Solving Integral Equations

[Albert Chan]

Numerically getting F(x) is expensive, required possibly many f(x) calls.
We can use third order iterative formulas instead.

Redo your 2nd example (Mathematica code):

In[1]:= f[x_] := Exp[x^2]
In[2]:= F[x_] := NIntegrate[f[t], {t, 0, x}] - 0.95

In[3]:= order2[x_] := x - F[x]/f[x]         (* newton's method *)
In[4]:= NestList[order2, 2., 8]
Out[4]= {2., 1.71606, 1.39774, 1.07527, 0.84433, 0.772609, 0.768049, 0.768033, 0.768033}

In[5]:= order3[x_] := order3[x, F[x], f[x]]
In[6]:= order3[x_, Fx_, fx_] := x - Fx/mean[fx, f[x-Fx/fx]]

In[7]:= mean[a_, b_] := (a + b)/2         (* algorithm 7 *)
In[8]:= NestList[order3, 2., 6]
Out[8]= {2., 1.57877, 1.1098, 0.803199, 0.768074, 0.768033, 0.768033}

In[9]:= mean[a_, b_] := 2/(1/a + 1/b)   (* algorithm 9 *)
In[10]:= NestList[order3, 2., 5]
Out[10]= {2., 1.45024, 0.908868, 0.769107, 0.768033, 0.768033}

In[11]:= Last[%] // CForm
Out[11]//CForm= 0.768032819933785