Usage of newton

Suppose we want to solve \(e^x=x+c\) for multiple values of \(c\). We can create functions for \(f\) and \(f'\) in each case.

using FundamentalsNumericalComputation

for c = [2,4,7.5,11]
    f = x -> exp(x) - x - c;
    dfdx = x -> exp(x) - 1;
    x = FNC.newton(f,dfdx,1.0);  r = x[end];
    println("root with c = $c is $r")
end

root with c = 2.0 is 1.1461932206205836
root with c = 4.0 is 1.7490313860127016
root with c = 7.5 is 2.2803781488230648
root with c = 11.0 is 2.610868638149876

There’s a subtlety about the definition of f. It uses whatever value is assigned to c at the moment f is called. If we later change the value assigned to c, the function is changed also.

c = 11;  f = x -> exp(x) - x - c;
@show f(0);

f(0) = -10.0

c = 100; 
@show f(0);

f(0) = -99.0
Convergence of Newton’s method Using the secant line