1. Polynomials UC Berkeley Fall 2004, E77 http://jagger.me.berkeley.edu/~pack/e77 Copyright 2005, Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. http://jagger.me.berkeley.edu/~pack/e77http://creativecommons.org/licenses/by-sa/2.0/

2. Polynomials An n th order polynomial in variable x is written as It is natural to associate a row vector A p with p, namely A few examples: Row vector representations of are >> Ap = [ 6 5 -3 7 ]; >> Aq = [ 9 -2 4 ]; But [ 0 0 0 9 -2 4 ] also represents q… “leading zeros”

3. Functions operating on polynomials Let’s spend the lecture writing several useful functions for polynomial manipulation, including –Polynomial addition –Polynomial subtraction –Polynomial multiplication –Polynomial evaluation (use polyval ) –Plotting the graph of a polynomial –Roots of a polynomial (use roots ) The inputs to the functions will be row vectors, representing polynomials.

4. Polynomial Addition Adding two polynomials requires adding their coefficients. The sum of is simply In terms of the row-vector representation of the polynomials, we simply add them, element-by-element. But the row vectors may be different lengths, and we need to “ align ” them.

5. Polynomial Addition The row vector representations of are >> A = [ 6 5 -3 7 ]; >> B = [ 9 2 -4 ]; but obviously >> C = A + B will give an error. We need to “pad” B with a zero >> C = A + [ 0 B ]; or add B only to the correct elements of A. >> C = A; >> C(2:end) = C(2:end) + B;

6. addpoly.m function C = addpoly(A,B) % This adds two row vectors, interpreting % them as polynomials. % Should check that A and B are row vectors lenA = length(A); lenB = length(B); if lenA==lenB C = A+B; elseif lenA<lenB C = [zeros(1,lenB-lenA) A] + B; else C = A + [zeros(1,lenA-lenB) B]; end Inserting the right number of leading zeros to A

7. subpoly.m function C = subpoly(A,B) % This subtracts two row vectors, % interpreting them as polynomials. % Should check that A and B are row vectors C = addpoly(A,-B);

8. Polynomial Multiplication Multiplying a(x) = 6x 3 + 5x 2 – 3x + 7 b(x) = 3x 2 + 2x – 4 Express the product as (6x 3 + 5x 2 – 3x + 7)(3x 2 + 2x – 4) = (6x 3 + 5x 2 – 3x + 7)*3x 2 + (6x 3 + 5x 2 – 3x + 7)*2x + (6x 3 + 5x 2 – 3x + 7)*(-4) The result is 5 th order, so we need a 1-by-6 array for the result C = [ 0 0 0 0 0 0 ]

9. Polynomial Multiplication Multiplying a(x) = 6x 3 + 5x 2 – 3x + 7 A = [6 5 -3 7] b(x) = 3x 2 + 2x – 4 B = [3 2 -4] Express the product as (6x 3 + 5x 2 – 3x + 7)(3x 2 + 2x – 4) = (6x 3 + 5x 2 – 3x + 7)*3x 2 + (6x 3 + 5x 2 – 3x + 7)*2x + (6x 3 + 5x 2 – 3x + 7)*(-4) The result is 5 th order, so we need a 1-by-6 array for the result C = [ 0 0 0 0 0 0 ]

10. Polynomial Multiplication Multiplying a(x) = 6x 3 + 5x 2 – 3x + 7 A = [6 5 -3 7] b(x) = 3x 2 + 2x – 4 B = [3 2 -4] Express the product as (6x 3 + 5x 2 – 3x + 7)(3x 2 + 2x – 4) = (6x 3 + 5x 2 – 3x + 7)*3x 2 add A*B(1) to C(1:4) + (6x 3 + 5x 2 – 3x + 7)*2x + (6x 3 + 5x 2 – 3x + 7)*(-4) The result is 5 th order, so we need a 1-by-6 array for the result C = [ 0 0 0 0 0 0 ]

11. Polynomial Multiplication Multiplying a(x) = 6x 3 + 5x 2 – 3x + 7 A = [6 5 -3 7] b(x) = 3x 2 + 2x – 4 B = [3 2 -4] Express the product as (6x 3 + 5x 2 – 3x + 7)(3x 2 + 2x – 4) = (6x 3 + 5x 2 – 3x + 7)*3x 2 add A*B(1) to C(1:4) + (6x 3 + 5x 2 – 3x + 7)*2x + (6x 3 + 5x 2 – 3x + 7)*(-4) The result is 5 th order, so we need a 1-by-6 array for the result C = [ 18 15 -9 21 0 0 ]

