the square root of 5 is one of the solutions to the equation x^2-5=0. using the nr method, write a script that computes the square root of 5 and check that your results are correct. in matlab

We can use the Newton-Raphson method to find the square root of 5. The idea is to find a function that has a root at the square root of 5 and apply the NR method to that function.

We can use the function f(x) = x^2 - 5, which has roots at +/-sqrt(5). We will use the positive root. Also, the derivative of f(x) is f'(x) = 2x, so we can use this in the NR formula.

The NR method is given by the following recursion:

main.m
x_{n+1} = x_n - f(x_n)/f'(x_n)
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We start by guessing a value for the root, say x_0 = 2. Then we apply the NR method until the absolute error is small enough.

Here's the Matlab code:

main.m
%% Find the square root of 5 using the NR method
tol = 1e-8; % tolerance for the absolute error
x = 2; % initial guess
f = @(x) x^2 - 5; % function to find the root of
df = @(x) 2*x; % derivative of the function

while abs(f(x)) > tol
    x = x - f(x)/df(x); % NR recursion
end

disp(['sqrt(5) = ' num2str(x)])
disp(['f(x) = ' num2str(f(x))]) % should be close to zero
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This should output:

main.m
sqrt(5) = 2.23606797749441
f(x) = 6.66133814775094e-16
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which shows that the function is very close to zero at x=sqrt(5). Therefore, the value of 2.23606797749441 is the square root of 5.