Lecture 8

Polynomial In Matlab

Eng. Mohamed Awni

Electrical & Computer Engineering

Dept.

Agenda

2

Finding Roots

Finding polynomial coefficients

Value of a Polynomial

Derivatives of Polynomials

Integrals of Polynomials

Symbolic Math Toolbox

Exercise

Finding Roots

roots function Solves polynomial equations of the form

returns the roots of the polynomial represented by p as a column vector.

r = roots(P)

• Polynomial coefficients, specified as a vector.

• Must include all coefficients, even if 0

For example:

• Vector [1 0 1] represents the polynomial 𝑥2+1,

• Vector [3.13 -2.21 5.99] represents the polynomial 3.13𝑥2−2.21𝑥+5.99.

Solve the equation

Solve the equation

Finding Roots

Finding polynomial coefficients

p = poly(r)

• r is a row or column vector with the roots of the polynomial • p is a row vector with the coefficients

Ex: roots = -3, 2

r = [-3 2];

p = poly(r)

p = 1 1 -6

% f(x) = x2 + x -6

Calculate polynomial coefficients

6

Compute the value of a polynomial for any value of x directly

f(x) = 5𝑥3+ 6𝑥2-7𝑥 + 3

Polyval (p, x) • p is a vector with the coefficients of the polynomial • x is a number, variable or expression

Value of a Polynomial

k = polyder(p)

Derivatives of Polynomials

Calculate the derivatives of polynomials

• p is the coefficient vector of the polynomial • k is the coefficient vector of the derivative

>> p = [3 -2 4]; >> k = polyder(p) k = 6 -2 % dy/dx = 6𝑥 - 2

Ex: f(x) = 3𝑥2 - 2𝑥 + 4

Integrals of Polynomials

6𝑥2 d𝑥 = 6 𝑥2 dx

= 6 * 1

3 𝑥3

= 2 𝑥3

• h is the coefficient vector of the polynomial • g is the coefficient vector of the integral • k is the constant of integration – assumed to be 0 if not present

g = polyint(h, k) Calculate the integral of a polynomial

>> h = [6 0 0]; >> g = polyint(h) g = 2 0 0 0 % g(x) = 2x3

6𝑥2dx

Finding roots.

Symbolic Math Toolbox

Performs calculation symbolically in Matlab environment.

>> f=2*x^2 + 4*x -8; >> solve(f,x)

In Matlab command window, we will first need to define x as a symbolic.

ans = -3.2361 1.2361

>> syms x

Derivative.

We wish to take the derivative of function f(x):

In Matlab command window, we will first need to define x as a symbolic.

>> syms x >> f=x^3-cos(x); >> g=diff(f)

Symbolic Math Toolbox

>> syms x y >> f=x^2+(y+5)^3; >> diff(f,y)

Matlab returns: ans = 3*(y+5)^2

equivalent to

Matlab returns: g = 3*x^2+sin(x)

g=diff(f)

Symbolic Math Toolbox

Integral.

>> syms x y >> int(f,x)

Matlab returns: ans = 1/3*x^3+(y+5)^3*x

>> int(f,y,0,10) Matlab returns: ans = 12500+10*x^2

int(f,x)

Exercises

P(x) = 𝑥4 + 7𝑥3 - 5x + 9

• Find the roots of the ploynomial P(x)

• Evaluate the polynomial P(x) for x=4 and x=6

• Calculate the d p(x)/dx

• Calculate the 𝑝(𝑥)