2.3 Approximating Nonlinear Multidimensional Functions with Multidimensional Arguments
For many tracking applications, estimation methods require the evaluation of integrals containing nonlinear multidimensional functions with multidimensional arguments weighted by a probability density. In this section, we will review methods for numerically approximating such nonlinear functions.
This section begins with a review of scalar methods that are then extended into multiple dimensions. All of the approximations to a nonlinear function are essentially expansions of the nonlinear function into a polynomial with arbitrary coefficients. Specification of the exact form of the coefficients depends on the application, the desired accuracy and other computational considerations. Issues related to the specification of these coefficients for estimation and tracking applications will be addressed in a later chapters on specific methods for the evaluation of density-weighted integrals.
In the first subsection we review methods of approximating scalar nonlinear functions with a scalar arguments. We begin with a discussion of a general polynomial expansion of a scalar function. This leads directly to the derivation of a scalar Taylor polynomial approximation for a nonlinear function. A numerical approximation of the Taylor polynomial is then derived by replacing the differentials of the Taylor polynomial by their finite difference equivalents, resulting in Stirling's interpolation formula (Stirling's ...