Octave Programming Tutorial/Polynomials

In Octave, a polynomial is represented by its coefficients (arranged in descending order). For example, the vector

octave:1> p = [-2, -1, 0, 1, 2];

represents the polynomial

$-2x^{4}-x^{3}+x+2.$

You can check this by displaying the polynomial with the function polyout.

octave:2> polyout(p, 'x')
-2*x^4 - 1*x^3 + 0*x^2 + 1*x^1 + 2

The function displays the polynomial in the variable specified (x in this case). Note that the ^ means raised to the power of much like the Octave operator.

Functions

• Evaluating a polynomial:

y = polyval(p, x)

This returns $p(x)$. If x is a vector or matrix, the polynomial is evaluated at each of the elements of x.

• Multiplication:

r = conv(p, q)

Here, p and q are vectors containing the coefficients of two polynomials and the result, r, will contain the coefficients of their product.

(As an aside, the reason this function is called conv is that we use vector convolution to do polynomial multiplication.)

• Division:

[b, r] = deconv(y, a)

This returns the coefficients of the polynomials b and r such that

$y=ab+r.$

So, b contains the coefficients of the quotient and r the coefficients of the remainder of y and a .

• Root finding:

roots(p)

This returns a vector containing all the roots of the polynomial with coefficients in p.

• Derivative:

q = polyder(p)

This returns the coefficients of the derivative of the polynomial whose coefficients are given by vector p.

• Integration:

q = polyint(p)

This returns the coefficients of the integral of the polynomial whose coefficients are represented by the vector p. The constant of integration is set to 0.

• Data fitting:

p = polyfit(x, y, n)

This returns the coefficients of a polynomial $p(x)$ of degree $n$ that best fits the data $(x_{i},y_{i})$ in the least squares sense.

Warning: Adding polynomials

Since polynomials are represented by vectors of their coefficients, adding polynomials is not straightforward in Octave. For example, we define the polynomials $p(x)=x^{2}-1$ and $q(x)=x+1$.

octave:1> p = [1, 0, -1];
octave:2> q = [1, 1];

If we try to add them, we get

octave:3> p + q
error: operator +: nonconformant arguments (op1 is 1x3, op2 is 1x2)
error: evaluating binary operator `+' near line 22, column 3

This happens because Octave is trying to add two vectors (p and q) of different lengths. This operation is not defined. To work around this, you have to add some leading zeroes to q.

octave:4> q = [0, 1, 1];

Note that adding leading zeroes does not change the polynomial. Next, we add

octave:5> p + q
ans =

  1  1  0

octave:6> polyout(ans, 'x')
1*x^2 + 1*x^1 + 0

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