Difference between revisions of "Numerical analysis"
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| − | '''Numerical analysis''' is the study of [[Algorithm|algorithms]] that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from [[discrete mathematics]]). | + | '''Numerical analysis''' is the study of [[Algorithm|algorithms]] that use numerical approximation (as opposed to [[general symbolic manipulations]]) for the [[Problem|problems]] of [[mathematical analysis]] (as distinguished from [[discrete mathematics]]). |
| − | (TO DO: expand, organize, cross-reference, illustrate.) | + | (TO DO: fix math, expand, organize, cross-reference, illustrate.) |
== Description == | == Description == | ||
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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. | 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. | ||
| − | 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 equation|differential equations]]. | + | == 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, [[Computer|computers]] [[Computation|calculate]] the required functions instead. | ||
| + | |||
| + | These same interpolation formulas nevertheless continue to be used as part of the [[software algorithms]] for solving [[Differential equation|differential equations]]. | ||
== See also == | == See also == | ||
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* [[Differential equation]] | * [[Differential equation]] | ||
* [[Mathematics]] | * [[Mathematics]] | ||
| + | * [[Newton's method]] | ||
* [[Number]] | * [[Number]] | ||
* [[Round-off error]] | * [[Round-off error]] | ||
Revision as of 07:00, 15 September 2015
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).
(TO DO: fix math, expand, organize, cross-reference, illustrate.)
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.
See also
- Computational physics
- Computer science
- Differential equation
- Mathematics
- Newton's method
- Number
- Round-off error
