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## 27.2 Finding Roots

Octave can find the roots of a given polynomial. This is done by computing the companion matrix of the polynomial (see the `compan` function for a definition), and then finding its eigenvalues.

Function File: roots (v)

For a vector v with N components, return the roots of the polynomial

 ```v(1) * z^(N-1) + … + v(N-1) * z + v(N) ```

As an example, the following code finds the roots of the quadratic polynomial

 ```p(x) = x^2 - 5. ```
 ```c = [1, 0, -5]; roots(c) ⇒ 2.2361 ⇒ -2.2361 ```

Note that the true result is +/- sqrt(5) which is roughly +/- 2.2361.

Function File: compan (c)

Compute the companion matrix corresponding to polynomial coefficient vector c.

The companion matrix is

 ``` _ _ | -c(2)/c(1) -c(3)/c(1) … -c(N)/c(1) -c(N+1)/c(1) | | 1 0 … 0 0 | | 0 1 … 0 0 | A = | . . . . . | | . . . . . | | . . . . . | |_ 0 0 … 1 0 _| ```

The eigenvalues of the companion matrix are equal to the roots of the polynomial.

Function File: [multp, indx] = mpoles (p)
Function File: [multp, indx] = mpoles (p, tol)
Function File: [multp, indx] = mpoles (p, tol, reorder)

Identify unique poles in p and associates their multiplicity, ordering them from largest to smallest.

If the relative difference of the poles is less than tol, then they are considered to be multiples. The default value for tol is 0.001.

If the optional parameter reorder is zero, poles are not sorted.

The value multp is a vector specifying the multiplicity of the poles. multp(:) refers to multiplicity of p(indx(:)).

For example,

 ```p = [2 3 1 1 2]; [m, n] = mpoles(p); ⇒ m = [1; 1; 2; 1; 2] ⇒ n = [2; 5; 1; 4; 3] ⇒ p(n) = [3, 2, 2, 1, 1] ```

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