Reduction (mathematics)
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

 into one with the smallest whole-number denominator possible (while keeping the numerator an integer) is called "reducing a fraction". Rewriting a radical
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

 (or "root") expression with the smallest possible whole number under the radical symbol is called "reducing a radical".

In linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, reduction refers to applying simple rules to a series of equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

s or matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 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" or "row-echelon form"; this is the goal of Gaussian elimination
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...

.

In calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, reduction refers to using the technique of integration by parts
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...

 to evaluate a whole class of integrals 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 FEA
FEA
Fea was the smallest unit of land a fief could own under the feudal system.-Acronyms and abbreviations:FEA is an acronym or abbreviation for:...

analysis to simplify 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


Where K and F are known and K, x and F are divided into submatrices as shown above. If F2 contains only zeros, and only x1 is desired, K can be reduced to yield the following system of equations


K11,reduced is obtained by writing out the set of equations as follows



Equation (2) can be rearranged


And substituting into (1)


In matrix form


And


In a similar fashion, any row/column i of F with a zero value may be eliminated if the corresponding value of xi is not desired. A reduced K may be reduced again. As a note, since each reduction requires an inversion, and each inversion is a n3 most large matrices are pre-processed to reduce calculation time.
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