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 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 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 Graphical Methods For two equations: Solve both equations for x 2:
5 Graphical Methods
6 Plot x 2 vs. x 1 on rectilinear paper, the intersection of the lines present the solution.
7 ill-Condition System
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 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 For a square matrix of order 3
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 Example
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15 Exercise 15
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