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Reduction (mathematics)
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==Static (Guyan) reduction== In dynamic analysis, '''static reduction''' refers to reducing the number of degrees of freedom. Static reduction can also be used in [[Finite element method|finite element analysis]] to refer to simplification of 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: :<math>\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}</math> where ''K'' and ''F'' are known and ''K'', ''x'' and ''F'' are divided into submatrices as shown above. If ''F''<sub>2</sub> contains only zeros, and only ''x''<sub>1</sub> is desired, ''K'' can be reduced to yield the following system of equations :<math>\begin{bmatrix} K_{11,\text{reduced}} \end{bmatrix}\begin{bmatrix} x_1 \end{bmatrix} = \begin{bmatrix} F_1 \end{bmatrix}</math> <math>K_{11,\text{reduced}}</math> is obtained by writing out the set of equations as follows: {{NumBlk|:|<math>K_{11}x_1 + K_{12}x_2 = F_1</math>|{{EquationRef|1}}}} {{NumBlk|:|<math>K_{21}x_1 + K_{22}x_2 = 0</math>|{{EquationRef|2}}}} Equation ({{EquationNote|2}}) can be solved for <math>x_2</math> (assuming [[invertible matrix|invertibility]] of <math>K_{22}</math>): :<math>-K_{22}^{-1} K_{21} x_1 = x_2. </math> And substituting into ({{EquationNote|1}}) gives :<math>K_{11}x_1 - K_{12} K_{22}^{-1} K_{21} x_1 = F_1.</math> Thus :<math>K_{11,\text{reduced}} = K_{11} - K_{12} K_{22}^{-1} K_{21}.</math> In a similar fashion, any row or column ''i'' of ''F'' with a zero value may be eliminated if the corresponding value of ''x''<sub>''i''</sub> is not desired. A reduced ''K'' may be reduced again. As a note, since each reduction requires an inversion, and each inversion is an operation with computational cost [[Big O notation|''O''(''n''<sup>3</sup>)]], most large matrices are pre-processed to reduce calculation time.
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