write the algorithms of newton raphson method,false position method in numerical analysis

Answer 1

Newton-Raphson Method Algorithm in Numerical Analysis:

1. Start with an initial guess x0 for the root of the function f(x).
2. Calculate the value of the function f(x0) and its derivative f'(x0) at the initial guess.
3. Calculate the next guess for the root using the formula: x1 = x0 - f(x0) / f'(x0).
4. Repeat steps 2 and 3 until the desired level of accuracy is reached, i.e., |x1 - x0| < ε, where ε is a small positive number.
5. The final value of x1 is the approximate root of the function f(x).

False Position Method Algorithm in Numerical Analysis:

1. Choose two initial guesses x0 and x1 such that f(x0) and f(x1) have opposite signs.
2. Calculate the next guess for the root using the formula: x2 = x1 - f(x1) * (x0 - x1) / (f(x0) - f(x1)).
3. Calculate the value of the function f(x2) at the new guess.
4. If f(x2) is approximately zero (i.e., |f(x2)| < ε, where ε is a small positive number), then x2 is the approximate root of the function f(x).
5. Otherwise, update x0 and x1 based on the signs of f(x0) and f(x2) and repeat steps 2-4 until the desired level of accuracy is reached.