Newton Raphson Method Calculation

Assume the equation or functions whose roots are to be calculated as f(x) = 0.

In order to prove the validity of Newton Raphson method following steps are followed:

Step 1: Draw a graph of f(x) for different values of x as shown below:

Step 2: A tangent is drawn to f(x) at x₀. This is the initial value.

Step 3:This tangent will intersect the X- axis at some fixed point (x₁,0) if the first derivative of f(x) is not zero i.e. f'(x₀) ≠ 0.

Step 4: As this method assumes iteration of roots, this x₁ is considered to be the next approximation of the root.

Step 5: Now steps 2 to 4 are repeated until we reach the actual root x^*.

Now we know that the slope-intercept equation of any line is represented as y = mx + c,

Where m is the slope of the line and c is the x-intercept of the line. 

Using the same formula we, get

y = f(x₀) + f'(x₀) (x − x₀)

Here f(x₀) represents the c and f'(x₀) represents the slope of the tangent m. As this equation holds true for every value of x, it must hold true for x₁. Thus, substituting x with x₁, and equating the equation to zero as we need to calculate the roots, we get:

0 = f(x₀) + f'(x₀) (x₁ − x₀)

x₁ = x₀ – f(x₀)/f'(x₀)

Which is the Newton Raphson method formula.

Thus, Newton Raphson’s method was mathematically proved and accepted to be valid.

Convergence of Newton Raphson Method

The Newton-Raphson method tends to converge if the following condition holds true:

|f(x).f”(x)| < |f'(x)|²

It means that the method converges when the modulus of the product of the value of the function at x and the second derivative of a function at x is lesser than the square of the modulo of the first derivative of the function at x. The Newton-Raphson Method has a convergence of order 2 which means it has a quadratic convergence.

Note:

Newton Raphson’s method is not valid if the first derivative of the function is 0 which means f'(x) = 0. It is only possible when the given function is a constant function.

Articles related to Newton Raphson Method:

• Newton’s Method for Finding Roots
• Difference between Newton Raphson Method and Regular Falsi Method
• Difference between Bisection Method and Newton Raphson Method
• Root Finding Algorithm

Newton Raphson Method or Newton Method is a powerful technique for solving equations numerically. It is most commonly used for approximation of the roots of the real-valued functions. Newton Rapson Method was developed by Isaac Newton and Joseph Raphson, hence the name Newton Rapson Method. 

Newton Raphson Method involves iteratively refining an initial guess to converge it toward the desired root. However, the method is not efficient to calculate the roots of the polynomials or equations with higher degrees but in the case of small-degree equations, this method yields very quick results. In this article, we will learn about Newton Raphson Method and the steps to calculate the roots using this method as well.