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# Lecture8

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Polynomials In Matlab

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### Lecture8

1. 1. Lecture 8 Polynomial In Matlab Eng. Mohamed Awni Electrical & Computer Engineering Dept.
2. 2. Agenda 2  Finding Roots  Finding polynomial coefficients  Value of a Polynomial  Derivatives of Polynomials  Integrals of Polynomials  Symbolic Math Toolbox Exercise
3. 3. 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.
4. 4. Solve the equation Solve the equation Finding Roots
5. 5. 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. 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
7. 7. 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
8. 8. Integrals of Polynomials 6 𝑥2 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 6𝑥2 dx
9. 9. 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
10. 10. 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)
11. 11. 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)
12. 12. 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)