Reduction (mathematics)
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, reduction refers to the
rewriting In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a well-formed formula, formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewr ...
of an
expression Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphorical expression, a particular word, phrase, o ...
into a simpler form. For example, the process of rewriting a
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
into one with the smallest whole-number denominator possible (while keeping the numerator a whole number) is called "
reducing a fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
". Rewriting a
radical Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
(or "root") expression with the smallest possible whole number under the radical symbol is called "reducing a radical". Minimizing the number of radicals that appear underneath other radicals in an expression is called denesting radicals.


Algebra

In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, ''reduction'' refers to applying simple rules to a series of
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
s or
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
to change them into a simpler form. In the case of matrices, the process involves manipulating either the rows or the columns of the matrix and so is usually referred to as ''row-reduction'' or ''column-reduction'', respectively. Often the aim of reduction is to transform a matrix into its "row-reduced
echelon form In linear algebra, a Matrix (mathematics), matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form ...
" or "row-echelon form"; this is the goal of
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
.


Calculus

In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, ''reduction'' refers to using the technique of
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
to evaluate
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s by reducing them to simpler forms.


Static (Guyan) reduction

In dynamic analysis, static reduction refers to reducing the number of degrees of freedom. Static reduction can also be used in
finite element analysis The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
to refer to simplification of a linear algebraic problem. Since a static reduction requires several inversion steps it is an expensive matrix operation and is prone to some error in the solution. Consider the following system of linear equations in an FEA problem: :\begin K_ & K_ \\ K_ & K_ \end \begin x_1 \\ x_2 \end = \begin F_1 \\ F_2 \end where ''K'' and ''F'' are known and ''K'', ''x'' and ''F'' are divided into submatrices as shown above. If ''F''2 contains only zeros, and only ''x''1 is desired, ''K'' can be reduced to yield the following system of equations :\begin K_ \end\begin x_1 \end = \begin F_1 \end K_ is obtained by writing out the set of equations as follows: Equation () can be solved for x_2 (assuming invertibility of K_): :-K_^ K_ x_1 = x_2. And substituting into () gives :K_x_1 - K_ K_^ K_ x_1 = F_1. Thus :K_ = K_ - K_ K_^ K_. In a similar fashion, any row or column ''i'' of ''F'' with a zero value may be eliminated if the corresponding value of ''x''''i'' is not desired. A reduced ''K'' may be reduced again. As a note, since each reduction requires an inversion, and each inversion is an operation with computational cost ''O''(''n''3), most large matrices are pre-processed to reduce calculation time.


History

In the 9th century, Persian mathematician
Al-Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
's ''
Al-Jabr ''The Compendious Book on Calculation by Completion and Balancing'' ( ar, كتاب المختصر في حساب الجبر والمقابلة, ; la, Liber Algebræ et Almucabola), also known as ''Al-Jabr'' (), is an Arabic mathematical treati ...
'' introduced the fundamental concepts of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation and the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as ''al-jabr''. The name "
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
" comes from the "''al-jabr''" in the title of his book.


References

{{DEFAULTSORT:Reduction (Mathematics) Mathematical terminology Linear algebra Calculus Iranian inventions