High-order Fitting Research Articles

On quantal bound state solutions and potential energy surface fitting. A comparison of the finite element, Numerov-Cooley and finite difference methods

This paper investigates several factors affecting the accuracy and efficiency of numerical determination of the bound state energy eigenvalues of the one dimensional Schrödinger equation. The efficiencies of the finite element method (FEM), the Numerov-Cooley method, and the finite difference method are compared. From this comparison, it is concluded that for potentials containing a single energy minimum, the Numerov-Cooley method is the most efficient, while for the most complex potentials the finite element method is superior due to its better numerical stability in the classically forbidden regions. The effects of various polynomial interpolation schemes on the calculated eigenvalues of potentials known only on a small number of points is examined. It is found that while higher order fits are superior to lower ones when the potential points are known accurately, they can introduce spurious information into the potential for inaccurately known points, and thus produce poor eigenvalues. Likewise, for accurately known potentials, a spline or Hermitian interpolation is better than a Lagrangian fit, but the Lagrangian functions are less susceptible to noise in a less well known case.

A study of LPC analysis of speech in additive noise

A study of the autocorrelation LPC analysis of speech in additive noise is presented. In the noise-free case it is shown that finite word length implementation of the analysis may produce stable but poor spectral estimates. The beneficial effects of proper preemphasis are reaffirmed in terms of decreased numerical error as well as decreased LPC order needed for a good spectral fit. For the ease of noisy input speech the conditions for severe distortion of the spectral estimate are presented. A proper LPC spectral analysis of speech in additive noise is shown to require a higher order fit than currently used, a more precise implementation, and a more accurate parameter quantization for transmission.

Maximum-power validation of models without higher-order fitting

A new solution is presented to the problem of validating optimally a given dynamic model against given long-sample observations. If the model can be parametrized and cast into a general innovations structure, i.e. if expressions for the one-step predictor and the prediction error covariances are available, a test can be constructed that has asymptotic maximum discriminating power, for the least favourable case that the difference to be detected between model and observed system is small. A class of alternative models must be specified, but, unlike in other optimal tests, it is not required also to fit a best model within this class. Since the alternative class may include models more complicated than that to be validated, the test can be used for recursive determination of structure and order. For linear transfer-function or polynomial-operator models the asymptotic maximum-power test does not require much more computing, and sometimes less, than the conventional tests of auto- and cross-correlation. Generally, the latter tests are less efficient, even for linear models, if these is some a priori knowledge about the structure. A simple example demonstrates that there are realistic cases where the asymptotic maximum-power test may be considerably better.