Higher-order Partial Derivatives Research Articles

Numerical Solution of Singular Poisson Problems via the Sinc–Galerkin Method

The Galerkin method is used with sinc functions for the basis elements to approximate the solution of the Poisson problem. Particular attention is paid to singular problems (a first or higher order partial derivative of the solution is undefined on the boundary). The linear system is derived and a spectral decomposition technique of solution is used. Explicit parameter selections that enhance the performance of the method are given. The method is thoroughly tested with the numerical results presented.

Automatic differentiation of functions of derivatives

In a recent paper an algorithm FEED was introduced for the systematic exact evaluation of higher-order partial derivatives of functions of many variables. The present paper demonstrates that FEED can be applied to a much broader class of functions than envisioned previously. Specifically, FEED can be used to evaluate the higher-order partial derivatives of functions which are defined in terms of the derivatives of other functions, a task required in many applications.

Nonlocal sensitivity analysis, automatic derivative evaluation, and sequential nonlinear estimation

The present paper summarizes recent work by the authors on computational methods for nonlinear processes. Section 2 develops a complete set of ordinary differential equations for generating solutions to parameterized systems of nonlinear equations over parameter intervals of interest. Section 3 presents a simple finite algorithm for the systematic exact evaluation of higher-order partial derivatives. Section 4 obtains an exact sequential characterization for the solution to a general nonlinear least-squares estimation problem as the duration of the process increases and additional observations are made.

Automatic Derivative Evaluation in the Optimization of Nonlinear Models

This paper describes the for exact evaluation of higher-order partial derivatives of functions of many variables without explicitly using the analytical expressions for those derivatives. We present and test the table method together with optimization methods which employ first, second and third order derivatives. The actual use of these methods proved to be easy and accurate.

Nonlinear least squares via automatic derivative evaluation

This paper further develops the “table algorithm” (Kalaba et al., 1981) for exact evaluation of higher-order partial derivatives of functions of many variables. We present and apply the table algorithm in the least-squares estimation of the parameters of a CES production function. The actual use of the algorithm proved to be easy and accurate.

Extremal tests for scalar functions of several real variables at degenerate critical points

It is, of course, well known to students of calculus that the extremal nature of a function f at a critical point is decided if the so-called discriminant (or Hessian) given by the expression A =f2-fxxfyy evaluated at the critical point is nonzero. It is also known through simple examples that, in the so-called degenerate case when A = 0 at the point, f may have either an extremum or a saddle point. Consequently, the extremal nature of fin this case is indeterminate from a knowledge of the second derivatives at the point in question alone and higher order partial derivatives at the point must be considered. (It is interesting that during the last century a certain confusion existed concerning the degenerate case, even apparently in the minds of some renowned mathematicians. For a short account of this history of the degenerate case see [1].) Systematic, yet straightforward and simple methods by which to take into account the higher order derivatives seem, however, difficult to come by. In fact, the only method known to the author which offers an essentially complete account of this case is due to Freedman [2]. (His techniques are concerned with the solution of the equationf (x, y)= 0 for x = x (y) but implicitly yield information about extrema as well. He also considers cases other than the degenerate case. Also in a recent paper I-3] Butler and Freedman consider the case when the lowest order terms off are cubic or higher; as stated below, we do not consider this case here.) The purpose of this note is to present a complete method for determining the extremal nature of f on the basis of its derivatives at the point in question under the two assumptions that (i)f possesses the necessary number of partial derivatives and (ii) the lowest order terms in its Taylor expansion with remainder at the point are quadratic. Under these conditions we will show how the extremal nature off may be decided in the degenerate case through a sequence of tests each involving a discriminant and each having a degenerate case, whose occurence, however, can be followed by the next test of the sequence. Each test has the same format as the standard discriminant test using A, which itself may be considered as simply the first test of the sequence. Although they accomplish more or less the same ends, the details

A computer program for graphical and iterative fitting of low and high order transfer functions to biological data

A program, written for a Digital Equipment Corporation LAB-11 minicomputer (PDP-11/20), is described which enables the user to fit frequency response functions (equivalent to transfer functions for linear systems) to data resulting from frequency domain analysis of biological signals. The program is interactive, and the user may utilize graphical or numerical curve fitting techniques to obtain a transfer function equation and display it (evaluated at prescribed points) on a two color, point-plot display against a background of the experimental data points. The numerical curve fit of the transfer function uses a non-linear least squares method. A simple and efficient method for the calculation of high order partial derivatives permits high order as well as low order transfer functions to be fit to the experimental points. The program also allows “known” or predetermined transfer function equations to be “forced” through the experimental points. Statistical comparisons are made in the program as to “goodness of fit” between transfer functions of different orders, forms and origin (i.e., “forced” vis-à-vis “fit”). The resident driver program and the overlayable segments are written in standard FORTRAN IV; only the Bode diagram plot routine is written in assembly language.

A modified Bairstow method for multiple zeros of a polynomial

A modification of Bairstow’s method to find multiple quadratic factors of a polynomial is presented. The nonlinear system of equations of the Bairstow method is replaced by high order partial derivatives of that system. The partials are computed by a repetition of the Bairstow recursion formulas. Numerical results demonstrate that the modified method converges in many cases where the Bairstow method fails due to the multiplicity of the quadratic factor. Rall [4] has described a generalization of Newton’s method for simultaneous nonlinear equations with multiple roots. This may be applied to solve the nonlinear Bairstow equations; however, it fails in some cases due to near-zero divisors. Examples are presented which illustrate the behavior of the author’s algorithm as well as the methods of Rall and Bairstow.

Analytically Derived Unit Graph and Runoff

A technique has been programed to solve analytically for measures of precipitation excess and unit graph shape parameters. The optimization procedures achieved a very high degree of adjustment of the model to data, but the results were inconsistent. The paper describes the development of an objective fitting technique. A two-step fitting procedure was devised which substantially increased the objectivity of the fitting technique. The two shape parameters were first individually corrected by a univariate technique to near-optimum values. This reduced the effect of the higher-order partial derivatives with respect to the shape parameters. The simultaneous differential correction technique was then used to obtain corrections for both the shape parameters and the runoff parameters. While this two-step solution technique does not quite yield exact results, they are close enough to the desired values and can be obtained consistently from different starting points.