- A numerical method is a complete and definite set of procedures for the solution of a problem, together with computable error estimates. Various computational techniques are used to find the solution of a problem.
- The study and implementation of such methods is the field of numerical analysis or numerical methods. In this article different methods used in Numerical analysis are explained in brief.
Computing Values of Functions
- One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula. It is sometimes not very efficient.
- For example, Horner scheme, since it reduces the necessary number of multiplications and additions.
Solution of equations / Finding roots of equations
- Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not.
- Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero).
- For example, Bisection method, Regula-falsi method (False position method), Secant method, Newton-Raphson method, Iteration method (Successive approximation method) are some of the methods which are used to find the root of different polynomial equations or transcendental equations.
Solving Systems of Equations
- Much effort has been put in the development of methods for solving systems of linear equations.
- Standard direct methods, i.e., methods that use some matrix decompositions are Gaussian elimination, LU decomposition, Cholesky decomposition
- Iterative methods such as the Jacobi method, Gauss–Seidel method are usually preferred for large systems.
Interpolation and Extrapolation
- Interpolation is the technique of determining the values of the functions for any intermediate point when the values of two points are given.
- For example, Observing the population of city if the population of 1991,2001 and 2011 is known and if we want to calculate the population for 1995 or 2008 i.e. any population in between 1991 to 2011.
- Following are various interpolation methods used in numerical analysis.
- Polynomial interpolation
a) Vandermonde polynomial
b) Lagrange polynomial
c) Newton polynomial
2. Piecewise linear interpolation
3. Cubic spline interpolation
4. Multidimensional interpolation
- Extrapolation is very similar to interpolation, except that now the value of the unknown function at a point which is outside the given points must be found.
- For example, Observing the population of city if the population of 1991,2001 and 2011 is known and if we want to calculate the population for 1990 or 2015 i.e., any population outside of 1991 to 2011.
Regression
- Regression is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), the unknown function can be found.
- The least squares-method is one way to achieve this. Least-square method is commonly used to fit a parameterized curve to experimental data.
- In interpolation the curve must pass through the data points. In general, the fitting curve is not expected to pass through the data points, making this problem substantially different from the one of interpolation.
Optimization
- Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints.
- For example, say mango juice is sold at a stand, at Rs.40. If 150 glasses of can be sold per day, giving the profit of Rs. 15 on each glass so overall profit of Rs.2250 per day. But for the price of Rs.50 only 80 glasses can be sold giving overall profit Rs.2000 per day. Here due to constraint of charge profit will be less.
- Linear programming deals with the case that both the objective function and the constraints are linear.
- A famous method in linear programming is the simplex method.
- The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.
Numerical Integration
- Numerical integration is used to evaluate a definite integral when there is no closed-form expression for the integral or when the explicit function is not known and the data is available in tabular form only.
- Numerical integration (or quadrature) consists of methods to find the approximate area under the graph of the function f(x) between two x-values.
- Techniques such as Trapezoid rule, Monte-Carlo Techniques, Simpson rule are used to for the calculation of numerical integrations.
Numerical Differentiation
- Numerical differentiation is the process of finding the numerical value of a derivative of a given function at a given point.
- There are many applications where derivatives need to be computed numerically. The simplest approach simply uses the definition of the derivative.
Derivative is defined as,
- For example, Euler’s method, Modified Euler’s method, Taylor’s series, Runge-Kutta method are used to calculate numerical differentiation.
Solution of Ordinary Differential equations (ODE)
- Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations.
- For solution of differential equations Euler method, Runge-Kutta methods, Finite difference method, Shooting method are used.
