1. 1 Faculty of Applied Engineering and Urban Planning Civil Engineering Department ECGD3110 Numerical Analysis Lecture 11 1 st Semester 2009/2010 UoP Copyrights 2009 Solution of Linear Systems Graphical Methods Cramer's Method

2. 2 Noncomputer Methods for Solving Systems of Equations For small number of equations (n ≤ 3) linear equations can be solved readily by simple techniques such as “ method of elimination. ” Linear algebra provides the tools to solve such systems of linear equations. Nowadays, easy access to computers makes the solution of large sets of linear algebraic equations possible and practical.

3. 3 Solving Small Numbers of Equations There are many ways to solve a system of linear equations: –Graphical methods –Cramer ’ s rule –Method of elimination –Computer methods For n ≤ 3 Noncomputer Methods for Solving Systems of Equations

4. 4 Graphical Methods For two equations: Solve both equations for x 2:

5. 5 Graphical Methods

6. 6 Plot x 2 vs. x 1 on rectilinear paper, the intersection of the lines present the solution.

7. 7 ill-Condition System

8. 8 Determinants and Cramer ’ s Rule Determinant can be illustrated for a set of three equations:Determinant can be illustrated for a set of three equations: Where A is the coefficient matrix:Where A is the coefficient matrix:

9. 9 Assuming all matrices are square matrices, there is a number associated with each square matrix A called the determinant, D, of A. If [A] is order 1, then [A] has one element: A=[a 11 ] D=a 11 For a square matrix of order 2, A= the determinant is a 11 a 12 a 21 a 22 Determinants and Cramer ’ s Rule

10. 10 For a square matrix of order 3

11. 11 Cramer’s rule expresses the solution of a systems of linear equations in terms of ratios of determinants of the array of coefficients of the equations. For example, x 1 would be computed as:

12. 12 Example

13. 13

14. 14

15. 15 Exercise 15

16. 16 n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, …. F(n) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, …. Fibonacci Numbers