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In Linear Algebra , ''reduction'' refers to applying simple rules to a series of Equation s 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 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 ''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.