Question

must be done in matlab

Part B (Based off Week 3 Content) Newtons Method approximates a root of a function by iterating through the equation where n is the nth estimate for the root of the function f(z). In order to it- erate through this method, we need to provide an initial guess for the root, For example, if we apply this method to f(z) = sin(z) using note that f(cos(r) = 1, we sin(1) =-0.5574 cos(1) sin(-0.5574 ) cos(-0.5574) Z3 -0.5574- 0.0659 sin(0.0659) 0.0659--(0.0659) = We can see that as we iterate through Newtons Method we are converging towards r = 0, which is one of the roots of f(z) = sin(z) Questions: 1. Create a function that can numerically solve f(r) given f(x) and I are provided as inputs You can do this using the central finite difference formula: 2h where h = 10-10 2. Create a function that can apply a single iteration of Newtons Method given f(x) and r are provided as inputs. The output should be r2- 3. Using a for loop, create a function that can apply N iterations of New- tons Method given f(x), r and N are provided as inputs. The output should be a vector containing all iterations of n (should have a length of N +1) Useful Control Structures and Functions: for

Answer 1

Derivative function:

START CODE

function der = func_der(f,x)
der=(f(x+10^(-10))-f(x-10^(-10)))/(2*10^(-10)); %numerically calculates derivative
end

END CODE

Newton's next iteration function:

START CODE

function newt = newt_one(f,x)
newt=(x-f(x)/func_der(f,x)); %next iteration
end

END CODE

Main function:

START CODE

function [x] = Q_newt_final(f,x1,N)
x=zeros(1,N+1); %initialize
x(1)=x1; %initialize
for i=2:N+1
    x(i)=newt_one(f,x(i-1)); %for loop
end
end

END CODE

Sample usage:

Command Window >> Q newt final ( (x) x.*2-3,1,5) ans - 1.0000 2.0000 1.7500 1.7321 1.7321 17321

$\blacksquare$

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