Computational mathematics

A black and white rendition of the Yale Babylonian Collection's Tablet YBC 7289 (c. 1800–1600 BCE), showing a Babylonian approximation to the square root of 2 (1 24 51 10 w: sexagesimal) in the context of Pythagoras' Theorem for an isosceles triangle. The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888.

Computational mathematics may refer to two different aspect 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).

Computational applied mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic computations. Computation in research is prominent.^[1] 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, for example stochastic finite elements
• the mathematics of scientific computation^[3] (the theoretical side involving mathematical proofs^[4]), in particular numerical analysis, the theory of numerical methods (but theory of computation and complexity of algorithms belong to theoretical computer science)
• symbolic computation and computer algebra systems
• computer-assisted research in various areas of mathematics, such as logic (automated theorem proving), discrete mathematics (search for mathematical structures such as groups), number theory (primality testing and factorization), cryptography, and computational algebraic topology
• 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
• use of mathematics in economics, finance and to certain extents of accounting i.e. use of differential and integral calculus(newton's method) and financial maths to solve real life problems.

References

Further reading

• 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.
• 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.

External links

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