Reduction (mathematics)

In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible (while keeping the numerator an integer) is called "reducing a fraction". Rewriting a radical (or "root") expression with the smallest possible whole number under the radical symbol is called "reducing a radical".

There are many different procedures, operations, and systems relating to reduction in mathematics. An incomplete list of these can be found in the article on reduction.

Algebra

In linear algebra, reduction refers to applying simple rules to a series of equations or matrices 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.

In calculus, reduction refers to using the technique of integration by parts 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 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

$\begin{bmatrix} K_{11} & K_{12} \\ K_{21} & K_{22} \end{bmatrix}\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}=\begin{bmatrix} F_{1} \\ F_{2} \end{bmatrix}$

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

$\begin{bmatrix} K_{11,reduced} \end{bmatrix}\begin{bmatrix} x_{1} \end{bmatrix}=\begin{bmatrix} F_{1} \end{bmatrix}$

K_11,reduced is obtained by writing out the set of equations as follows

$K_{11}x_{1}%2BK_{12}x_{2}=F_{1}$

$K_{21}x_{1}%2BK_{22}x_{2}=0$

Equation (2) can be rearranged

$-K_{22}^{-1}K_{21}x_{1}=x_{2}$

And substituting into (1)

$K_{11}x_{1}-K_{12}K_{22}^{-1}K_{21}x_{1}=F_{1}$

In matrix form

$\begin{bmatrix} K_{11}-K_{12}K_{22}^{-1}K_{21} \end{bmatrix}\begin{bmatrix} x_{1} \end{bmatrix}=\begin{bmatrix} F_{1} \end{bmatrix}$

And

$K_{11,reduced}=K_{11}-K_{12}K_{22}^{-1}K_{21}$

In a similar fashion, any row/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 a n³ most large matrices are pre-processed to reduce calculation time.