Representing Polynomials

Transcription

1 Lab 4

2 Representing Polynomials A polynomial of nth degree looks like: a n s n +a n 1 a n a 2 s 2 +a 1 s+a 0 The coefficients a n, a n-1,, a 2, a 1, a 0 are the coefficients of decreasing powers of s. MATLAB has some powerful built-in functions to work with polynomials. To describe a polynomial to MATLAB, we enter the coefficients in an array: p = [ a n a n-1 a 2 a 1 a 0 ] If a term is missing in the polynomial, its coefficient is replaced with a zero it should not be skipped. Examples: Equation Polynomial s 2 + 5s + 6 [1 5 6] s + 1 [1 1] s 4-2s [ ] 2s 6-3s 4 [ ] Roots of an equation Once a polynomial has been represented as a vector in MATLAB, it's roots can simply be found using: r = roots(p) Example: For s 2 + 5s + 6, p = [1 5 6] r = roots(p) produces the output: r = r is a vector containing all the roots. To convert roots back into a polynomial, you can use the poly function: p = poly(r) p = which are the coefficients representing s 2 + 5s + 6.

3 Exercise 1) Find the roots of the cubic equation: s 3 4s 2 + s + 6 Use the disp() function and vector indexing r( ) to make it print the three roots in a message like this: Root 1:?, Root 2:?, Root 3:? 2) Determine the coefficients of the polynomial whose roots are: 2, 3, 4-5j, 4+5j Also type in the polynomial equation in your document (using s^ and *).

4 Symbolic Math MATLAB comes with a symbolic math toolbox which allows for solving and manipulating expressions in their everyday human readable form. For example, the equation s 2 + 5s + 6 can be entered using the symbolic math toolbox simply by: First defining 's' as a symbol, so that MATLAB does not think it is a variable. syms s Then typing out the expression that uses the symbol s. p = s^2 + 5*s + 6 Note the use of ^ to denote raising to a power, and * to represent product. Writing s2 + 5s + 6 won't work. Roots of an equation Roots of an equation defined using symbolic math can be found using the solve function: syms s p = s^2 + 5*s + 6 r = solve(p) produces the output: r = -3-2 Exercise 1) Repeat the previous exercise (1) using the symbolic math toolbox: Find the roots of the cubic equation: s 3 4s 2 + s + 6 Use the disp() function and vector indexing r( ) to make it print the three roots in a message like this: Root 1:?, Root 2:?, Root 3:?

5 Partial fractions Given a ratio of polynomials, if the numerator is represented by the vector b and the denominator by the vector a, the partial fraction expansion is given by: [r p k] = residue(b, a) For example, 4s+8 s 2 +6s+8 = b (s) a (s) = r 1 + r 2 +k(s) s p 1 s p 2 Here, b = [ -4 8 ] and a = [ ]. The residue function returns: [r p k] = residue(b, a) r = p = -4-2 k = [] Therefore, 4s+8 s 2 +6s+8 = 12 s s+2 The residue function also converts r, p, k back to the rational form when called like this: [b a] = residue(r, p, k) Using the symbolic math toolbox First declare the symbol s. syms s; Define the equation making appropriate use of parenthesis. f = (-4*s + 8)/(s^2 + 6*s + 8) Simplify the expression: >> u = simplify(f) u = 8/(s + 2) - 12/(s + 4) Depending on the expression, you may not get a partial fraction expansion. For example, it does not work for 1/(s^3-4*s^2+s+6). In that case, open MuPad: mupad Enter the following command in your notebook: partfrac(1/(s^3-4*s^2+s+6))

6 Exercise Write a script to obtain the partial fraction expansions of a) b) s 2 +5s+6 s 3 +s 2 +s+1 s 3 +s 2 +s+1 s 2 +5s+6 Type out the answer in terms of s^ in your document.

7 Anonymous functions We have written two types of programs: scripts and functions. For example, add.m was a function, and the main.m that goes with it was a script. Functions are identified by the presence of a function declaration line at the beginning of the file. We were also required to put each function in its own file. Anonymous functions are simple functions, usually one liners, that can be declared anywhere without creating a new file for them. This can be helpful when we wish to write small or temporary functions which do not merit creation of a new m-file. An anonymous function is able to access all the variables defined in its enclosing function/script, so we need to pass fewer arguments to it. Anonymous functions can be created in one line as: function_name arguments) expression; For example: clc; % clears the command window clear all; % clears the workspace of existing variables close all; % closes any open figure window num1 = 20; num2 = 5; % definition of an anonymous function called divide with two inputs divide x/y; num4 = divide(num1, num2); % this is the function call disp(['num1 = ' num2str(num1)]); disp(['num2 = ' num2str(num2)]); disp(['num4 = num1 / num2 = ' num2str(num4)]); We will be using anonymous functions to perform integrations and other computaions in the next lab. Exercise 1) Write a script that defines two variables x=20 and y=4, defines an anonymous function called multiply, then calls it to multiply them together. Follow the style given in the example above. 2) Write an anonymous function that has the input t, and returns: y=(2cos(4 π t)) 2