A numerical method refers to a technique or algorithm used to solve mathematical problems that involve numerical calculations. It involves approximating the solution to a mathematical problem by using iterative procedures and discrete steps.

  Numerical methods are generally applied when analytical or exact solutions to mathematical problems are either unavailable or too difficult to obtain. These methods are particularly useful in solving complex equations, systems of equations, differential equations, optimization problems, and many other mathematical models.

  Various numerical methods exist, each designed to address different types of mathematical problems. Some commonly used numerical methods include:

  1. Newton-Raphson Method: This method is used to find the roots of a nonlinear equation. It involves an iterative process that converges to the root through successive approximations.

  2. Finite Difference Method: This method is used to approximate derivatives of a function. It replaces the continuous derivative with a set of discrete differences calculated at selected points.

  3. Euler's Method: This method is used for numerical integration of ordinary differential equations. It approximates the solution by stepping forward in small increments along the solution curve.

  4. Gaussian Elimination: This method is used to solve systems of linear equations. It involves manipulating the augmented matrix of the system to reduce it to an upper triangular form.

  5. Monte Carlo Method: This method is a stochastic simulation technique used for solving problems that involve randomness or uncertainty. It involves generating random numbers and using statistical sampling methods to estimate unknown quantities.

  Numerical methods play a crucial role in many fields, including engineering, physics, computer science, finance, and data analysis. They provide practical and efficient approaches to solving complex mathematical problems and enable us to obtain approximate solutions when analytical solutions are not feasible.