SYSTEMS OF EQUATIONS

General Purpose

Introduction

CONCEPT OF SYSTEM OF LINEAR EQUATIONS.System of m equations with n unknowns. It is a set of algebraic expressions of the form

xj, are the unknowns , (j = 1,2 ,..., n).aij, are the coefficients (i = 1,2 ,..., m) (j = 1,2 ,..., n).ci, are independent terms, (i = 1,2 ,..., m).

When n is small, it is usual to designate the unknown with the letters x, y, z, t, ...Note that the number of equations need not be equal to the number of Incognico ites.When ci = 0 for all i, the system is called homogeneous.

3 is a system of linear equations with four unknowns.The coefficients of the first equation of the system are the numbers 3, -2, 1, -1.The term itself is independent of 2.

graphic methods

The Gauss we just saw is simple and effective way of solving a system of linear equations. But it hasa drawback. If a system of 300 unknowns we are interested only in July, following the method of Gauss, we wouldfollow the process of triangulation as if we were interested all of them.Cramer's rule, now we shall see, cleverly uses the properties of matrices and determinantsclear, separate mind, any one of the unknowns of an equation system is linear.

CRAMER'S RULE.

Cramer System. It is a system in which: m = n and 0 * A * .... That is: The matrix A is square and regular.In this case, A has an inverse A-1, so multiplying [2] on the left by A -1:

CRAMER'S RULE.

which are the Cramer formulas, which are reflected in the following rule:Cramer's rule. The value of the unknown in a system xj Cramer is a fraction whose numerator is adetermining which ob is to replace the column by the column j are the independent terms, and whosedenominator is * A *.

Example

Solve the system: Then, Cramer is a system.

Therefore, the solution of the system is

Solve the system: Then, Cramer is a system.

Therefore, the solution of the system is:

disposal methods

METHOD OF DISPOSAL OF GAUSS-JORDAN

Is the method of solving systems of equationslinear, which is to arrive at a "staggered" by transforming the expanded matrix in a phased arrayby rows.The following examples explain in detail the process forward.

Solve the system:We consider the augmented matrix associated with the system, spreading a little column of the independent terms

Then, the system has been as follows:

Solving the equations, beginning with the last is:

This is a compatible system determined.

Solve the system: The augmented matrix is,

Exchanging the first row with the third is:

Then, the system has been as follows:

Solving the last equation, z = 1 +2 t; if we do t = ", is: z = 1 +2

System solutions are giving arbitrary values to the parameter. "It is a compatible system undetermined.

Solve the system: The augmented matrix is,

We exchanged the first two rows is:

After the system has been as follows:It is noted that the system is incompatible.

Method Thomas

The tridiagonal matrix algorithm (TDMA), also known as the Thomas algorithm, is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system may be written as

The following algorithm performs the TDMA, overwriting the original arrays. In some situations this is not desirable, so some prefer to copy the original arrays beforehand.Forward elimination phaseFor k = 2 step until n do

Variants

In some situations, particularly those involving periodic boundary conditions, a slightly perturbed form of the tridiagonal system may need to be solved

In this case, we can make use of the Sherman-Morrison formula to avoid the additional operations of Gaussian elimination and still use the Thomas algorithm. We will now solve

bibliography.

http://www.campusoei.org/cursos/centrocima/matematica/si_ec_li.pdf

http://www.slideshare.net/andrespla/lu-decomposition-gauss-jordan-thomas-4830213?from=ss_embed