**Computational mathematics** involves mathematical research in mathematics as well as in areas of science where computing plays a central and essential role, and emphasizes algorithms, numerical methods, and symbolic computations.^{ [1] }

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).

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,
^{ [2] }such as Monte Carlo methods and other representations of uncertainty in scientific computation - The mathematics of scientific computation,
^{ [3] }^{ [4] }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.

**Algebraic geometry** is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

**Discrete mathematics** is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

**Mathematics** includes the study of such topics as quantity, structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition.

**Analysis** is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

**Computational physics** is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science.

**Computer science** is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. One well known subject classification system for computer science is the ACM Computing Classification System devised by the Association for Computing Machinery.

An academic discipline or field of study is a branch of knowledge, taught and researched as part of higher education. A scholar's discipline is commonly defined by the university faculties and learned societies to which he/she belongs and the academic journals in which he/she publishes research.

**Theoretical computer science** (**TCS**) is a subset of general computer science that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.

**Lists of mathematics topics** cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects and reference tables. They also cover equations named after people, societies, mathematicians, journals and meta-lists.

**Computational science**, also known as **scientific computing** or **scientific computation** (**SC**), is a rapidly growing field that uses advanced computing capabilities to understand and solve complex problems. It is an area of science which spans many disciplines, but at its core, it involves the development of models and simulations to understand natural systems.

Mathematics encompasses a growing variety and depth of subjects over its history, and comprehension of it requires a system to categorize and organize these various subjects into a more general **areas of mathematics**. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different purposes they serve.

**Computational finance** is a branch of applied computer science that deals with problems of practical interest in finance. Some slightly different definitions are the study of data and algorithms currently used in finance and the mathematics of computer programs that realize financial models or systems.

**Computational science and engineering** (**CSE**) is a relatively new discipline that deals with the development and application of computational models and simulations, often coupled with high-performance computing, to solve complex physical problems arising in engineering analysis and design as well as natural phenomena. CSE has been described as the "third mode of discovery".

**Computational statistics**, or **statistical computing**, is the interface between statistics and computer science. It is the area of computational science specific to the mathematical science of statistics. This area is also developing rapidly, leading to calls that a broader concept of computing should be taught as part of general statistical education.

The **Faculty of Mathematics and Computer Science** is one of twelve faculties at the University of Heidelberg. It comprises the Institute of Mathematics, the Institute of Applied Mathematics, the School of Applied Sciences, and the Institute of Computer Science. The faculty maintains close relationships to the Interdisciplinary Center for Scientific Computing (IWR) and the Mathematics Center Heidelberg (MATCH). The first chair of mathematics was entrusted to the physician Jacob Curio in the year 1547.

The following outline is provided as an overview of and topical guide to formal science:

**Peter Bürgisser** is a Swiss mathematician and theoretical computer scientist who deals with algorithmic algebra and algebraic complexity theory.

* Complexity and Real Computation* is a book on the computational complexity theory of real computation. It studies algorithms whose inputs and outputs are real numbers, using the Blum–Shub–Smale machine as its model of computation. For instance, this theory is capable of addressing a question posed in 1991 by Roger Penrose in

- ↑ National Science Foundation, Division of Mathematical Science, Program description PD 06-888 Computational Mathematics, 2006. Retrieved April 2007.
- ↑ "NSF Seeks Proposals on Stochastic Systems". SIAM News. August 19, 2005. Archived from the original on February 5, 2012. Retrieved February 2, 2015.
- ↑ Future Directions in Computational Mathematics, Algorithms, and Scientific Software, Report of panel chaired by R. Rheinbold, 1985. Distributed by SIAM.
- ↑ Mathematics of Computation, Journal overview. Retrieved April 2007.

- Cucker, F. (2003).
*Foundations of Computational Mathematics: Special Volume*. Handbook of Numerical Analysis. North-Holland Publishing. ISBN 978-0-444-51247-5. - Harris, J. W.; Stocker, H. (1998).
*Handbook of Mathematics and Computational Science*. Springer-Verlag. ISBN 978-0-387-94746-4. - Hartmann, A.K. (2009).
*Practical Guide to Computer Simulations*. World Scientific. ISBN 978-981-283-415-7. Archived from the original on February 11, 2009. Retrieved May 3, 2012. - Nonweiler, T. R. (1986).
*Computational Mathematics: An Introduction to Numerical Approximation*. John Wiley and Sons. ISBN 978-0-470-20260-9. - Gentle, J. E. (2007).
*Foundations of Computational Science*. Springer-Verlag. ISBN 978-0-387-00450-1. - White, R. E. (2003).
*Computational Mathematics: Models, Methods, and Analysis with MATLAB*. Chapman and Hall. ISBN 978-1584883647. - Yang, X. S. (2008).
*Introduction to Computational Mathematics*. World Scientific. ISBN 978-9812818171. - Strang, G. (2007).
*Computational Science and Engineering*. Wiley. ISBN 978-0961408817.

- Foundations of Computational Mathematics, a non-profit organization
- International Journal of Computer Discovered Mathematics

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