In Octave, a polynomial is represented by its coefficients (arranged in descending order). For example, a vector of length corresponds to the following polynomial of order
The companion matrix is
The eigenvalues of the companion matrix are equal to the roots of the polynomial.
y = conv (a, b) returns a vector of length equal to
length (a) + length (b) - 1.
If a and b are polynomial coefficient vectors,
returns the coefficients of the product polynomial.
[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)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.
If two output arguments are requested, the second contains the values of the polynomial for each value of x.
The constant of integration is set to zero.
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) 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 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,
b = [1, 1, 1]; a = [1, -5, 8, -4]; [r, p, k, e] = residue (b, a); => r = [-2, 7, 3] => p = [2, 2, 1] => k = (0x0) => e = [1, 2, 1]
which implies the following partial fraction expansion
For a vector v with N components, return the roots of the polynomial
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