MATLAB Polynomials

MATLAB represents polynomials as row vectors, containing coefficients arranged in descending order of powers. For example, the equation P(x) = X ⁴ + 7× ³ - 5×+ 9Can be expressed as-

p = [1 7 0 -5 9]

Evaluate polynomial

polyvalThis function is used to evaluate a polynomial for a specific value. For example, at x = 4 When calculating the polynomial p we mentioned earlier, please enter-

1　7　0　　-5　9]
polyval(p,4)

MATLAB executes the above statement and returns the following result-

ans =　693

MATLAB also provides the polyvalm function for calculating matrix polynomials. A matrix polynomial is a polynomial with matrices as variables.

For example, let's create a square matrix X and calculate the polynomial p at X-

1　7　0　　-5　9]
X　=　[1　2　-3　4;　2　-5　6　3;　3　1　0　2;　5　-7　3　8]
polyvalm(p,　X)

MATLAB executes the above statement and returns the following result-

ans =
　　　　　　2307　　　　　　　-1769　　　　　　　　-939　　　　　　　　4499
　　　　　　2314　　　　　　　-2376　　　　　　　　-249　　　　　　　　4695
　　　　　　2256　　　　　　　-1892　　　　　　　　-549　　　　　　　　4310
　　　　　　4570　　　　　　　-4532　　　　　　　-1062　　　　　　　　9269

to find the roots of a polynomial

rootsThe function calculates the roots of a polynomial. For example, to calculate the roots of the polynomial p, please enter-

1　7　0　　-5　9]
r = roots(p)

MATLAB executes the above statement and returns the following result-

r =
　　　-6.8661　+　0.0000i
　　　-1.4247　+　0.0000i
　　　0.6454　+　0.7095i
　　　0.6454　-　0.7095i

This functionpolyis the inverse function of the root function and returns to the polynomial coefficients. For example-

p2　= poly(r)

MATLAB executes the above statement and returns the following result-

p2　=
　　　Columns　1　through　3:
　　　　　　1.00000　+　0.00000i　　　7.00000　+　0.00000i 0.00000　+　0.00000i
　　　Columns　4　and　5:
　　　　　　-5.00000　-　0.00000i　　　9.00000　+　0.00000i

Polynomial Curve Fitting

polyfitThe function finds the coefficients of a polynomial that fits a set of data in the least squares sense. If x and y are two vectors containing the x and y data to be fitted as a polynomial of degree n, then we can write-to fit the data with a polynomial-

p = polyfit(x,y,n)

Instance

Create a script file and enter the following code-

x = [1　2　3　4　5　6); y = [5.5　43.1　128　290.7　498.4　978.67); % data
p = polyfit(x,y,4) % Get the polynomial
% Calculate the polyfit estimate over a smaller range
% And plot the estimated values based on actual data for comparison
x2　=　1:.1:6;　　　　　　　　　　
y2　= polyval(p,x2);
plot(x,y,'o',x2,y2)
grid on

When running the file, MATLAB displays the following result-

p =
　　　4.1056　　-47.9607　　222.2598　-362.7453　　191.1250

and draw the following figure-