Numerical Analysis Questions Long
Newton's method, also known as Newton-Raphson method, is an iterative numerical technique used to find the roots of a nonlinear equation. It is based on the idea of approximating the roots by using tangent lines to the curve of the equation.
The method starts with an initial guess for the root, denoted as x0. Then, it iteratively improves this guess by finding the tangent line to the curve at the current guess and determining where it intersects the x-axis. This intersection point becomes the new guess, denoted as x1. The process is repeated until a desired level of accuracy is achieved.
Mathematically, the Newton's method can be expressed as follows:
x_(n+1) = x_n - f(x_n)/f'(x_n)
Where:
- x_(n+1) is the new guess for the root
- x_n is the current guess for the root
- f(x_n) is the value of the function at the current guess
- f'(x_n) is the derivative of the function at the current guess
To illustrate the concept, let's consider an example equation: f(x) = x^2 - 4. We want to find the root of this equation using Newton's method.
1. Choose an initial guess, let's say x0 = 2.
2. Calculate the value of the function at the current guess: f(x0) = (2)^2 - 4 = 0.
3. Calculate the derivative of the function at the current guess: f'(x0) = 2x0 = 4.
4. Apply the Newton's method formula to find the new guess:
x1 = x0 - f(x0)/f'(x0) = 2 - 0/4 = 2.
Since the new guess is the same as the previous guess, we have reached the root of the equation. In this case, the root is x = 2.
The process can be repeated with the new guess to further refine the root if desired. Newton's method is a powerful tool for solving nonlinear equations, but it may not always converge or may converge to a different root depending on the initial guess and the behavior of the function. Therefore, careful consideration of the initial guess and the properties of the equation is necessary for successful application of Newton's method.