## Finite Difference Methods for the Eigenvalues of Laplace's Operator |

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ALGOL algorithm angle basic five-point boundary conditions calculated Chebyshev polynomial coefficient constant continuous problem corners of G defined on G denote derivatives diagonal Dirichlet Dirichlet integral Dirichlet problem discrete eigenvalues distance double precision dxdy eigen eigenvalue problem eigenvectors equation error example figure finite difference methods Forsythe and Wasow Forsythe's G requires h term hence hypothesis independent of h integral interpolation is required involving iteration L-shaped domain Lemma Let G linear matrix MAXRES mesh lines mesh points neighbor number of points obtain orthonormal points of G polation polynomial positive definite precision arithmetic proof of Theorem proved rate of convergence Rayleigh quotient reentrant corners requires no interpolation Saul'ev second degree extrapolation second degree operator self-adjoint operators separation of variables simple eigenvalue STEP symmetric matrix symmetrized second degree Theorem 2.1 union of squares values of h vector vertices Wasow 9 zeroth degree