First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 84
In [5]: fixedpt(x->x-(x^2-2)/4,-5.,10^(-5),15)
Did not converge. The last estimate is p=-Inf.
Let’s verify the linear convergence of the fixed-point iteration numerically in this example.
We write another version of the fixed-point code, fixedpt2, and we compute pn−√ 2
p n−1 − √ for each
2
n. The code uses some functions we have not seen before:
• global list defines the variable list as a global variable, which allows us to access list
outside the function definition, where we plot it;
• Float64[ ] creates an empty array of dimension 1, with a float type;
• append!(list,x) appends x to the list.
In [6]: using PyPlot
In [7]: function fixedpt2(g::Function,pzero,eps,N)
n=1
global list
list=Float64[]
error=1.
while n10^-5.
pone=g(pzero)
error=abs(pone-pzero)
append!(list,(pone-2^0.5)/(pzero-2^0.5))
pzero=pone
n=n+1
end
return(list)
end
Out[7]: fixedpt2 (generic function with 1 method)
In [8]: fixedpt2(x->x-(x^2-2)/4,1.5,10^(-7),15)
Out[8]: 9-element Array{Float64,1}:
0.27144660940672544