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Solving Nonlinear Equation(s) in MATLAB
1 Introduction
This tutorial helps you use MATLAB to solve nonlinear algebraic equations of single ormultiple variables.
2 Writing MATLAB functions
In order to use the MATLAB solvers, you must first be able to write MATLAB functions.There are two different methods to create a function - (a) inline command, and (b) Matlab editor
2.1 The โ inline โ command
The inline command can be used for simple, one-line functions. For example, to create f ( x ) = x (^3) - 5 x (^2) -x + 2 :
>> f = inline (โ x^ 3 -5x^2 - x + 2โ) _f = Inline function: f(x) = x^3-5x^2-x+_ You can now evaluate the function value at any given x. For example, to evaluate the function value at x = 4, simply type โ f (4)โ at Matlab command line. EDU>> f(4) ans =
-
2.2 The MATLAB editor
The editor allows the user to write functions of any length and/or complexity.
- Set the current working directory to your diskspace e.g. โc:\CHEE222\Matlab\Iamhappy\Temp\โ
- (a) type \โedit funโ at the command prompt - enter yes to create file ALTERNATIVELY (b) go to โFileโ, select โNewโ; select โM-Fileโ. type the following: function y = fun ( x ) y = x^3 - 5 x^2 -x + 2 ; NOTE have chosen โ: The filename and the function name should be the same. In the previous example, we fun โ as the filename and the function name.
- Save the file as โ fun.m โ in the working directory
- MATLAB function FZERO fzero can be used to solve a single variable nonlinear equation of the form f ( x ) = 0. The equation must first be programmed as a function (either inline or m-file).
3.1 Using FZERO for a function defined by inline command
The following command solves the equationinitial guess of x = 4. y = f ( x ) = x^3 - 5 x^2 -x + 2 ;, starting from an
EDU>> fzero(f,4) MATLAB returns the answer: ans = 5. Changing the initial guess to x = 2 EDU>> fzero(f,2) gives ans =
f 2 ( x 1 , x 2 ) = 2 x 2 โ x^22 โ 3 x 1 x 2
f 1 ( x 1 , x 2 ) = x 1 โ 4 x 12 โ x 1 x 2
xo = [ 1 1 ]^ '
NOTE: In utilizing ROOTS function, all coefficients of the polynomial must bespecified.
e.g. f ( x ) = x^4 - 3 x^2 + 2. The function in the full polynomial form must be expressed as: f ( x ) = 1w x^4 - 0wx^3 +3w x^2 -0 w x + 2. Accordingly, the polynomial must be defined in MATLAB as follows: p = [1 0 -3 0 2]:
5 FSOLVE The MATLAB routine fsolve is used to solve sets of nonlinear algebraic equations usinga quasi-Newton method. The user must supply a routine to evaluate the function vector. Consider the following system of nonlinear equations, and solve for x 1 and x 2 :
The m-file used to solve the above problem using fsolve is:
which is placed in a m-file called nle.m. Enter the initial guess
Note: x (^) o is the TRANSPOSE of a row vector Now, solve with x = fsolve ( โ nleโ ; x 0 )
which gives us the results x = [0_._ 25 0_._ 00] โ.
