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*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).

A **computer** is a machine that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming. Modern computers have the ability to follow generalized sets of operations, called *programs.* These programs enable computers to perform an extremely wide range of tasks. A "complete" computer including the hardware, the operating system, and peripheral equipment required and used for "full" operation can be referred to as a **computer system**. This term may as well be used for a group of computers that are connected and work together, in particular a computer network or computer cluster.

**Applied mathematics** is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.

In computational mathematics, **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 mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes *exact* computation with expressions containing variables that have no given value and are manipulated as symbols.

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.^{ [1] }

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

**Computation** is any type of calculation that includes both arithmetical and non-arithmetical steps and follows a well-defined model, for example an algorithm.

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.

**Computational science** is a rapidly growing multidisciplinary 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.

*Not to be confused with computer engineering.*

**Computer simulation** is the reproduction of the behavior of a system using a computer to simulate the outcomes of a mathematical model associated with said system. Since they allow to check the reliability of chosen mathematical models, computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics, astrophysics, climatology, chemistry, biology and manufacturing, as well as human systems in economics, psychology, social science, health care and engineering. Simulation of a system is represented as the running of the system's model. It can be used to explore and gain new insights into new technology and to estimate the performance of systems too complex for analytical solutions.

**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.

**Mathematical 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.

The following outline is provided as an overview of and topical guide to academic disciplines:

**Theoretical computer science** (**TCS**) is a subset of general computer science and mathematics that focuses on more mathematical topics of computing and includes the theory of computation.

This article itemizes the various **lists** of **mathematics topics**. 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.

Mathematics encompasses a growing variety and depth of subjects over history, and comprehension requires a system to categorize and organize the many subjects into 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. In addition, as mathematics continues to be developed, these classification schemes must change as well to account for newly created areas or newly discovered links between different areas. Classification is made more difficult by some subjects, often the most active, which straddle the boundary between different areas.

**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 mechanics** is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science as a "third way" besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science.

**Kurt Mehlhorn** is a German theoretical computer scientist. He has been a vice president of the Max Planck Society and is director of the Max Planck Institute for Computer Science.

**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:

This is a glossary of terms that are or have been considered areas of study in mathematics.

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

- ↑ 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. 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.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

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

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