# Polynomial Manipulations

In Octave, a polynomial is represented by its coefficients (arranged in descending order). For example, a vector c of length n+1 corresponds to the following n-th order polynomial

`compan (c)`
Compute the companion matrix corresponding to polynomial coefficient vector c. The companion matrix is The eigenvalues of the companion matrix are equal to the roots of the polynomial.
`conv (a, b)`
Convolve two vectors. `y = conv (a, b)` returns a vector of length equal to `length (a) + length (b) - 1`. If a and b are polynomial coefficient vectors, `conv` returns the coefficients of the product polynomial.
`deconv (y, a)`
Deconvolve two vectors. `[b, r] = deconv (y, a)` solves for b and r such that `y = conv (a, b) + r`. If y and a are polynomial coefficient vectors, b will contain the coefficients of the polynomial quotient and r will be a remander polynomial of lowest order.
`poly (a)`
If a is a square n-by-n matrix, `poly (a)` is the row vector of the coefficients of `det (z * eye (n) - a)`, the characteristic polynomial of a. If x is a vector, `poly (x)` is a vector of coefficients of the polynomial whose roots are the elements of x.
`polyderiv (c)`
Returns the coefficients of the derivative of the polynomial whose coefficients are given by vector c.
`polyinteg (c)`
Returns the coefficients of the integral the polynomial whose coefficients are represented by the vector c. The constant of integration is set to zero.
`polyreduce (c)`
Reduces a polynomial coefficient vector to a minimum number of terms by stripping off any leading zeros.
`polyval (c, x)`
Evaluate a polynomial. `polyval (c, x)` will evaluate the polynomial at the specified value of x. If x is a vector or matrix, the polynomial is evaluated at each of the elements of x.
`polyvalm (c, x)`
Evaluate a polynomial in the matrix sense. `polyvalm (c, x)` will evaluate the polynomial in the matrix sense, i.e. matrix multiplication is used instead of element by element multiplication as is used in polyval. The argument x must be a square matrix.
`residue (b, a, tol)`
If b and a are vectors of polynomial coefficients, then residue calculates the partial fraction expansion corresponding to the ratio of the two polynomials. The function `residue` returns r, p, k, and e, where the vector r contains the residue terms, p contains the pole values, k contains the coefficients of a direct polynomial term (if it exists) and e is a vector containing the powers of the denominators in the partial fraction terms. Assuming b and a represent polynomials we have: where M is the number of poles (the length of the r, p, and e vectors) and N is the length of the k vector. The argument tol is optional, and if not specified, a default value of 0.001 is assumed. The tolerance value is used to determine whether poles with small imaginary components are declared real. It is also used to determine if two poles are distinct. If the ratio of the imaginary part of a pole to the real part is less than tol, the imaginary part is discarded. If two poles are farther apart than tol they are distinct. For example, Example:
``` b = [1, 1, 1];
a = [1, -5, 8, -4];

[r, p, k, e] = residue (b, a)
```
returns
```r = [-2, 7, 3]

p = [2, 2, 1]

k = [](0x0)

e = [1, 2, 1]
```
which implies the following partial fraction expansion
`roots (v)`
For a vector v with n components, return the roots of the polynomial