residu

Arguments

P, Q1, Q2

polynomials or matrix polynomials with real or complex coefficients.

Description

V=residu(P,Q1,Q2) returns the matrix V such that V(i,j) is the sum of the residues of the rational fraction P(i,j)/(Q1(i,j)*Q2(i,j)) calculated at the zeros of Q1(i,j).

Q1(i,j) and Q2(i,j) must not have any common root.

Examples

s = poly(0,'s');
H = [s/(s+1)^2, 1/(s+2)];
N = H.num;
D = H.den;
w = residu(N.*horner(N,-s), D, horner(D,-s));  //N(s) N(-s) / D(s) D(-s)
sqrt(sum(w))  //This is H2 norm
h2norm(tf2ss(H))

p = (s-1)*(s+1)*(s+2)*(s+10);
a = (s-5)*(s-1)*(s*s)*((s+1/2)**2);
b = (s-3)*(s+2/5)*(s+3);
residu(p, a, b) + 531863/4410    //Exact
z = poly(0,'z');
a = z^3 + 0.7*z^2 + 0.5*z - 0.3;
b = z^3 + 0.3*z^2 + 0.2*z + 0.1;
atild = gtild(a, 'd');
btild = gtild(b, 'd');
residu(b*btild, z*a, atild) - 2.9488038   //Exact
a = a + 0*%i;
b = b + 0*%i;
real(residu(b*btild, z*a, atild) - 2.9488038) //Complex case

See also

• pfss — partial fraction decomposition
• bdiag — block diagonalization, generalized eigenvectors
• roots — roots of a polynomial
• poly — Polynomial definition from given roots or coefficients, or characteristic to a square matrix.
• gtild — tilde operation