Computational mathematics is the study of the interaction between mathematics and calculations done by a computer.

National Science Foundation The U.S. National Science Foundation (NSF) is an Independent agencies of the United States government#Examples of independent agencies, independent agency of the Federal government of the United States, United States federal government that su ...

, Division of Mathematical Science
Program description PD 06-888 Computational Mathematics
2006. Retrieved April 2007. A large part of computational mathematics consists roughly of using mathematics for allowing and improving

computer A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic set ...

computation in areas of

science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...

and

engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...

where mathematics are useful. This involves in particular

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

design, computational complexity, numerical methods and

computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating expression (mathematics), ...

. Computational mathematics refers also to the use of computers for mathematics itself. This includes mathematical experimentation for establishing

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

s (particularly in

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

), the use of computers for proving theorems (for example the four color theorem), and the design and use of proof assistants.

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 Computational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science, and more specifically the Computer Sciences, which uses advanced computing capabilities to understand and s ...

s, also known as scientific computation or computational engineering * Systems sciences, for which directly requires the mathematical models from

Systems engineering Systems engineering is an interdisciplinary field of engineering and engineering management that focuses on how to design, integrate, and manage complex systems over their Enterprise life cycle, life cycles. At its core, systems engineering uti ...

* Solving mathematical problems by

computer simulation Computer simulation is the running of a mathematical model on a computer, the model being designed to represent the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determin ...

as opposed to traditional engineering methods. * Numerical methods used in scientific computation, for example numerical linear algebra and numerical solution of

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...

Stochastic Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; i ...

methods, such as

Monte Carlo methods Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on Resampling (statistics), repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve pr ...

and other representations of

uncertainty Uncertainty or incertitude refers to situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown, and is particularly relevant for decision ...

in scientific computation * The mathematics of scientific computation,Future Directions in Computational Mathematics, Algorithms, and Scientific Software, Report of panel chaired by R. Rheinbold, 1985
Distributed by SIAM

Mathematics of Computation ''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and Other Aids to Computation'', obtaining its current name in 1960. Articles older than f ...

Journal overview
Retrieved April 2007. in particular

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

, the theory of numerical methods * Computational complexity *

Computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating expression (mathematics), ...

and

computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...

s * Computer-assisted research in various areas of mathematics, such as

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

(

automated theorem proving Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a majo ...

discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

, and computational

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

Cryptography Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), ...

and

computer security Computer security (also cybersecurity, digital security, or information technology (IT) security) is a subdiscipline within the field of information security. It consists of the protection of computer software, systems and computer network, n ...

, which involve, in particular, research on primality testing, factorization, elliptic curves, and mathematics of

blockchain The blockchain is a distributed ledger with growing lists of Record (computer science), records (''blocks'') that are securely linked together via Cryptographic hash function, cryptographic hashes. Each block contains a cryptographic hash of th ...

Computational linguistics Computational linguistics is an interdisciplinary field concerned with the computational modelling of natural language, as well as the study of appropriate computational approaches to linguistic questions. In general, computational linguistics ...

, the use of mathematical and computer techniques in

natural language A natural language or ordinary language is a language that occurs naturally in a human community by a process of use, repetition, and change. It can take different forms, typically either a spoken language or a sign language. Natural languages ...

s * Computational algebraic geometry * Computational group theory * Computational geometry *

Computational number theory In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithm ...

* Computational topology * Computational statistics * Algorithmic information theory * Algorithmic game theory *

Mathematical economics Mathematical economics is the application of Mathematics, mathematical methods to represent theories and analyze problems in economics. Often, these Applied mathematics#Economics, applied methods are beyond simple geometry, and may include diff ...

, the use of mathematics in economics, finance and, to certain extents, of accounting. * Experimental mathematics

Journals

Journals that publish contributions from computational mathematics include * ACM Transactions on Mathematical Software *

* SIAM Journal on Scientific Computing * SIAM Journal on Numerical Analysis

References

External links

Foundations of Computational Mathematics
a non-profit organization

{{Computational science Applied mathematics Computational science

Areas of computational mathematics

Journals

See also

References

Further reading

External links