CHAPTER 4: Direct Methods for Solving Linear Equations Systems

Erika Villarreal

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SOLVING A SYSTEM OF LINEAR EQUATIONS

1. Gauss Elimination

For simplicity, we assume that the coefficient matrix A in Eq. (2.0.1) is a nonsingular3 ×3 matrix with M = N = 3. Then we can write the equation as

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SOLVING A SYSTEM OF LINEAR EQUATIONS

1. Gauss Elimination

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SOLVING A SYSTEM OF LINEAR EQUATIONS

1. Gauss Elimination

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SOLVING A SYSTEM OF LINEAR EQUATIONS

1. Gauss Elimination

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SOLVING A SYSTEM OF LINEAR EQUATIONS

2. Gauss-Jordan Elimination

We can use this technique to determine if the system has a unique solution, infinite solutions, or no solution.

Echelon Form and Reduced Echelon Form:

1. Echelon FormA matrix is in echelon form if it has leading ones on the main diagonal andzeros below the leading ones. Here are some examples of matrices that are in echelon form.

2. Reduced Echelon FormA matrix is in reduced echelon form if it has leading ones on the main diagonal and zeros above and below the leading ones. Here are some examples of matrices that are in reduced echelon form.

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SOLVING A SYSTEM OF LINEAR EQUATIONS

2. Gauss-Jordan Elimination

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SOLVING A SYSTEM OF LINEAR EQUATIONS

2. Gauss-Jordan EliminationGaussian Elimination: Gauss-Jordan Elimination:

Gaussian Elimination puts a matrix in echelon form. Example: Solve the system by using Gaussian Elimination.

1. Put the matrix in augmented matrix form.

2. Use row operations to put the matrix in echelon form

3.Write the equations form the echelon form matrix and solve the equations.

Gauss-Jordan Elimination puts a matrix in reduced echelon form.Example: Solve the system by using Gauss-Jordan Elimination.

1. Put the matrix in augmented matrix form.

2. Use row operations to put the matrix in echelon form

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SOLVING A SYSTEM OF LINEAR EQUATIONS

3. LU DECOMPOSITION

It shows how an original matrix is decomposed into two triangular matrices, an upper and lower.

LU decomposition involves only operations on the coefficient matrix [A], providing an efficient means to calculate the inverse matrix or solving systems of linear algebra.

The first step is to break down or transform [A] [L] and [U], ie to obtain the lower triangular matrix [L] and the upper triangular matrix [U].

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SOLVING A SYSTEM OF LINEAR EQUATIONS

3. LU DECOMPOSITION

STEPS TO FINDING THE UPPER TRIANGULAR MATRIX (MATRIX [U])

That is:- + Factor * pivot position to change

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SOLVING A SYSTEM OF LINEAR EQUATIONS

3. LU DECOMPOSITION

STEPS TO FINDING THE LOWER TRIANGULAR MATRIX (MATRIX [L])

To find the lower triangular matrix seeks to zero values above each pivot, as well as become an every pivot. It uses the same concept of "factor" described above and are located all the "factors" below the diagonal as appropriate for each.

Schematically they look for the following:

  

Originally it was:

Because [A] = [T] [U] to find [L] and [U] from [A] not alter the equation and have the following:

Therefore, if Ax = b, then Lux = b, so that Ax = LUX = b.

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SOLVING A SYSTEM OF LINEAR EQUATIONS

3. LU DECOMPOSITIONSTEPS TO SOLVE A SYSTEM OF EQUATIONS BY THE LU DECOMPOSITION METHOD

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• http://www.utdallas.edu/dept/abp/PDF_Files/LinearAlgebra_Folder/Gauss_Jordan.pdf

• http://www.monografias.com/trabajos45/descomposicion-lu/descomposicion-lu.shtml

• C. Chapra, S.; P. Canale, R. Métodos Numéricos para Ingenieros. (3ª ed.). McGrawHill.

• Factorización LU. Wikipedia. Extraído el 22 Enero, 2007, de

•  http://es.wikipedia.org/wiki/Factorizaci%C3%B3n_LU

• MÉTOTO DE GAUSS-SEIDEL. Universidad Autónoma de Ciudad Juárez (UACJ). Extraído el 22 Enero, 2007,

• http://docentes.uacj.mx/gtapia/AN/Unidad3/Seidel/SEIDEL.htm

BIBLIOGRAPY