Newton’s Technique to Fish Out the Roots

Newton-Raphson method, a fundamental tool in numerical analysis, is renowned for its efficient root-finding capability. Developed independently by Isaac Newton and Joseph Raphson, it uses derivatives to iteratively converge to a root of a real-valued function. Starting from an initial guess, it refines this guess through successive approximations. Its versatility extends to solving nonlinear equations, optimization, and machine learning. Praised for rapid convergence, it’s pivotal in mathematical and engineering applications.

Isaac Newton and Joseph Raphson developed the method independently, but not simultaneously. Newton first introduced the method in a less generalized form in the mid-17th century, as part of his work on calculus and the theory of fluxions. Joseph Raphson then published a more generalized form of the method in 1690. Their contributions were separate and at different times, with Raphson refining and generalizing Newton’s earlier work.

Here’s how it works mathematically:

1. Starting Point: Choose an initial guess \( x_0 \) for the root of the function \( f(x) \).
2. Iteration Formula: Apply the Newton-Raphson iteration formula:
   \( x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)} \)
   Here, \( x_n \) is the current approximation, \( f(x_n) \) is the value of the function at \( x_n \), and \( f'(x_n) \) is the value of its derivative at \( x_n \).
3. Convergence: Repeat the iteration until the value of \( x \) converges to a stable value within a desired level of accuracy. This means that the change in \( x \) from one iteration to the next becomes negligibly small.
4. Root Approximation: The value of \( x \) at which the process stabilizes is taken as the approximate root of the function.

Explanation:

• The method uses the idea of linear approximation. At each iteration, the function \( f(x) \) is approximated by a tangent line at \( x_n \), and the x-intercept of this tangent (where the tangent cuts the x-axis) is taken as the next approximation \( x_{n+1} \).
• The derivative \( f'(x_n) \) in the formula represents the slope of the tangent line. If \( f'(x_n) \) is very small, the method may not work properly because it involves dividing by a small number, which can lead to large errors or division by zero.
• The method assumes that the function is differentiable in the neighborhood of the root and that the initial guess is close enough to the actual root for the process to converge.
• The method can converge very rapidly, especially if the initial guess is close to the true root, but if the initial guess is not well chosen, the method may fail to converge or converge to a different root.

Let’s try solving for a root of a function using the Newton-Raphson method. Consider the function \( f(x) = \cos(x) – x \), a function which does not have a straightforward algebraic solution for its roots.

The derivative of \( f(x) \) is \( f'(x) = -\sin(x) – 1 \).

We’ll start with an initial guess. Let’s say our initial guess is \( x_0 = 1 \).

Now, we apply the Newton-Raphson formula:
\( x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)} \)

In the first iteration of the Newton-Raphson method for the function \( f(x) = \cos(x) – x \) with an initial guess of \( x_0 = 1 \), the values are:

• Initial guess, \( x_0 \): \(1\)
• Function value at \( x_0 \), \( f(x_0) \): \(-0.45969769413186023\)
• Derivative value at \( x_0 \), \( f'(x_0) \): \(-1.8414709848078965\)
• The next approximation, \( x_1 \): \(0.7503638678402439\)

After performing a few iterations of the Newton-Raphson method on the function \( f(x) = \cos(x) – x \) with an initial guess of \( x_0 = 1 \), the approximations for the root are as follows:

1. 0.7503638678402439
2. 0.7391128909113617
3. 0.739085133385284
4. 0.7390851332151607
5. 0.7390851332151607

As we can see, the approximation quickly converges to around 0.7390851332151607, which is a solution to the equation \( \cos(x) = x \).