Newton’s Method Calculator

Newton’s technique calculator permits you to determine an approximation of the foundation of a actual function. Thia calculator makes use of the Newton's technique formula to display the generation of the incremental calculation.

what's Newton’s method?

In calculus, Newton's approach (also called Newton Raphson method), is a root-finding algorithm that offers a greater accurate approximation to the basis (or 0) of a actual-valued feature.

Newton's method is based totally on tangent strains. The basic idea is if x is close sufficient to the basis of f(x), the tangent of the graph will intersect the x-axis at a point (x, f(x)) at a factor that's in the direction of the basis than x.

Newton's technique system:

If x_n is an estimation solution of the function f(x) which is identical to 0 and if f’(x_n) isn't always equal to the 0, then the next estimation is given via,

x_n+1 = x_n – f(x_n) / f’(x_n)

This newtons method system is utilized by the newton’s approach calculator for finding the root of a actual-valued feature.

Example:

Find an approximation to x using Newton's method to solve \(x^2 - 4 = 0\) for 3 iterations, starting from \(x_0 = 1\) with 4 significant figures. How many decimal places is the estimated solution accurate?

Solution:

First, apply the power rule:

Where,

\(f(x) = x^2 - 4\)

So,

\(f'(x) = 2x\)

Iteration 1:

\(f(x_0) = f(1) = (1)^2 - 4 = 1 - 4 = -3\)

\(f'(x_0) = f'(1) = 2(1) = 2\)

Now, Newton's method formula:

\(x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}\)

\(x_1 = 1 - \frac{-3}{2}\)

\(x_1 = 1 + 1.5 = 2.5\)

Iteration 2:

\(f(x_1) = f(2.5) = (2.5)^2 - 4 = 6.25 - 4 = 2.25\)

\(f'(x_1) = f'(2.5) = 2(2.5) = 5\)

Now, using Newton's method formula:

\(x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}\)

\(x_2 = 2.5 - \frac{2.25}{5}\)

\(x_2 = 2.5 - 0.45 = 2.05\)

Iteration 3:

\(f(x_2) = f(2.05) = (2.05)^2 - 4 = 4.2025 - 4 = 0.2025\)

\(f'(x_2) = f'(2.05) = 2(2.05) = 4.1\)

Now, using Newton's method formula:

\(x_3 = x_2 - \frac{f(x_2)}{f'(x_2)}\)

\(x_3 = 2.05 - \frac{0.2025}{4.1}\)

\(x_3 = 2.05 - 0.0494 = 2.0006\)

The estimate solution after 3 iterations is \(x = 2.0006\), accurate to 4 significant figures.

How Our Calculator Works?

Newton's technique calculator implements Newton's method to find the foundation of a real characteristic and provide iterations by means of following those commands:

Input:

• Firstly, substitute a actual-valued feature and its derivative (non-compulsory).
• Now, plug within the initial price and most iterations as according to requirements.
• Then, add the enormous determine within the applicable area.
• click the calculate button, to locate the iterations of a given feature.

Output:

• The newton approach calculator presentations the given feature and its derivative.
• The calculator applies the power rule to the actual feature and provides an iterations desk in keeping with given values.
• It offers a step-with the aid of-step solution for all iterations in a fraction of a second.

FAQ:

Will Newton's technique continually converge?

Newton's approach does not continually converge. His idea of convergence refers to "local" convergence, which means it have to begin close to the basis, and "approximately" refers back to the function you need to deal with.

Why is the Newton method quicker than the bisection technique?

The characteristic f ought to have a non-stop by-product. if you start too far from the foundation, Newton's technique won't converge. but, while it converges, it's far quicker than the bisection technique and is usually quadratic.