Computational mathematics may refer to two 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 (and cannot) be computerized in mathematics (effective methods), which computations may be done with present technology (complexity theory), and 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, also known as scientific computation or 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.
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