A priori estimation is presented for a computational complexity of the homotopy method applying to a certain class of strongly monotone nonlinear equations. In the present papers, a condition is presented for a certain class of uniquely solvable equations, under which an upper bound of a computational complexity of the Newton type homotopy method can be a priori estimated. In this paper, a condition is considered in a case of linear homotopy equations including the Newton type homotopy equations. In the first place, the homotopy algorithm based on the simplified Newton method is introduced. Then by using Urabe type theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, a condition is presented under which an upper bound of a computational complexity of the algorithm can be a priori estimated, when it is applied to a certain class of strongly monotone nonlinear equations. The presented condition is demonstrated by numerical experiments.

A priori estimation is presented for a computational complexity of the homotopy method applying to a certain class of uniquely solvable nonlinear equations. In the first place, the reason is explained why a computational complexity of the homotopy method can not be a priori estimated in general. In this paper, the homotopy algorithm is considered in which a numerical path following algorithm is executed based on the simplified Newton method. Then by introducing Urabe's theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, it is shown that a computational complexity of the algorithm can be a priori estimated, when it is applied to a certain class of uniquely solvable nonlinear equation. In this paper, two types of path following algorithms are considered, one with a numerical error estimation in the domain of a nonlinear operator and another with one in the range of the operator.

In this paper, in the first place, a slightly version upped Urabe type theorem of convergence criterion is presented for the modified Newton method. Then, based on this theorem, a posteriori stopping criterion is presented for a class of numerical methods of calculating solutions including the simplicial approximate homotopy method for nonlinear equations. By this criterion it is estimated whether an approximate solution satisfies the conditions of the Urabe theorem or not. Finally, it is shown that under a certain mild condition a class of simplicial approximate homotopy methods such as Merrill's method generate an approximate solution which satisfies our stopping criterion in restarting finite steps.

In this paper, we present a numerical method with guaranteed accuracy to solve initial value problems (IVPs) of normal form simultaneous first order ordinary differential equations (ODEs) which have wide domain. Our method is based on the algorithm proposed by Kashiwagi, by which we can obtain inclusions of exact values at several discrete points of the solution curve of ODEs. The method can be regarded as an extension of the Lohner's method. But the algorithm is not efficient for equations which have wide domain, because the error bounds become too wide from a practical point of view. Our purpose is to produce tight bounds even for such equations. We realize it by combining Kashiwagi's algorithm with the mean value form. We also consider the wrapping effects to obtain tighter bounds.

A type of infinite dimensional homotopy method is considered for numerically calculating a solution curve of a nonlinear functional equation being a Fredholm operator with index 1 and an A-proper operator. In this method, a property of so-called A-proper homotopy plays an important role.

Affine arithmetic is a kind of interval arithmetic defined by Stolfi et al. In affine arithmetic, it is difficult to realize the efficient nonlinear binomial operations. The purpose of this letter is to propose a new dividing method which is able to supply more suitable evaluation than the old dividing method. And this letter also shows the efficiency of the new dividing method by numerical examples.

In this paper, we are concerned with a problem of obtaining an approximate solution of a finite-dimensional nonlinear equation with guaranteed accuracy. Assuming that an approximate solution of a nonlinear equation is already calculated by a certain numerical method, we present computable conditions to validate whether there exists an exact solution in a neighborhood of this approximate solution or not. In order to check such conditions by computers, we present a method using rational arithmetic. In this method, both the effects of the truncation errors and the rounding errors of numerical computation are taken into consideration. Moreover, based on rational arithmetic we propose a new modified Newton interation to obtain an improved approximate solution with desired accuracy.

In this letter, a constructive simplified Newton method is presented for calculating solutions of infinite dimensional nonlinear equations, which uses a projection scheme from an infinite dimensional space to finite dimensional subspaces. A convergence theorem of the method is shown based on Urabe's theorem.

Interval arithmetic is able to be applied when we include the ranges of various functions. When we include them applying the interval arithmetic, the serious problem that the widths of the range inclusions increase extremely exists. In range inclusion of polynomials particularly, Horner's method and Alefeld's method are well known as the conventional methods which mitigates this problem. The purpose of this paper is to propose the new methods which are able to mitigate this problem more efficiently than the conventional methods. And in this paper, we show and compare the efficiencies of the new methods by some numerical examples.

An estimation method of region is presented, in which a solution path of the so-called Newton type homotopy equation in guaranteed to exist, it is applied to a certain class of uniquely solvable nonlinear equations. The region can be estimated a posteriori, and its upper bound also can be estimated a priori.