Improved error estimates for numerical solutions of symmetric integral equations

Abstract:

The most widely employed method for a numerical solution of a symmetric integral equation with kernel $K(x,t)$ in interval $I \equiv [a,b]$ is the replacement of the original problem by the sequence of eigenproblems \[ {K^{(n)}}y_i^{(n)} = {\mu _{in}}y_i^{(n)},\quad {K^{(n)}} \equiv \{ {w_{jn}}K({x_{in}},{x_{jn}})\} ,\quad i = 1, \ldots ,n,\] with ${w_{jn}} > 0$ and ${x_{jn}} \in I,j = 1, \ldots ,n$. The eigenvectors $y_i^{(n)}$ are further used to obtain an approximation, with improved error estimates, of the numerical eigensolution for some $N > n$, with no necessity of computing ${\mu _{iN}}$ and $y_i^{(N)},i = 1, \ldots ,N$, and of constructing another matrix.