The Exact Condition Number of the Truncated Singular Value Solution of a Linear Ill-Posed Problem

Abstract

The main result of this paper is the formulation of an explicit expression for the condition number of the truncated least squares solution of $Ax=b$. This expression is given in terms of the singular values of $A$ and the Fourier coefficients of $b$. The result is derived using the notion of the Fréchet derivative together with the product norm on the data $[A,b]$ and the 2-norm on the solution. Numerical experiments are given to confirm our results by comparing them to those obtained by means of a finite difference approach.