Numerical analysis: Difference between revisions

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The numerical point of view goes back to the earliest mathematical writings. A tablet from the [[Yale Babylonian Collection]] ([[YBC 7289]]), gives a [[sexagesimal]] numerical approximation of the [[square root of 2]], the length of the [[diagonal]] in a [[unit square]].

Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used.Numerical methods are techniques used to solve mathematical problems that are difficult or impossible to solve analytically (exactly). These methods provide approximate solutions by performing numerical calculations. Common numerical methods include:

1. **Root-Finding Methods**: Techniques like the bisection method, Newton-Raphson method, and secant method are used to find the roots (solutions) of equations.

2. **Numerical Integration**: Methods like the trapezoidal rule, Simpson's rule, and Gaussian quadrature approximate the integral of a function.

3. **Numerical Differentiation**: Techniques to estimate the derivative of a function based on discrete data points.

4. **Solving Linear Systems**: Methods such as Gaussian elimination, LU decomposition, and iterative methods like Jacobi and Gauss-Seidel are used to solve systems of linear equations.

5. **Optimization**: Techniques like gradient descent and the simplex method are used to find the maximum or minimum of functions.

6. **Solving Differential Equations**: Methods like Euler's method, Runge-Kutta methods, and finite difference methods are used to approximate solutions to ordinary and partial differential equations.

Numerical methods are widely used in engineering, physics, finance, and other fields where exact solutions are difficult to obtain.

==Applications==