Modified-Moments Method: Applications to Harmonic Solids

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Abstract
Appropriately chosen modified moments of the frequency distribution provide valuable information about harmonic solids. They contain the same information as exact power moments but much more efficiently. They stably determine quadrature formulas which have been shown to give thermal and dynamic properties of harmonic solids with great accuracy. They can be calculated either directly or from exact power moments when these are available. In this paper we give the motivation for a particularly suitable choice of modified moments for harmonic solids and show how the modified moments can be obtained from exact power moments for solids with finite-ranged forces. The modified moments can be stably transformed to power moments and to other choices of modified moments by methods which are described. We show how these transformations can be used to obtain the recursion coefficients of the orthogonal polynomials defined by the frequency distribution. We use the recursion coefficients to obtain the various Gaussian-quadrature formulas of use in the harmonic-solid problem. The transformation from modified moments to quadrature formulas is extremely stable, in striking contrast to the behavior encountered with power moments for which the transformation is exponentially ill conditioned. A particularly useful feature of the modified-moments method for harmonic solids is that additional information about the form of the spectral density function can be incorporated in order to improve the accuracy of averages of singular functions without loss of stability. We illustrate this feature of the method by presenting new results for singular averages such as inverse-power moments.