.90230070479447e-11 relative error = 1.520953400410707e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.04000000000000003 y[1] (closed_form) = 2.576721677085934 y[1] (numeric) = 2.57672167713795 absolute error = 5.201528097131813e-11 relative error = 2.018661209469206e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.05000000000000004 y[1] (closed_form) = 2.587737149360984 y[1] (numeric) = 2.587737149425986 absolute error = 6.500178173496352e-11 relative error = 2.51191593207274e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.06000000000000005 y[1] (closed_form) = 2.598740236234604 y[1] (numeric) = 2.598740236312585 absolute error = 7.798073298204145e-11 relative error = 3.000712879830966e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.07000000000000005 y[1] (closed_form) = 2.609730960027214 y[1] (numeric) = 2.609730960118167 absolute error = 9.095213471255192e-11 relative error = 3.485115366512856e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.08000000000000006 y[1] (closed_form) = 2.620709342863813 y[1] (numeric) = 2.620709342967729 absolute error = 1.03915986926495e-10 relative error = 3.965185502522744e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.09000000000000007 y[1] (closed_form) = 2.631675406676253 y[1] (numeric) = 2.631675406793127 absolute error = 1.168736218915001e-10 relative error = 4.441034847800962e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1000000000000001 y[1] (closed_form) = 2.642629173205489 y[1] (numeric) = 2.642629173335314 absolute error = 1.298254836967772e-10 relative error = 4.912739366276648e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1100000000000001 y[1] (closed_form) = 2.653570664003778 y[1] (numeric) = 2.653570664146549 absolute error = 1.427711282531163e-10 relative error = 5.380340165416944e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1200000000000001 y[1] (closed_form) = 2.664499900436843 y[1] (numeric) = 2.664499900592554 absolute error = 1.557109996497275e-10 relative error = 5.84391088264626e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1300000000000001 y[1] (closed_form) = 2.675416903686012 y[1] (numeric) = 2.675416903854654 absolute error = 1.686424333513514e-10 relative error = 6.303407634115156e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1400000000000001 y[1] (closed_form) = 2.686321694750294 y[1] (numeric) = 2.686321694931865 absolute error = 1.815712025177163e-10 relative error = 6.759101222781666e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1500000000000001 y[1] (closed_form) = 2.69721429444846 y[1] (numeric) = 2.697214294642953 absolute error = 1.944937544351433e-10 relative error = 7.210912193200962e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1600000000000001 y[1] (closed_form) = 2.708094723421051 y[1] (numeric) = 2.708094723628461 absolute error = 2.074100891036323e-10 relative error = 7.658893439355684e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1700000000000001 y[1] (closed_form) = 2.718963002132381 y[1] (numeric) = 2.718963002352702 absolute error = 2.203215387908131e-10 relative error = 8.103145891210111e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1800000000000001 y[1] (closed_form) = 2.729819150872486 y[1] (numeric) = 2.729819151105714 absolute error = 2.332281034966854e-10 relative error = 8.543719953834404e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.1900000000000001 y[1] (closed_form) = 2.74066318975906 y[1] (numeric) = 2.740663190005189 absolute error = 2.461293391320396e-10 relative error = 8.980648919274083e-09 % Desired digits = 8 Estimated correct digits = 13 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2000000000000001 y[1] (closed_form) = 2.751495138739343 y[1] (numeric) = 2.75149513899837 absolute error = 2.590265779645051e-10 relative error = 9.414030005634819e-09 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2100000000000002 y[1] (closed_form) = 2.762315017591991 y[1] (numeric) = 2.76231501786391 absolute error = 2.719193759048721e-10 relative error = 9.843894493319376e-09 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 11 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2200000000000002 y[1] (closed_form) = 2.773122845928907 y[1] (numeric) = 2.773122846213715 absolute error = 2.848077329531407e-10 relative error = 1.02702890847859e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2300000000000002 y[1] (closed_form) = 2.783918643197048 y[1] (numeric) = 2.783918643494738 absolute error = 2.97690760930891e-10 relative error = 1.069322775140524e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2400000000000002 y[1] (closed_form) = 2.794702428680198 y[1] (numeric) = 2.794702428990767 absolute error = 3.105693480165428e-10 relative error = 1.111278770968148e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2500000000000002 y[1] (closed_form) = 2.80547422150072 y[1] (numeric) = 2.805474221824166 absolute error = 3.234457146561454e-10 relative error = 1.152909237865411e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2600000000000002 y[1] (closed_form) = 2.816234040621273 y[1] (numeric) = 2.816234040957591 absolute error = 3.363180844928593e-10 relative error = 1.194212127407799e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2700000000000002 