12. Polynomial Multiplication Multiplying a(x) = 6x 3 + 5x 2 – 3x + 7 A = [6 5 -3 7] b(x) = 3x 2 + 2x – 4 B = [3 2 -4] Express the product as (6x 3 + 5x 2 – 3x + 7)(3x 2 + 2x – 4) = (6x 3 + 5x 2 – 3x + 7)*3x 2 add A*B(1) to C(1:4) + (6x 3 + 5x 2 – 3x + 7)*2x add A*B(2) to C(2:5) + (6x 3 + 5x 2 – 3x + 7)*(-4) The result is 5 th order, so we need a 1-by-6 array for the result C = [ 18 15 -9 21 0 0 ]

13. Polynomial Multiplication Multiplying a(x) = 6x 3 + 5x 2 – 3x + 7 A = [6 5 -3 7] b(x) = 3x 2 + 2x – 4 B = [3 2 -4] Express the product as (6x 3 + 5x 2 – 3x + 7)(3x 2 + 2x – 4) = (6x 3 + 5x 2 – 3x + 7)*3x 2 add A*B(1) to C(1:4) + (6x 3 + 5x 2 – 3x + 7)*2x add A*B(2) to C(2:5) + (6x 3 + 5x 2 – 3x + 7)*(-4) The result is 5 th order, so we need a 1-by-6 array for the result C = [ 18 27 1 15 14 0 ]

14. Polynomial Multiplication Multiplying a(x) = 6x 3 + 5x 2 – 3x + 7 A = [6 5 -3 7] b(x) = 3x 2 + 2x – 4 B = [3 2 -4] Express the product as (6x 3 + 5x 2 – 3x + 7)(3x 2 + 2x – 4) = (6x 3 + 5x 2 – 3x + 7)*3x 2 add A*B(1) to C(1:4) + (6x 3 + 5x 2 – 3x + 7)*2x add A*B(2) to C(2:5) + (6x 3 + 5x 2 – 3x + 7)*(-4) add A*B(3) to C(3:6) The result is 5 th order, so we need a 1-by-6 array for the result C = [ 18 27 1 15 14 0 ]

15. Polynomial Multiplication Multiplying a(x) = 6x 3 + 5x 2 – 3x + 7 A = [6 5 -3 7] b(x) = 3x 2 + 2x – 4 B = [3 2 -4] Express the product as (6x 3 + 5x 2 – 3x + 7)(3x 2 + 2x – 4) = (6x 3 + 5x 2 – 3x + 7)*3x 2 add A*B(1) to C(1:4) + (6x 3 + 5x 2 – 3x + 7)*2x add A*B(2) to C(2:5) + (6x 3 + 5x 2 – 3x + 7)*(-4) add A*B(3) to C(3:6) The result is 5 th order, so we need a 1-by-6 array for the result C = [ 18 27 -23 -5 26 28 ] A*B(1) + C(1:1+length(A)-1) A*B(2) + C(2:2+length(A)-1) A*B(3) + C(3:3+length(A)-1) A*B(i) + C(i:i+length(A)-1) Do this for every element of B ( for loop)

16. multpoly.m function C = multpoly(A,B) % This computes the product of the inputs, % interpreting them as polynomials. lenA = length(A); lenB = length(B); lenC = lenA + lenB - 1; C = zeros(1,lenC); for i=1:lenB idx = i:i+lenA-1; C(idx) = A*B(i) + C(idx); end

17. Polynomial Evaluation Use the Matlab function polyval >> A = [ 1 1 -2 ]; >> x = 2.6; >> y = polyval(A,x) polyval works for vectors too >> x = linspace(-3,3,200); >> y = polyval(A,x); >> plot(x,y)

18. Roots of Polynomials An n th order polynomial (with nonzero leading coefficient) It is a fundamental theorem of algebra that the equation has n solutions, called the roots of p. Label these roots r 1, r 2,…, r n. Another way to say this is that p can be factored into the for Even if the coefficients of p are real numbers, the roots may be complex.

19. The roots command The command roots computes the roots of a polynomial, and returns them as a column vector. Compute the roots of >> roots([1 1 -2]) >> roots([1 3 5]) >> roots([2 -1 4 1 -3])