This document discusses several MATLAB functions for working with polynomials, including: - Using polyfit to find the polynomial that best fits a set of data points - Evaluating polynomials with polyval - Storing and running MATLAB code from text files called M-files - Performing operations on polynomials like multiplication with conv and division with deconv

1. EngMahmoud Hussein

2. APPLICATION

3. Solving linear equation
 Suppose that you would like to solve the following system of equations:
 2X+Y-Z=6
 X-Y-Z=-3
 X+2Y-3Z=-9
 In matrix form, you can write:

4. Solving linear equation

5. Quiz
• Write a vector start from zero and increment with 5 and end with 43.
• Write a vector with the following sequence[1 3 5 7 9 13 17 21 -2 -5 -7 -9]
• Create a vector and sort it.
• Create matrix [1 2 3;4 5 6;7 9 8] and delete second row.
• Solve the following equations
 2X+Y-Z=6
 X-Y-Z=-3
 X+2Y-3Z=-9
 Write a vector [1 2 3] and get the transpose.

6. Curve fitting
 Given a set of Inputs and outputs for a particular system; MATLAB can
find a polynomial function the fit the output to input.
 This is done using the polyfit instruction.
 p = polyfit(x,y,n) 􀃆 finds the coefficients of a polynomial p(x) of degree
n that fits the data, p(x(i)) to y(i), in a least squares sense.

7. Curve fitting
 Polynomial evaluation.
y = polyval(p,x)
Returns the value of a polynomial of degree n evaluated at x.
The input argument p is a vector of length n+1 whose elements are
the coefficients in descending powers of the polynomial to be
evaluated.

8. Curve fitting
 Polynomial evaluation.􀃆
Example:

9. Curve fitting
Example
p=polyfit(month,temp,2);

10. Curve fitting
 Graphical comparison
>>month=1:1:12;
 >>temp=[17 19 22 24 27 30 34 35 31 26 23 20];
 >>n=2;
 >>p=polyfit(month,temp,n);
 >>x=1:0.1:12;
 >>y=polyval(p,x);
 >>plot(month,temp,’r’,x,y,’g’)

11. Curve fitting

12. M-Files
• Text files containing MATLAB code.
• Useful for complex tasks.
 we can store the typed input into a file and
 tell MATLAB to get its input from that file.
 Extension: .m
 We can put comments on program to improve program quality and
enable to another programmer to edit.

13. M-Files
 If an m-􀃆le contains MATLAB statements just as you would type them
into MATLAB, theyare called scripts.
 M-􀃆les can also accept input and produce output, in which case they
are called functions.
 The MATLAB executes the instructions, just as if you had typed them in
the command window.

14. Polynomials
 Sometimes you need to:
Write a polynomial,
Find its roots,
Multiply it by another one,
Divide it by another polynomial,
Differentiate it,
Integrate it,
Substitute by a value in it, or
Put it in partial fractions form.

15. Polynomials
 MATLAB provides us with a complete set to handle polynomials.
 Polynomials are identified by their coefficients or by its roots.
 N th order polynomial is defined by a row array of length N+1
containing the coefficients of the polynomial ordered by descending
powers.
 Example:
 p = [1 0 -2 -5];

16. Operations on Polynomials

17. Operations on Polynomials

18. Convolution and deconvolution
 Polynomial multiplication 􀃆convolution
 Polynomial division 􀃆deconvolution.
 Example:
 C = conv(a,b) a*b (a,b are polynomials)
 􀃆 [Z,r] = deconv(C,a) Z = C/a = b, r = [0]

19. Partial fraction expansion
 residue finds the partial fraction expansion of the ratio of two
polynomials.
 r is a column vector of residues,
 􀃆 p is a column vector of pole locations, and
 􀃆 k is a row vector of direct terms.
 [r,p,k] = residue(b,a)

20. Partial fraction expansion
 Note: Given three input arguments (r, p, k), residue converts back to
polynomial form.
 [b2,a2] = residue(r,p,k)
 Try this:

21. Basic analysis data

22. relation operators

23. logical operators
 The output of these operators is Boolean.
 1= True.
 0= False.
 Operators are:

24. logical operators
 The logical operators AND & OR can be
 represented by symbols or functional form
 operators
 >>a=1; b=0;

25. Examples

26. Examples

27. plotting two vectors