Polynomials in MATLAB

Expressions that contain some number of variables and constants combined using operations such as addition, subtraction, multiplication and whole-number exponents are called POLYNOMIALS.

For example,

x² + x + 1, x³ + x² + x + 1

In MATLAB, polynomials can be represented as row vectors.

To declare a polynomial, say P(x) = 3x² + 2x + 1, we use:

P = [3, 2, 1];

Or

P = [3, 2, 1];

Note that the coefficients of the variables are taken in the row vector and not the powers.

To declare a polynomial such as Q(x) = 13x³ + 4, we use

Q = [13, 0, 0, 4];

(Because 13 x³ + 4 can be written as 13x³ + 0x² + 0x + 4.)

Ø  To evaluate the value of a polynomial for a given value, polyval() function is used.

For example, to evaluate the value of the polynomial

Y = x³ + 5x² – 3x + 1 at x = 2

               

Y = [1, 5, -3, 1];

Z=polyval(Y, 2);

Will give the result:

Z = 23

Ø  To find the roots of a polynomial, roots() function is used.

For example, to find the roots of x² + 3x + 2 = 0

S = [1,3,2];

T = roots(s);

Will give the result:

T = -2, -1

Ø  Addition of polynomials

Polynomials can be added by using the arithmetic + operator

For example,

X = [1, 1, 1];

Y = [3, 2, 4];

Z = X + Y;

Z = 4, 3, 5;

This is equivalent to -

X = s² + s + 1;

Y = 3s² + 2s + 4;

Z = X + Y;

Z = 4s² + 3s + 5

Ø  Subtraction of a polynomial

Polynomials can be subtracted by using the arithmetic - operator

For example,

X = [6, 4, 3];

Y = [3, 2, 1];

Z = X - Y;

Z = 3           2              2

This is equivalent to -

X = 6s² + 4s + 3;

Y = 3s² + 2s + 1;

Z = X - Y;

Z = 3s² + 2s + 2

Ø  Multiplication of a polynomial

Polynomials can be multiplied using the function conv().

For example, to multiply (s+1) and (s+9),

X = [1, 1];

Y = [1, 9];

Z = conv(X, Y);

Z = 1     10     9

This is equivalent to –

Z = (s+1) (s+9)

= s² + s + 9s + 9

= s² + 10s + 9

Ø  Division of polynomials

Polynomials can be divided using the function deconv().

For example, to divide (s³ + 3s² + 3s + 1) from (s+1),

X = [1, 3, 3, 1];

Y = [1, 1];

(P, Q) = deconv(X, Y);

        Here, p gets the resultant polynomial and q gets the remainder.

        Hence,

                P = 1, 2, 1

                Q = 0

Ø  Formation of Polynomials from roots

Roots of a polynomial are represented by a column vector.

R= [2; 2];

To form a polynomial from the given roots, the function poly() is used.

For example,

X = poly(R);

X =1          4          4

Hence the formed polynomial is s² + 4s + 4

Ø  Differentiation/integration of polynomials

To perform Differentiation of the polynomial, we use a function called polyder().

For example,

X = [1, 4, 4];

Y = polyder[X];

Y = 2     4

This is equivalent to

X² + 4X + 4 = 0

Differentiating,

2x+4=0

To perform Integration of the polynomial, we use the function polyint().

For example,

Y = [1, 1];

X = polyint(Y, 6);

Where 6 is the constant of integration

X = 1         1          6

This is equivalent to

X + 1= 0

Integrating,

X² + X + 6 = 0