% Class 11 - Root Finding Open Methods
% Coded by NFR on 10.1.2019
% 
% Five different methods
% Method 1: Fixed Point Iteration (FPI)
f1 = @(x) sin(sqrt(x))-x;
% To use FPI, add x to the give function
h1 = @(x) f1(x) + x;
% Call the FPI algorithm to solve
root = FPI(h1,0.5,0.01)
% Method 2: Newton Raphson Method (use analytical slope)
% Remember this method can diverge
% make your initial guess close to the root
f2 = @(x) -2 + 6*x -4*x^2 + 0.5*x^3;
d2 = @(x) 6 - 8*x + 1.5*x^2;
% Use newton raphson algorithm
root1 = newtraph(f2,d2,4.5)
root2 = newtraph(f2,d2,4.43)
% Let's investigate by plotting
fplot(f2,[0 8])
% Method 3: Secant method
% Requires two initial x guesses
root1 = secantmethod(f2,5,5.2)
% Method 4: Modified secant method
root1 = modsecantmethod(f2,5,.05)
% Method 5: Brent's Method --> Matlab's function
% fzero
% fzero is a blend of open and bracket methods
% makes it efficient and (fairly) robust
root1 = fzero(f2,6)
% Two other functions for roots that might
% come in handy (only for polynomial functions)
root_ans = roots([0.5 -4 6 -2])
% Remember to insert coefficients in descending order (matlab syntax)
poly_coef = poly(root_ans)
% Extra Problem 6.14
f3 = @(x) x/(1-x)*sqrt(2*3/(2+x))-0.05
root_x = fzero(f3,0.01)