> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/900-index/980-hierarchy/classification/computational-mathematics.md).

# Computational mathematics

<https://en.wikipedia.org/wiki/Computational\\_mathematics>

Computational mathematics may refer to 2 different aspects of the relation between computing and mathematics.

**Computational applied mathematics** consists roughly of using mathematics for allowing and improving computer computation in *applied mathematics*.

Computational mathematics may also refer to the use of computers for mathematics itself.

This includes the use of computers for:

* mathematical computations (computer algebra)
* the study of what can be computerized in mathematics (effective methods)
* which computations may be done with present technology (complexity theory)
* which proofs can be done on computers (proof assistants)

Both aspects of computational mathematics involves *mathematical research* in mathematics as well as in areas of science where computing plays a central and essential role, that is, almost all sciences, and emphasize algorithms, numerical methods, and symbolic computations.

## Areas of computational mathematics

Computational mathematics emerged as a distinct part of *applied mathematics* by the early 1950s. Currently, computational mathematics can refer to or include:

* *computational science*, scientific computation, computational engineering
* solving mathematical problems by *computer simulation* as opposed to analytic methods of applied mathematics
* *numerical methods* used in scientific computation, for example numerical linear algebra and numerical solution of partial differential equations
* *stochastic methods*\*, such as Monte Carlo methods and other representations of uncertainty in scientific computation
* the mathematics of scientific computation, in particular numerical analysis, the theory of numerical methods
* *computational complexity*
* *computer algebra* and *computer algebra systems*
* *computer-assisted research* in various areas of mathematics, such as logic (automated theorem proving), discrete mathematics, combinatorics), number theory, and computational algebraic topology
* *cryptography* and *computer security*, which involve, in particular, research on primality testing, factorization, elliptic curves, and mathematics of blockchain
* *computational linguistics*, the use of mathematical and computer techniques in natural languages
* *computational algebraic geometry*
* *computational group theory*
* *computational geometry*
* *computational number theory*
* *computational topology*
* *computational statistics*
* *algorithmic information theory*
* *algorithmic game theory*
* *mathematical economics*, the use of mathematics in economics, finance and, to certain extents, of accounting.