y[1] (closed_form) = 2.826981904846505 y[1] (numeric) = 2.826981905195693 absolute error = 3.491873457051042e-10 relative error = 1.235194838376809e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2800000000000002 y[1] (closed_form) = 2.83771783282472 y[1] (numeric) = 2.837717833186773 absolute error = 3.620530542036704e-10 relative error = 1.27585995343052e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.2900000000000002 y[1] (closed_form) = 2.848441843049515 y[1] (numeric) = 2.848441843424431 absolute error = 3.749160981669775e-10 relative error = 1.316214684466213e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3000000000000002 y[1] (closed_form) = 2.859153953861397 y[1] (numeric) = 2.859153954249172 absolute error = 3.877755894166057e-10 relative error = 1.356259913506581e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3100000000000002 y[1] (closed_form) = 2.869854183449371 y[1] (numeric) = 2.869854183850003 absolute error = 4.00631972041765e-10 relative error = 1.396001143027526e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3200000000000002 y[1] (closed_form) = 2.880542549852506 y[1] (numeric) = 2.880542550265991 absolute error = 4.134852460424554e-10 relative error = 1.435442243557997e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3300000000000002 y[1] (closed_form) = 2.89121907096147 y[1] (numeric) = 2.891219071387806 absolute error = 4.263362995970965e-10 relative error = 1.474590092044873e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3400000000000002 y[1] (closed_form) = 2.90188376452005 y[1] (numeric) = 2.901883764959235 absolute error = 4.391846886164785e-10 relative error = 1.513446865054281e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3500000000000003 y[1] (closed_form) = 2.912536648126649 y[1] (numeric) = 2.912536648578679 absolute error = 4.520299690113916e-10 relative error = 1.552014699290182e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3600000000000003 y[1] (closed_form) = 2.923177739235748 y[1] (numeric) = 2.923177739700622 absolute error = 4.648748053170948e-10 relative error = 1.590306326835379e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3700000000000003 y[1] (closed_form) = 2.933807055159358 y[1] (numeric) = 2.933807055637075 absolute error = 4.777165329983291e-10 relative error = 1.628316123100947e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3800000000000003 y[1] (closed_form) = 2.944424613068448 y[1] (numeric) = 2.944424613559006 absolute error = 4.905573725011436e-10 relative error = 1.666055127795998e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.3900000000000003 y[1] (closed_form) = 2.955030429994348 y[1] (numeric) = 2.955030430497743 absolute error = 5.033959915579089e-10 relative error = 1.703522192016384e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4000000000000003 y[1] (closed_form) = 2.965624522830128 y[1] (numeric) = 2.965624523346361 absolute error = 5.162332783470447e-10 relative error = 1.740723663339544e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4100000000000003 y[1] (closed_form) = 2.976206908331969 y[1] (numeric) = 2.976206908861037 absolute error = 5.290679006009213e-10 relative error = 1.777658331212733e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4200000000000003 y[1] (closed_form) = 2.986777603120494 y[1] (numeric) = 2.986777603662397 absolute error = 5.419034110332177e-10 relative error = 1.814341350581488e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4300000000000003 y[1] (closed_form) = 2.997336623682104 y[1] (numeric) = 2.99733662423684 absolute error = 5.547358128410451e-10 relative error = 1.850762468446321e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4400000000000003 y[1] (closed_form) = 3.007883986370268 y[1] (numeric) = 3.007883986937836 absolute error = 5.675677705596627e-10 relative error = 1.886933715301198e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4500000000000003 y[1] (closed_form) = 3.018419707406812 y[1] (numeric) = 3.018419707987211 absolute error = 5.803992841890704e-10 relative error = 1.92285811931603e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4600000000000004 y[1] (closed_form) = 3.028943802883184 y[1] (numeric) = 3.028943803476414 absolute error = 5.932303537292682e-10 relative error = 1.958538660124977e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4700000000000004 y[1] (closed_form) = 3.039456288761698 y[1] (numeric) = 3.039456289367759 absolute error = 6.060600910018366e-10 relative error = 1.993975347639399e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4800000000000004 y[1] (closed_form) = 3.049957180876763 y[1] (numeric) = 3.049957181495654 absolute error = 6.188911605420344e-10 relative error = 2.029179833810399e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 TOP MAIN SOLVE Loop x[1] = 0.4900000000000004 y[1] (closed_form) = 3.060446494936092 y[1] (numeric) = 3.060446495567812 absolute error = 6.317204537253929e-10 relative error = 2.064144740875741e-08 % Desired digits = 8 Estimated correct digits = 12 Correct digits = 10 h = 0.001 Finished! diff ( y , x , 1 ) = arccos ( sqrt ( 0.1 * x + 0.2 ) ) ; Iterations = 500 Total Elapsed Time = 0.0 Seconds Elapsed Time(since restart) = 0.0 Seconds Time to Timeout = 3 Minutes 0.0 Seconds Percent Done = 100.2 %