Newton’s Method Calculator

Newton’s method calculator allows you to determine an approximation of the root of a real function. Thia calculator uses the Newton’s method formula to display the iteration of the incremental calculation.

What is Newton’s Method?

In calculus, Newton’s method (also known as Newton Raphson method), is a root-finding algorithm that provides a more accurate approximation to the root (or zero) of a real-valued function.

Newton’s method is based on tangent lines. The basic idea is that if x is close enough to the root of f(x), the tangent of the graph will intersect the x-axis at a point (x, f(x)) at a point which is closer to the root than x.

Newton’s Method Formula:

If x_n is an estimation solution of the function f(x) which is equal to zero and if f’(x_n) is not equal to the zero, then the next estimation is given by,

x_n+1 = x_n – f(x_n) / f’(x_n)

This newtons method formula is used by the newton’s method calculator for finding the root of a real-valued function.

Example:

Find an approximation to x with newton’s method to solve x^2 for 3 iterations, starting from x_0 = 1 with 4 significant figures. So, how many decimal places is the estimate solution accurate?

Solution:

First apply the power rule:

Where,

x^2 = 2x

So,

Iteration 1:

F(x_0) = f(5) = (5)^2 = 25

F’(x_0) = f’(5) = 2 (5) = 10

Now, newton’s method calculator uses the formula

X_1 = x_0 – f(x_0) / f’(x_0)

X_1 = 5 – 25/10

X_1 = 2.5

Iteration 2:

F(x_1) = f(2.5) = (2.5)^2 = 6.25

F’(x_1) = f’(2.5) = 2 (2.5) = 5

Now, newtons method calculator uses the newton method formula:

X_2= x_1 – f(x_1) / f’(x_1)

X_2 = 2.5 – 6.25/5

X_2 = 1.25

Iteration 3:

F(x_2) = f(1.25) = (1.25)^2 = 1.5625

F’(x_2) = f’(1.25) = 2 (1.25) = 2.5

Now, using thenewton’s method formula:

X_3= x_2 – f(x_2) / f’(x_2)

X_3 = 1.25 – 1.5625/2.5

X_3 = 0.625

Hence, newton’s method calculator gives an Iterations Table for the same values:

Step	x	f(x)	f'(x)
1	2.5	25	10
2	1.25	6.25	5
3	0.625	1.5625	2.5

However, an Online Derivative Calculator allows you to determine the derivative of the function with respect to a given variable.

How Our Calculator Works?

Newton’s method calculator implements Newton’s method to find the root of a real function and provide iterations by following these instructions:

Input:

• Firstly, substitute a real-valued function and its derivative (optional).
• Now, plug in the initial value and maximum iterations as per requirements.
• Then, add the significant figure in the relevant field.
• Click the calculate button, to find the iterations of a given function.

Output:

• The newton method calculator displays the given function and its derivative.
• The calculator applies the power rule to the real function and provides an iterations table according to given values.
• It gives a step-by-step solution for all iterations in a fraction of a second.

FAQ:

Will Newton’s method always converge?

Newton’s method does not always converge. His theory of convergence refers to “local” convergence, which means it must start near the root, and “about” refers to the function you want to deal with.

Why is the Newton method faster than the bisection method?

The function f must have a continuous derivative. If you start too far from the root, Newton’s method may not converge. However, when it converges, it is faster than the bisection method and is usually quadratic.

Reference:

From the source of Wikipedia: Newton’s method, Difficulty in calculating derivative of a function, Failure of the method to converge to the root, Overshoot.