A
with(NumericalMethods);
Digits:=25;
ODE:=diff(y(x),x)=-2*x*y(x);
IC:=y(0)=1;
sol:=exp(-x^2);
h:=0.05;
k:=0.005;
K:=0.0005;
Eh1:=ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,3], stepsize=h, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,3], stepsize=h, solution=sol, output=plot);
Ek1:=ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,3], stepsize=k, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,3], stepsize=k, solution=sol, output=plot);
EK1:=ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,3], stepsize=K, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,3], stepsize=K, solution=sol, output=plot);
q11:=ln(Eh1/Ek1)/ln(h/k);
q12:=ln(Ek1/EK1)/ln(k/K);
Eh2:=ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,4], stepsize=h, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,4], stepsize=h, solution=sol, output=plot)
Ek2:=ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,4], stepsize=k, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,4], stepsize=k, solution=sol, output=plot);
EK2:=ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,4], stepsize=K, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..3, method=[taylor,4], stepsize=K, solution=sol, output=plot)
q21 := ln(Eh2/Ek2)/ln(h/k);
q22 := ln(Ek2/EK2)/ln(k/K);
B
h := 0.5e-1;
k := 0.5e-2;
K := 0.5e-3
Eh1 := ODENumeric(ODE,yinit=[1],x=0..8,method=[taylor,3],stepsize = h,solution = sol,output = totalerror);
ODENumeric(ODE,yinit=1,x=0..8,method=[taylor,3],stepsize = h,solution = sol,output = plot);
Ek1 := ODENumeric(ODE,yinit=[1],x=0..8,method=[taylor,3],stepsize = k,solution = sol,output = totalerror);
ODENumeric(ODE,yinit=1,x=0..8,method=[taylor,3],stepsize = k,solution = sol,output = plot);
EK1 := ODENumeric(ODE,yinit=[1],x=0..8,method=[taylor,3],stepsize = K,solution = sol,output = totalerror);
ODENumeric(ODE,yinit=1,x=0..8,method=[taylor,3],stepsize = K,solution = sol,output = plot);
q11 := ln(Eh1/Ek1)/ln(h/k);
q12 := ln(Ek1/EK1)/ln(k/K);
q13 := ln(Eh1/EK1)/ln(h/K);
Eh1 := ODENumeric(ODE,yinit=[1],x=0..8,method=[taylor,4],stepsize = h,solution = sol,output = totalerror);
ODENumeric(ODE,yinit=1,x=0..8,method=[taylor,4],stepsize = h,solution = sol,output = plot);
Ek1 := ODENumeric(ODE,yinit=[1],x=0..8,method=[taylor,4],stepsize = k,solution = sol,output = totalerror);
ODENumeric(ODE,yinit=1,x=0..8,method=[taylor,4],stepsize = k,solution = sol,output = plot);
EK1 := ODENumeric(ODE,yinit=[1],x=0..8,method=[taylor,4],stepsize = K,solution = sol,output = totalerror);
ODENumeric(ODE,yinit=1,x=0..8,method=[taylor,4],stepsize = K,solution = sol,output = plot);
q21 := ln(Eh2/Ek2)/ln(h/k);
q22 := ln(Ek2/EK2)/ln(k/K);
q23 := ln(Eh2/EK2)/ln(h/K);
Eh1:=ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,3], stepsize=h, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,3], stepsize=h, solution=sol, output=plot);
Ek1:=ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,3], stepsize=k, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,3], stepsize=k, solution=sol, output=plot);
EK1:=ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,3], stepsize=K, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,3], stepsize=K, solution=sol, output=plot);
q11:=ln(Eh1/Ek1)/ln(h/k);
q12:=ln(Ek1/EK1)/ln(k/K);
Eh2:=ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,4], stepsize=h, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,4], stepsize=h, solution=sol, output=plot);
Ek2:=ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,4], stepsize=k, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,4], stepsize=k, solution=sol, output=plot);
EK2:=ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,4], stepsize=K, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..8, method=[taylor,4], stepsize=K, solution=sol, output=plot);
q21 := ln(Eh2/Ek2)/ln(h/k);
q22 := ln(Ek2/EK2)/ln(k/K);
Eh1:=ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,3], stepsize=h, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,3], stepsize=h, solution=sol, output=plot);
Ek1:=ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,3], stepsize=k, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,3], stepsize=k, solution=sol, output=plot);
EK1:=ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,3], stepsize=K, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,3], stepsize=K, solution=sol, output=plot);
q11:=ln(Eh1/Ek1)/ln(h/k);
q12:=ln(Ek1/EK1)/ln(k/K);
Eh2:=ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,4], stepsize=h, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,4], stepsize=h, solution=sol, output=plot);
Ek2:=ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,4], stepsize=k, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,4], stepsize=k, solution=sol, output=plot);
EK2:=ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,4], stepsize=K, solution=sol, output=totalerror);
ODENumeric(ODE, yinit=[1], x=0..1, method=[taylor,4], stepsize=K, solution=sol, output=plot);
q21 := ln(Eh2/Ek2)/ln(h/k);
q22 := ln(Ek2/EK2)/ln(k/K);