Difference between revisions of "Numerical analysis"

Revision as of 16:19, 27 April 2016

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).

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

• Approximation
• Computational physics
• Computer science
• Differential equation
• Euler method
• Finite difference method
• Mathematics
• Newton's method
• Number
• Numerical stability
• Rounding
• Round-off error
• Signal processing

External links

• Numerical analysis