The Newton-Raphson method is a powerful and widely used technique for finding approximate solutions to equations. It is particularly useful for finding the roots of nonlinear equations, which cannot be solved using traditional algebraic methods. In this essay, we will explore the Newton-Raphson method through a series of example problems.

The basic idea behind the Newton-Raphson method is to iteratively improve an initial guess for the solution to an equation by using information about the slope of the equation at that point. This process is repeated until the solution is found to within a specified tolerance.

To illustrate the Newton-Raphson method, consider the equation x^2 - 3x + 2 = 0. This equation has two roots, x = 1 and x = 2. To find these roots using the Newton-Raphson method, we first need to rewrite the equation in the form f(x) = 0, where f(x) is a differentiable function. In this case, we can simply let f(x) = x^2 - 3x + 2.

Next, we need to choose an initial guess for the solution, which we will call x0. This initial guess should be reasonably close to the actual solution, as the Newton-Raphson method relies on the assumption that the function is well-behaved in the vicinity of the solution. In this example, we will choose x0 = 1.5.

Using the initial guess x0, we can then calculate the next approximation for the solution, x1, using the following formula:

x1 = x0 - f(x0) / f'(x0)

Where f'(x0) is the derivative of f(x) evaluated at x0.

Substituting the values from our example, we get:

x1 = 1.5 - (1.5^2 - 3*1.5 + 2) / (2*1.5 - 3)

= 1.5 - (-0.25) / (-0.5)

= 1.5 - 0.5

= 1

We can repeat this process to find the second root of the equation. Choosing x0 = 2, we get:

x1 = 2 - (2^2 - 3*2 + 2) / (2*2 - 3)

= 2 - (2) / (-1)

= 2 + 2

= 4

Since the solution x = 4 is not within the specified tolerance of the actual solution x = 2, we can repeat the process using x1 as the new initial guess. Continuing this process, we will eventually converge on the solution x = 2 to within the desired tolerance.

This simple example illustrates the basic principles of the Newton-Raphson method. In practice, the method can be applied to a wide range of equations, including those with multiple roots and those with more complicated functional forms. It is an essential tool for solving nonlinear equations, and is widely used in fields such as engineering, physics, and finance.

## Real Work Application of Newthon

We can clearly see that a plateau at 20 appears at the maximum value of our function and is there for the entire asymptotic region of the exponential tail. So when faced with solving an equation that seems impossible, we need this method! Next, we will calculate the first derivative and substitute both the function and derivative into our formula. Convergence plot Converged solution Codes A snapshot of the Mathematica code for both solved problems are shown below, along with the download links to the Mathematica notebook. Well, you know that a root is synonymous with x-intercept, so you are being asked to find where the function crosses the x-axis. The result is the following figure: These red dots indicate areas of fast convergence, where they are low, and slow convergence, where they are high. For this purpose, a polynomial is kind of ideal: it is easily constructed, can have several roots at specific locations, and derivatives are simple to get, too.