Title :
A quasi-Newton adaptive algorithm for generalized symmetric eigenvalue problem
Author :
Mathew, George ; Reddy, V.U.
Author_Institution :
Dept. of Electr. Eng., Nat. Univ. of Singapore, Singapore
fDate :
10/1/1996 12:00:00 AM
Abstract :
We first recast the generalized symmetric eigenvalue problem, where the underlying matrix pencil consists of symmetric positive definite matrices, into an unconstrained minimization problem by constructing an appropriate cost function. We then extend it to the case of multiple eigen-vectors using an inflation technique. Based on this asymptotic formulation, we derive a quasi-Newton-based adaptive algorithm for estimating the required generalized eigen-vectors in the data case. The resulting algorithm is modular and parallel, and it is globally convergent with probability one. We also analyze the effect of inexact inflation on the convergence of this algorithm and that of inexact knowledge of one of the matrices (in the pencil) on the resulting eigenstructure. Simulation results demonstrate that the performance of this algorithm is almost identical to that of the rank-one updating algorithm of Karasalo (1986). Further, the performance of the proposed algorithm has been found to remain stable even over 1 million updates without suffering from any error accumulation problems
Keywords :
Newton method; adaptive signal processing; convergence of numerical methods; eigenvalues and eigenfunctions; error analysis; matrix algebra; minimisation; parallel algorithms; parameter estimation; asymptotic formulation; convergence; cost function; eigenstructure; error accumulation; generalized symmetric eigenvalue problem; inexact inflation; inexact knowledge; inflation technique; matrix pencil; multiple eigen-vectors; performance; quasi-Newton adaptive algorithm; symmetric positive definite matrices; unconstrained minimization problem; Adaptive algorithm; Algorithm design and analysis; Convergence; Cost function; Covariance matrix; Eigenvalues and eigenfunctions; Error correction; Iterative algorithms; Signal processing algorithms; Symmetric matrices;
Journal_Title :
Signal Processing, IEEE Transactions on