Gaussian Quadrature Applied to Eigenvalue Approximations

Abstract

We consider the eigenvalue problem

$$Kx = \lambda x,\left( {Kx} \right)\left( s \right) = \int\limits_I {k\left( {s,t} \right)x\left( t \right)dt,I = \left[ {0,1} \right]} $$

((1.1))

, with K : X → X, X = L₂ (I), a compact integral operator. In order to obtain approximations x_h resp. y_h for elements of $N\left( {K,\lambda } \right): = \left\{ {z \in X\left| {\left( {K - \lambda Id} \right)z = 0} \right.} \right\}$ resp.