## Numerical Methods

### Introduction to Numerical Methods

Numerical methods are mathematical techniques used to solve problems that cannot be solved analytically. They are used in various fields such as engineering, science, and finance to provide approximate solutions to complex problems. Numerical methods are based on the idea of breaking down complex problems into simpler parts and using mathematical algorithms to find solutions.

### Types of Numerical Methods

There are several types of numerical methods, including:

##### 1. Root finding methods

Root finding methods are used to find the root of an equation, i.e. the value of x for which the equation equals zero. Some common root finding methods include bisection method, secant method, and Newton-Raphson method.

##### 2. Integration methods

Integration methods are used to find the definite or indefinite integral of a function. Some common integration methods include trapezoidal rule, Simpson’s rule, and Gaussian quadrature.

##### 3. Differentiation methods

Differentiation methods are used to find the derivative of a function. Some common differentiation methods include forward difference method, backward difference method, and central difference method.

##### 4. Linear systems of equations

Linear systems of equations are a set of linear equations with multiple variables. Some common methods to solve linear systems of equations include Gaussian elimination, LU decomposition, and Jacobi method.

### Advantages of Numerical Methods

The following are some of the advantages of numerical methods:

- They provide approximate solutions to complex problems that cannot be solved analytically.
- They are computationally efficient and can be easily implemented on computers.
- They allow for the modeling of real-world problems in a numerical form.

### Disadvantages of Numerical Methods

The following are some of the disadvantages of numerical methods:

- They provide approximate solutions and may not be as accurate as analytical solutions.
- They may converge slowly or not converge at all.
- They may have stability issues and may become unstable for certain types of problems.

Numerical methods are essential tools for solving complex problems in various fields. They provide approximate solutions to problems that cannot be solved analytically and can be easily implemented on computers. However, it is important to keep in mind their limitations and disadvantages when using them for solving problems.