Reduction (mathematics)

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".

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.

tatic (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

:egin{bmatrix}K_{11} & K_{12} \K_{21} & K_{22}end{bmatrix}egin{bmatrix}x_{1} \x_{2}end{bmatrix}=egin{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

:egin{bmatrix}K_{11,reduced}end{bmatrix}egin{bmatrix}x_{1}end{bmatrix}=egin{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

:egin{bmatrix}K_{11}-K_{12}K_{22}^{-1}K_{21}end{bmatrix}egin{bmatrix}x_{1}end{bmatrix}=egin{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.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Reduction — Reduction, reduced, or reduce may refer to:cienceChemistry*Reduction – chemical reaction in which atoms have their oxidation number (oxidation state) changed. **Reduced gas – a gas with a low oxidation number **Ore reduction: see… …   Wikipedia

  • Reduction of order — is a technique in mathematics for solving second order ordinary differential equations. It is employed when one solution y 1(x) is known and a second linearly independent solution y 2(x) is desired.A Simple ExampleConsider the general second… …   Wikipedia

  • mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… …   Universalium

  • Mathematics of Sudoku — The class of Sudoku puzzles consists of a partially completed row column grid of cells partitioned into N regions each of size N cells, to be filled in using a prescribed set of N distinct symbols (typically the numbers {1, ..., N}), so that each …   Wikipedia

  • Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics       Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity.       Computer scientist Manindra Agrawal of the… …   Universalium

  • mathematics, East Asian — Introduction       the discipline of mathematics as it developed in China and Japan.       When speaking of mathematics in East Asia, it is necessary to take into account China, Japan, Korea, and Vietnam as a whole. At a very early time in their… …   Universalium

  • Reduction of the structure group — In mathematics, in particular the theory of principal bundles, one can ask if a G bundle comes from a subgroup H < G. This is called reduction of the structure group (to H), and makes sense for any map H o G, which need not be an inclusion… …   Wikipedia

  • Reduction system — In mathematics, a reduction system is a system where terms can be re written by using a finte list of rewriting rules.Examples of reduction systems include string rewriting systems, term rewriting systems, lambda calculus under lambda conversion …   Wikipedia

  • reduction of fractions — expression of fractions in their simplest forms (Mathematics) …   English contemporary dictionary

  • List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”