### 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".

## 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 *F*_{2} contains only zeros, and only *x*_{1} 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}+K_{12}x_{2}=F_{1}

- K_{21}x_{1}+K_{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 *n*^{3} most large matrices are pre-processed to reduce calculation time.