Newton Raphson Method

The Newton Raphson Method is referred to as one of the most commonly used techniques for finding the roots of given equations. It can be efficiently generalised to find solutions to a system of equations. Moreover, we can show that when we approach the root, the method is quadratically convergent. In this article, you will learn how to use the Newton Raphson method to find the roots or solutions of a given equation, and the geometric interpretation of this method.

Newton Raphson Method Formula

Let x0 be the approximate root of f(x) = 0 and let x1 = x0 + h be the correct root. Then f(x1) = 0

⇒ f(x0 + h) = 0….(1)

By expanding the above equation using Taylor’s theorem, we get:

f(x0) + hf1(x0) + … = 0

⇒ h = -f(x0) /f’(x0)

Therefore, x1 = x0 – f(x0)/ f’(x0)

Now, x1 is the better approximation than x0.

Similarly, the successive approximations x2, x3, …., xn+1 are given by

\(\begin{array}{l}x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}\end{array} \)

This is called Newton Raphson formula.

Other formulas include the following:

Newton’s Iterative Formula to Find bth Root of a Positive Real Number a The iterative formula is given by:

\(\begin{array}{l}x_{n+1}=\frac{1}{b}\left [(b-1)x_n+\frac{a}{x_n^{b-1}} \right]\end{array} \)
Newton’s Iterative Formula to Find a Reciprocal of a Number N The iterative formula is given by:

xi+1 = xi(2 – xiN)

Geometrical Interpretation of Newton Raphson Formula

The geometric meaning of Newton’s Raphson method is that a tangent is drawn at the point [x0, f(x0)] to the curve y = f(x).

Newton Raphson Method

It cuts the x-axis at x1, which will be a better approximation of the root. Now, drawing another tangent at [x1, f(x1)], which cuts the x-axis at x2, which is a still better approximation and the process can be continued till the desired accuracy is achieved.

Convergence of Newton Raphson Method

The order of convergence of Newton Raphson method is 2 or the convergence is quadratic. It converges if |f(x).f’’(x)| < |f’(x)|2. Also, this method fails if f’(x) = 0.

Solved Examples

Example 1:

Find the cube root of 12 using the Newton Raphson method assuming x0 = 2.5.

Solution:

We know that, the iterative formula to find bth root of a is given by:

\(\begin{array}{l}x_{n+1}=\frac{1}{b}\left[(b-1)x_n + \frac{a}{x_n^{b-1}} \right]\end{array} \)

From the given, a = 12, b = 3

Let x0 be the approximate cube root of 12, i.e., x0 = 2.5.

So, x1 = (⅓) [2x0 + 12/x02]

= (⅓) [2(2.5) + 12/(2.5)2]

= (⅓) [5 + 12/6.25]

= (⅓)(5 + 1.92)

= 6.92/3

= 2.306

Now,

x2 = (⅓)[2x1 + 12/x12]

= (1/3) [2(2.306) + 12/(2.306)2]

= (⅓) [4.612 + 12/5.3176]

= (⅓) [4.612 + 2.256]

= 6.868/3

= 2.289

Therefore, the approximate cube root of 12 is 2.289.

Example 2:

Find a real root of the equation -4x + cos x + 2 = 0, by Newton Raphson method up to four decimal places, assuming x0 = 0.5.

Solution:

Given equation: -4x + cos x + 2 = 0

x0 = 0/5

Let f(x) = -4x + cos x + 2

f’(x) = -4 – sin x

Now,

f(0) = -4(0) + cos 0 + 2 = 1 + 2 = 3 > 0

f(1) = -4(1) + cos 1 + 2 = -4 + 0.5403 + 2 = -1.4597 < 0

Thus, a root lies between 0 and 1.

Let us find the first approximation.

x1 = x0 – f(x0)/f’(x0)

= 0.5 – [-4(0.5) + cos 0.5 + 2]/ [-4 – sin 0.5]

= 0.5 – [(-2 + 2 + cos 0.5)/ (-4 – sin 0.4)]

= 0.5 – [cos 0.5/ (-4 – sin 0.5)]

= 0.5 – [0.8775/ (-4 – 0.4794)]

= 0.5 – (0.8775/-4.4794)

= 0.5 + 0.1958

= 0. 6958

Frequently Asked Questions on Newton Raphson Method

Where is the Newton Raphson method used?

Newton Raphson method is an efficient technique to solve the equations numerically. It gives us better approximations in terms of solutions.

Which is the correct formula of the Newton Raphson method?

xn+1 = xn – f(xn)/f'(xn)

Is the Newton Raphson method always convergent?

No, the Newton Raphson method is not always convergent. That means it cannot always guarantee that the condition is satisfied. However, this method fails when f'(x) is equal to 0.

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