Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
Contents
Description
One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection (YBC 7289), which gives a sexagesimal numerical approximation of \sqrt{2}, the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in astronomy, carpentry and construction.
Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of \sqrt{2}, modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.
Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century also the life sciences and even the arts have adopted elements of scientific computations.
- Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies)
- Numerical linear algebra is important for data analysis
- Stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology
Newton's method
See Newton's method.
History
Before the advent of modern computers, numerical methods often depended on hand interpolation in large printed tables.
Since the mid 20th century, computers calculate the required functions instead.
These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.
Numerical stability
See Numerical stability.
Signal processing
The principles of signal processing can be found in the classical numerical analysis techniques of the 17th century, according to Alan V. Oppenheim and Ronald W. Schafer.
See also
- Analysis of algorithms
- Approximation
- Computational physics
- Computational science
- Computer science
- Differential equation
- Euler method
- Finite difference method
- Function (mathematics) - a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
- List of numerical analysis topics
- Mathematics
- Newton's method
- Number
- Numerical differentiation - describes algorithms for estimating the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function.
- Numerical Recipes
- Numerical stability
- Numerical versus analytical mathematics
- Rounding
- Round-off error
- Signal processing
- Symbolic-numeric computation
