MATLAB: Simultaneous Equations

1. MATLAB: Simultaneous Equations Lecture 22 Winter Quarter

2. Simultaneous Equations with MATLAB • MATLAB is very powerful for solving systems of N equations with N unknowns. It is used extensively for analyzing electric power grids and circuits, fluid flow in piping systems, and other engineering applications. It can, of course, be used for any application with n equations and n unknowns. • For example, suppose that a Calculus text (C), a Physics text (P) and a Graphics text (G) can be purchased for $146. Then suppose the Calculus and Graphics texts can be purchased for $87 and the Physics and Calculus texts can be purchased for $105. How much does each text cost? Winter Quarter

3. The problem can be written as: 1P + 1C + 1G = 146 1C + 1G = 87 1P + 1C = 105 In MATLAB, the coefficients of the variables P, C & G are placed in a matrix and the Dollar values in a column vector. >>Books=[1,1,1;0,1,1;1,1,0] Books = 1 1 1 0 1 1 1 1 0 >>Dollars=[146;87;105] Dollars = 146 87 105 Simultaneous Equations with MATLAB Winter Quarter

4. Simultaneous Equations with MATLAB • Now, the inverse of the matrix Books is multiplied by the vector Dollars: >>Cost = inv (Books) * Dollars Cost = 59 46 41 • This vector contains the cost of P in row one, C in row 2 and G in row 3. Winter Quarter

5. Simultaneous Equations with MATLAB • Another MATLAB solution would be to use the reverse division sign as follows: >>Cost = Books \ Dollars Cost = 59 46 41 Winter Quarter

6. Simultaneous Equations with MATLAB • A third MATLAB solution can be obtained by using the Row Reduced Echelon Form (rref) function as follows. First combine the matrix and vector: EDU» COMBINED=[Books Dollars] COMBINED = 1 1 1 146 0 1 1 87 1 1 0 105 Winter Quarter

7. Simultaneous Equations with MATLAB • Then invoke the rref function with the resulting matrix. >> Cost = rref (COMBINED) Cost = 1 0 0 59 0 1 0 46 0 0 1 41 • The last column in the new matrix is the cost vector. Winter Quarter