Nima Rasakhoo

A new proof for Decell's finite algorithm for generalized inverses

A new proof for Decells finite algorithm for determining the Moore-Penrose generalized inverse of a rectangular matrix is given. In this proof we develop additional properties of the algorithm and point out how it can be used for further development of the algebraic properties of the Moore-Penrose generalized inverse.

Numerical experiments using Sukhanov's initial-value method for nonlinear two-point boundary value problems

In a recent paper, Sukhanov derived a new method for transforming a nonlinear two-point boundary-value problem into an initial-value problem. Sukhanovs equations involve only the solution of ordinary differential equations and not partial differential equations. An earlier paper by the authors presented their interpretation of Sukhanovs method. An alternative method is presented in this paper. Numerical results are given.

Analytical aspects of computational linear algebra

This paper deals with a system of ordinary differential equations with known initial conditions associated with a given square matrix. By using standard analytical and computational methods many of the important aspects of the given matrix can be determined. Among these are its determinant, its adjoint, its inverse (if it exist), the coefficients of the characteristic polynomial, the location of the roots ofthe characteristic polynomial, and the corresponding eigenvectors. Concomitently, the differential system yields a treatment of inhomogeneous linear algebraic systems associated with the given matrix, as in economic input-output analysis. In particular, new insights are provided into Faddeevs algorithm for the coefficients of the characteristic polynomial.

The most representative shape of a family of curves

Suppose that we are given a family of curves, either continuous or discrete, and we wish to determine a curve that is most representative of the family. Using least squares theory, we find that in both cases we seek the dominant eigenvalue and eigenvector of a symmetric matrix. Numerical approaches to the eigenvalue problem are then given. One involves the standard inverse power method, and the other involves a new differential equation approach. Faddeevs method for obtaining the characteristic polynomial is also employed. Two illustrative examples, one for a family of continuous curves and one for a family of discrete curves, are presented.

Nonlinear Least Squares

One of the most common problems encountered in practical data analysis involves the fitting of a theoretical model to experimental data. Frequently the model takes the form of a dependent variable expressed as a function of several independent variables. Often the model will contain one or more parameters which have to be estimated. This estimation takes place on the basis of fitting the model to observations using the least-squares concept.

Methods for Numerical Differentiation

It was in 1964 that R. Wengert (Reference 1) published a paper that showed how the partial derivatives of a given function could be evaluated by a computing machine without the user’s first having formed the analytical expressions for the desired derivatives. Furthermore, no use was made of symbol manipulation or finite difference approximations. A salient idea was that the computer was to do arithmetic, solely, and the user was to supply the needed “savvy” about differential calculus. In a companion paper (Reference 2) Wilkins discussed some practical considerations associated with the Wengert method. These two papers were studied intensely at the Rand Corporation for, at that time, there was great interest in orbit determination via quasilinearization (References 3 and 4). This resulted in a third early paper (Reference 3). A book was published in 1981 that described the situation to that date (Reference 5). A language known as PROSE also appeared.

Nonlinear Integral Equations

Integral equations appear in many engineering and physics problems. Numerical methods of solution for integral equations have been largely developed within the last 20 years (References 1–4). In this chapter a development involving an imbedding method for obtaining the numerical solution of nonlinear integral equations is described (References 5, 6). The numerical solution is obtained automatically from the initial value imbedding equations via the automatic derivative evaluation method described in the previous chapters (Reference 7). The derivatives required for the solution are computed automatically. The user need only enter the two known functions in equation (6.1) into the program. None of the derivatives associated with the imbedding method need be derived by hand. The subroutines are written in Basic here, rather than in Fortran, because the former has several advantages in the manipulation of the vectors and matrices which occur in using the method of lines. The vectors consisting of the variables or functions of the variables, and all their derivatives are defined as derivative vectors (instead of just vectors) in this chapter to distinguish them from the other vectors and matrices.

Sukhanov’s Variable Initial Value Method for Boundary Value Problems

In 1983, Sukhanov presented a new method for transforming a nonlinear two-point boundary value problem into an initial value problem (Reference 1). Sukhanov’s approach involves only the solution of ordinary differential equations and not partial differential equations as in other initial value methods for nonlinear two-point boundary value problems (References 2, 3, and 8). In this chapter, Sukhanov’s method is applied to second order and fourth-order nonlinear two-point boundary value problems.