Linear Equation Systems Direct Methods

Transcription

1 Linear Equation Systems Direct Methods

2 Content Solution of Systems of Linear Equations Systems of Equations Inverse of a Matrix Cramer s Rule Gauss Jordan Method Montante Method

3 Solution of Systems of Linear Equations The methods for solving systems of linear equations are: Direct methods: Inverse of a Matrix Cramer Method Gauss Jordan Montante Iterative methods: Jacobi Gauss Seidel

4 Systems of Equations Representation of a system of equations by matrices:

5 Inverse of a Matrix Example: Solve the system of equations using the inverse of the matrix:

6 Inverse of a Matrix Solution: From the example:

7 Cramer s Rule They are two linear equations with two unknowns, the solution using Cramer s rule is:

8 Cramer s Rule They are three linear equations in three unknowns, the solution using the Cramer s rule is: Cramer's rule can be generalized for a linear system

9 Cramer s Rule Example:

10 Gauss Jordan Method Augmented matrix (increased) To represent a linear system may be used a matrix of building first

11 Gauss Jordan Method and then combining these matrices to form the augmented matrix:

12 Gauss Jordan Method For a matrix of order the method consists of steps of calculating. The th step of calculating consists of two types of row operations: 1. Normalize the line with respect to element multiply the row by

13 Gauss Jordan Method Do column zeros, except the element for all Note. In the kth calculation step, before normalizing, it will search downward (i> k) an element aik that is greater than all of that column and is also greater than akk (in absolute value). If the item exists in a row i, then iand k rows are exchanged. If the last akk = 0, then the matrix is zero and there is not solution. This type of system of equations is called inconsistent. In cases where the system has a solution, It is known as consisting equation system.

14 Gauss Jordan Method Example: Solve the following systems of equations by the method of Gauss Jordan

15 Gauss Jordan Method Solution:

16 Montante Method Is a closed method that consists of steps of calculating and each calculation step consists of 4 operations. The th calculation step consists of: a) The line k is unchanged : b) Matching elements of the main diagonal to following restriction: element, but with the Note. The method was discovered in 1973 by René Mario Pardo Montante, graduated from the Faculty of Mechanical and Electrical Engineering (UANL).

17 Montante Method c) Do zeros elements of column except? d) The remaining elements are evaluated using the following recursion formula:

18 Montante Method Example Solve the following systems of equations by the Montante method:

19 Montante Method

20 Recursion Formulas for Linear Equations Systems Inverse of a Matrix 1. Get the inverse matrix of the coefficient matrix. Use the recursion formula of the inverse of a matrix. 2. Multiply the inverse matrix by the matrix of constants to get the values of the variable matrix. Use the recursion formula of matrix multiplication.

21 Recursion Formulas for Linear Equations Systems Cramer s Rule 1. Get the determinant of the coefficient matrix. Use the recursion formula for determinants. 2. Column 1 is replacing by the values of the constants and get its determinant. 3. Get the value of the first variable. 4. Column 2 is replacing by the values of the constants and obtain determinants. 5. Get the value of the second variable. 6. Repeat for the other variables.

22 Recursion Formulas for Linear Equations Systems Gauss Jordan Method Using row operations, the recursion formula of each stage, are: Note that the range may be, instead of the range, because when. This change saves time in the calculation for not recalculate elements were made zero in the previous stages.

23 Recursion Formulas for Linear Equations Systems Montante Method Observe recursion formulas of the section where the procedure of this method is explained.

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28 Linear Equation Systems Direct Methods