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Newton’s Method Calculator

Newton’s Method Calculator


f'(x) (if you know)

Initial value (x₀)

Maximum iterations

Significant Figure

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An online newton’s method calculator allows you to determine an approximation of the root of a real function. The calculator uses the Newtons method formula to display the iteration of the incremental calculation. Here you can learn more about Newton’s method, its formulas, and examples.

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.


However, an Online Tangent Line Calculator allows you to determine the tangent line to the implicit, parametric, polar, and explicit at a particular point.

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.


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?


First apply the power rule:


x^2 = 2x


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 Newton’s Method Calculator Works?

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


  • 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.


  • 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.


Why did Newton’s method fail?

If the derivative is zero, Newton’s method will not work. When the derivative is close to zero, the tangent is almost horizontal, so it may exceed the required root (numerical difficulty).

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.

Will Newton’s method diverge?

If the function cannot be continuously differentiated near the root, Newton’s method will always diverge and fail if the solution is not guessed in the first attempt.

What is the fastest method of convergence?

Newton method is a very good method. When the conditions are met, Newton’s method converges, and the convergence rate is faster than almost any other alternative iterative scheme that relies on the method of converting the original f(x) into a fixed-point function.


Use this online newton’s method calculator to find real roots of non-linear functions. Because this calculator provides a complete iterations table by using newton’s method formula. Finding iterations by hand is a lengthy and time-consuming technique. To make it convenient for you, our online newtons calculator performs all calculations related to the Newton method for free and fast.


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.

From the source of Paul’s online Notes: Stationary point, Poor initial estimate, Mitigation of non-convergence, Analysis, Basins of attraction.

From the source of AMSI: Finding a solution with geometry, The key calculation, The Algorithm, Using Newton’s method, Sensitive dependence on initial conditions.