Interpretation and solution of ill-conditioned equations arising in parameter estimation and system identification
A new method, named Epsilon Decomposition (E-D), for the efficient solution of ill-conditioned linear algebraic equations is presented. It represents a significant improvement over Singular Value Decomposition (SVD), which requires the user to obtain all the eigenvalues and eigenvectors of all ill-conditioned matrix. This can be quite costly if the matrix is large. The efficiency associated with E-D resides in the fact that the user need only obtain the zero, near-zero and the lowest non-zero eigenvalues of the coefficient matrix of the system of equations. The equivalence of E-D and SVD is based upon the fact that E-D produces a very small, \(\epsilon\), shift in the non-zero and near-zero eigenvalues without any need to compute them. The mathematical interpretation of the E-D and SVD solutions is that they produce the same minimum norm solutions. A numerical example is presented in detail.
ill-conditioned linear algebraic equations
scientific article; zbMATH DE number 4209967