A Quasi-Newton Adaptive Algorithm for Generalized Symmetric Eigenvalue Problem

In this paper, 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 eigenvectors using an inflation technique, Based on this asymptotic formulation, we derive a quasi-Newton-based adaptive algorithm for estimating the required generalized eigenvectors 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. 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.

Comment: Eigenvalue sensitivity method for optimal stabilisation of linear systems

In a letter RauA proposed a new method for designing statefeedback controllers using eigenvalue sensitivity matrices. However, there appears to be a conceptual mistake in the procedure, or else it is unduly restricted in its applicability. In particular the equation — BR~lBTK = A/.I, in which K is a positive-definite symmetric matrix.

Reproduction of World War I portrait of Dr. Nathan Wolf of the 2nd Company of the 1st Landsturm Infantry Regiment Portraits Men

Dr. Nathan Wolf's medals: (bottom left) silver Wound Badge; Iron Cross 2nd Class; Iron Cross 1st Class; Zaehringer Loewe (Baden); (on right) Medal of the Turkish Crescent

Reproduction of World War I portrait of Dr. Nathan Wolf of the 2nd Company of the 1st Landsturm Infantry Regiment Portraits Men

Dr. Nathan Wolf's medals: (bottom left) silver Wound Badge; Iron Cross 2nd Class; Iron Cross 1st Class; Zaehringer Loewe (Baden); (on right) Medal of the Turkish Crescent

Hebrew poem on the 78th anniversary of the consecration of the 1st Mill Street synagogue

Digital image