In this thesis some methods for solving systems of
nonlinear equations are described, which do not require
calculation of the Jacobian matrix. One of these methods
is programmed to solve a parametrized system with possible
singularities. The efficiency of this method and a modified
Newton's method are compared using experimental...
The existence of eigenvalues is shown for certain types of
integral equations with continuous kernels, the proofs utilizing some
basic results of collectively compact operator approximation theory.
We will consider the implementation of a computer program to
solve a nonlinear algebraic system of N equations and unknowns.
The program involves the use of a parameter, Newton's method, and
an automatic change of parameter. Also considered are rigorous
error bounds for the answer. The program was implemented and...
Two numerical methods are presented that can be used to solve
second order nonlinear ordinary differential equations with periodic
boundary conditions. One of these methods is a shooting method developed
solely for the periodic problem. The other, "quasilinearization,"
is a method applicable to a wide variety of problems. It is...
Interval arithmetic is applied to the problem of obtaining
rigorous solutions to integral equations on a computer. The
integral equations considered are the linear Fredholm equation of
the second kind and the nonlinear Urysohn equation. Techniques are
presented which enable the computer to find an approximate
solution, prove the existence...
The extension and convergence of positive operators is investigated by means of a monotone approximation technique. Some generalizations and extensions of Korovkin's monotone operator theorem on C[0, 1] are given. The concept of a regular set is introduced and it is shown that pointwise convergence is uniform on regular sets....
A function translator is presented which was designed for
interactive programs which allow functions to be defined on-line. The
translator handles functions which are specified by a formula and
functions which are specified as the solution to a system of differential
equations.