Symbolic computation

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. 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. Software applications that perform symbolic calculations are called computer algebra systems, with the term system alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the language used for the implementation), a dedicated memory manager, a user interface for the input/output of mathematical expressions, a large set of routines to perform usual operations, like simplification of expressions, differentiation using chain rule, polynomial factorization, indefinite integration, etc. Computer algebra is widely used to experiment in mathematics and to design the formulas that are used in numerical programs. It is also used for complete scientific computations, when purely numerical methods fail, as in public key cryptography, or for some non-linear problems. Some authors distinguish computer algebra from symbolic computation using the latter name to refer to kinds of symbolic computation other than the computation with mathematical formulas. Some authors use symbolic computation for the computer science aspect of the subject and 'computer algebra' for the mathematical aspect. In some languages the name of the field is not a direct translation of its English name. Typically, it is called calcul formel in French, which means 'formal computation'. This name reflects the ties this field has with formal methods. Symbolic computation has also been referred to, in the past, as symbolic manipulation, algebraic manipulation, symbolic processing, symbolic mathematics, or symbolic algebra, but these terms, which also refer to non-computational manipulation, are no longer used in reference to computer algebra. There is no learned society that is specific to computer algebra, but this function is assumed by the special interest group of the Association for Computing Machinery named SIGSAM (Special Interest Groupon Symbolic and Algebraic Manipulation). There are several annual conferences on computer algebra, the premier being ISSAC (International Symposium on Symbolic and Algebraic Computation), which is regularly sponsored by SIGSAM. There are several journals specializing in computer algebra, the top one being Journal of Symbolic Computation founded in 1985 by Bruno Buchberger. There are also several other journals that regularly publish articles in computer algebra. As numerical software is highly efficient for approximate numerical computation, it is common, in computer algebra, to emphasize exact computation with exactly represented data. Such an exact representation implies that, even when the size of the output is small, the intermediate data generated during a computation may grow in an unpredictable way. This behavior is called expression swell. To obviate this problem, various methods are used in the representation of the data, as well as in the algorithms that manipulate them.