In Octave, a polynomial is represented by its coefficients (arranged in descending order). For example, a vector $c$ of length N
p(x) = c(1) x^N + ... + c(N) x + c(N+1).
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.
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,
convreturns the coefficients of the product polynomial.
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.
If a is a square N-by-N matrix,
)is the row vector of the coefficients of
det (z * eye (N) - a), the characteristic polynomial of a. If x is a vector,
)is a vector of coefficients of the polynomial whose roots are the elements of x.
Return the coefficients of the derivative of the polynomial whose coefficients are given by vector c.
Return the coefficients of a polynomial p(x) of degree n that minimizes
sumsq (p(x(i)) - y(i)), to best fit the data in the least squares sense.
The polynomial coefficients are returned in a row vector.
If two output arguments are requested, the second is a structure containing the following fields:
- The Cholesky factor of the Vandermonde matrix used to compute the polynomial coefficients.
- The Vandermonde matrix used to compute the polynomial coefficients.
- The degrees of freedom.
- The norm of the residuals.
- The values of the polynomial for each value of x.
Return the coefficients of the integral of the polynomial whose coefficients are represented by the vector c.
The constant of integration is set to zero.
Reduces a polynomial coefficient vector to a minimum number of terms by stripping off any leading zeros.
Evaluate a polynomial.
)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.
Evaluate a polynomial in the matrix sense.
)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.
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
residuereturns 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 P (s) and Q(s) we have:P(s) M r(m) N ---- = SUM ------------- + SUM k(i)*s^(N-i) Q(s) m=1 (s-p(m))^e(m) i=1
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 expansions^2 + s + 1 -2 7 3 ------------------- = ----- + ------- + ----- s^3 - 5s^2 + 8s - 4 (s-2) (s-2)^2 (s-1)
For a vector v with N components, return the roots of the polynomialv(1) * z^(N-1) + ... + v(N-1) * z + v(N)