%% Quadratic Eigenvalue Problem with Mass, Damping, and Stiffness Matrices
% Solve a quadratic eigenvalue problem involving a mass matrix M, damping matrix C, and stiffness matrix K. This quadratic eigenvalue problem arises from the equation of motion:
% M \frac{d^2 y}{dt^2} + C \frac{dy}{dt} + Ky = f(t)
% This equation applies to a broad range of oscillating systems, including a dynamic mass-spring system or RLC electronic network. The fundamental solution is y(t) = x e^{\lambda t}, so both \lambda and x must solve the quadratic eigenvalue problem (QEP),
% \left( M \lambda^2 + C \lambda + K \right) x = 0
% Create coefficient matrices M, C, and K to represent a mass-spring system with four-degrees-of-freedom. The coefficient matrices are all symmetric and positive semidefinite, and M is a diagonal matrix.

M = diag([3 1 3 1])
C = [0.4 0 -0.3 0; 0 0 0 0; -0.3 0 0.5 -0.2; 0 0 -0.2 0.2]
K = [-7 2 4  0; 2 -4 2 0; 4 2 -9 3; 0 0 3 -3]

%
% Solve the QEP for the eigenvalues, eigenvectors, and condition numbers using polyeig.

[X,e,s] = polyeig(K,C,M)

%
% Check that the first eigenvalue, e(1), and first eigenvector, X(:,1), satisfy the QEP equation. The result is close to, but not exactly, zero.

lambda = e(1);
x = X(:,1);
(M*lambda^2 + C*lambda + K)*x

% Copyright 2015 The MathWorks, Inc.