Reduction (mathematics)

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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".

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.

[edit] 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 a 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 + 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 n³ most large matrices are pre-processed to reduce calculation time.