Question

Questions: 1. Write a MATLAB program to find all the roots of a given, twice continuously differentiable, function f E C2[a, b]. Your program should first probe the function f(x) on the given interval to find out where it changes sign. (Thus, the program has, in addition to f itself, four other input arguments: a, b, the number nprobe of equidistant values between a and b at which f is probed, and a tolerance tol.) For each subinterval [ai,b] over which the function changes sign, your program should then find a root as follows. Use either Newton's method or the secant method to find the root, monitoring decrease in lf(xx). If an iterate is reached for which there is no sufficient decrease (e.g., if If(x)I 2 0.5f(x1)1), then revert back to [a,bil, apply three bisection steps and restart the Newton or secant method. The ith root is deemed "found" as x if both hold a. Verify your program by finding the two roots of the function f(x) 2cosh(x/4)-x, starting your search with [a, b]- [0,10] and nprobe b. Find all the roots of the function 10. sin (x) f(x) f(x)-x x=0 in the interval [-10,10] for tol = 10-7.

Answer 1

function x_best_guess = probFun(f,a,b,nprobe)

dx = (b-a)/nprobe;
i=1;
test = a;
f_sign = sign(subs(f,test));
while i<nprobe
test = test+dx;
if(f_sign~=sign(subs(f,test)))
else
break;
end
i=i+1;
end

x_best_guess = test;

end

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%NEWTON-RAPHSON FUNCTION

function root = newtonRaphsonFun(f,xn,tol,a,b)

fp = diff(f);

f_prev = double(subs(f,xn));

while(true)
  
Xn = xn - double(subs(f,xn)/subs(fp,xn));

if abs(Xn-xn)<tol
root = Xn;
break;
end

xn = Xn;

f_iter = double(subs(f,xn));

if abs( f_iter-f_prev)<tol
another_best_guess = threeBisecFun(f,a,b,tol);
xn = another_best_guess;
end

f_prev = f_iter;
end

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function best_guess = threeBisecFun(f,a,b,f_tol)
%called by NewtonRaphsonFun at iteration when f absolute tolerance is not met

f_prev = double(subs(f,a));

while(true)
  
m = (a+b)/2;
fm = double(subs(f,m));

if fm<0
a = m;
else
b = m;
end

fm = double(subs(f,m));

if abs(fm-f_prev)<f_tol
break;
end

f_prev = fm;

end

best_guess = m;
end

-----------------------------------------------------------------------

%command

clc
clear

syms x; %symbolic to make differentiation adjustable when f changes instead of hard coding the derivative of f
f = 2*cosh(x/4)-x

nprobe = 10;
tol = 10^-4;
a = 0;
b = 10;

x_best_guess = probFun(f,a,b,nprobe)

root = newtonRaphsonFun(f,x_best_guess,tol,a,b)

f =

2*cosh(x/4) - x

x_best_guess =

1

root =

2.3576