Upcoming SlideShare
Loading in …5
×

# Chapter 4

1,772 views

Published on

0 Comments
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
Your message goes here
• Be the first to comment

• Be the first to like this

No Downloads
Views
Total views
1,772
On SlideShare
0
From Embeds
0
Number of Embeds
60
Actions
Shares
0
Downloads
36
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Chapter 4

1. 1. CHAPTER 4: Direct Methods for Solving Linear Equations Systems Erika Villarreal
2. 2. Here comes your footer  Page
3. 3. Here comes your footer  Page 3 planes intersect at a point part 2 of three illustrating  secret sharing  this version has added emphasis of the point created in  LightWave  by  stib
4. 4. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 1. Gauss Elimination For simplicity, we assume that the coefficient matrix A in Eq. (2.0.1) is a nonsingular 3 ×3 matrix with M = N = 3. Then we can write the equation as
5. 5. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 1. Gauss Elimination
6. 6. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 1. Gauss Elimination
7. 7. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 1. Gauss Elimination
8. 8. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 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 Form A matrix is in echelon form if it has leading ones on the main diagonal and zeros below the leading ones. Here are some examples of matrices that are in echelon form. 2. Reduced Echelon Form A 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.
9. 9. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 2. Gauss-Jordan Elimination
10. 10. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 2. Gauss-Jordan Elimination Gaussian 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
11. 11. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 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].
12. 12. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 3. LU DECOMPOSITION STEPS TO FINDING THE UPPER TRIANGULAR MATRIX (MATRIX [U]) That is: - + Factor * pivot position to change
13. 13. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 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 &quot;factor&quot; described above and are located all the &quot;factors&quot; 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.
14. 14. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 3. LU DECOMPOSITION STEPS TO SOLVE A SYSTEM OF EQUATIONS BY THE LU DECOMPOSITION METHOD
15. 15. Here comes your footer  Page Software and Tools for Microsoft PowerPoint. The website with innovative solutions. Save time and money by automating your presentations. www.presentationpoint.com <ul><li>http://www.utdallas.edu/dept/abp/PDF_Files/LinearAlgebra_Folder/Gauss_Jordan.pdf </li></ul><ul><li>http://www.monografias.com/trabajos45/descomposicion-lu/descomposicion-lu.shtml </li></ul><ul><li>C. Chapra, S.; P. Canale, R.  Métodos Numéricos para Ingenieros.  (3ª ed.). McGrawHill. </li></ul><ul><li>Factorización LU.  Wikipedia.  Extraído el 22 Enero, 2007, de </li></ul><ul><li>  http ://es.wikipedia.org/wiki/Factorizaci%C3%B3n_LU </li></ul><ul><li>MÉTOTO DE GAUSS-SEIDEL.  Universidad Autónoma de Ciudad Juárez (UACJ).  Extraído el 22 Enero, 2007, </li></ul><ul><li>http://docentes.uacj.mx/gtapia/AN/Unidad3/Seidel/SEIDEL.htm </li></ul>BIBLIOGRAPY