The classical condition number is a very rough measure of the effect of perturbations on the inverse of a square matrix. First, it assumes that the perturbation is infinitesimally small. Second, it does not take into account the perturbation structure (e.g., Vandermonde). Similarly, the classical notion of the inverse of a matrix neglects the possibility of large, structured perturbations. We d...

Some additive perturbation results for Drazin inverses are given. In particular, a formula is given for the Drazin inverse of a sum of two matrices, when one of the products of these matrices vanishes. Some special applications of this are also considered. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A09; 15A23

Because of the special structure of the equations AX XB = C the usual relation for linear equations "backward error = relative residual" does not hold, and application of the standard perturbation result for Ax = b yields a perturbation bound involving sep (A, B)x that is not always attainable. An expression is derived for the backward error of an approximate solution Y; it shows that the backw...

We find the probability distribution of the fluctuating parameters of a soliton propagating through a medium with additive noise. Our method is a modification of the instanton formalism (method of optimal fluctuation) based on a saddle-point approximation in the path integral. We first solve consistently a fundamental problem of soliton propagation within the framework of noisy nonlinear Schröd...

We prove an adiabatic theorem for the evolution of spectral data under a weak additive perturbation. For continuous functions of the unperturbed Hamiltonian the convergence is in norm while for a larger class functions, including the spectral projections associated to embedded eigenvalues, the convergence is in the strong operator topology.

in this paper, we investigate the application of the homotopy perturbation method (hpm) for solving a one-dimensional nonlinear inverse heat conduction problem. in this problem the thermal conductivity term is a linear function with respect to unknown heat temperature in bounded interval. furthermore, the temperature histories are unknown at the end point of the interval. this problem is ill-po...

We prove an adiabatic theorem for the evolution of spectral data under a weak additive perturbation in the context of a system without an intrinsic time scale. For continuous functions of the unperturbed Hamiltonian the convergence is in norm while for a larger class functions, including the spectral projections associated to embedded eigenvalues, the convergence is in the strong operator topol...

This paper presents a new method for estimating the ultimate bound in perturbed systems. The methodology, exploiting the perturbation structure and the system geometry, gives an implicit expression of the ultimate bound which can be solved by fixed point iterations. Thus, it is simpler than the classic Lyapunov analysis and, in many cases, less conservative. Keywords— Nonlinear Systems, Perturb...

Conditions are provided to conclude an error bound for truncations and perturbations of Markov decision problems. Both the average and finite horizon case are covered. The results are illustrated by a truncation of a Jacksonian queueing network with overflow control. An explicit error bound for this example is obtained. Key-words Markov Decision Problems * Truncation * Perturbation * Error Boun...