This is reprinted, with permission from the author and the publisher, from the original paper published in Research Association of Applied Geometry (RAAG) Memoirs, Vol. 2 (1958) pp. 29-46, published by Gakujutsu Bunken Fukyu-kai, Tokyo, Japan.

This paper surveys applications of Krawczyk--Moore--Jones's algorithm and presents its further developments. The application is focused onto nonlinear electric circuit analysis. The further developments are described on parallel KMJ algorithm, Gray code KMJ algorithm and KMJ processor.

This paper aims to survey fast methods of verifying the accuracy of a numerical solution of a linear system. For the last decade, a number of fast verification algorithms have been proposed to obtain an error bound of a numerical solution of a dense or sparse linear system. Such fast algorithms rely on the verified numerical computation using floating-point arithmetic defined by IEEE standard 754. Some fast verification methods for dense and sparse linear systems are reviewed together with corresponding numerical results to show the practical use and efficiency of the verified numerical computation as much as possible.

We survey a class of algorithms to evaluate polynomials with floating point coefficients and for computation performed with IEEE-754 floating point arithmetic. The principle is to apply, once or recursively, an error-free transformation of the polynomial evaluation with the Horner algorithm and to accurately sum the final decomposition. These compensated algorithms are as accurate as the Horner algorithm performed in $K$ times the working precision, for $K$ an arbitrary positive integer. We prove this accuracy property with an a priori error analysis. We also provide validated dynamic bounds and apply these results to compute a faithfully rounded evaluation. These compensated algorithms are fast. We illustrate their practical efficiency with numerical experiments on significant environments. Comparing to existing alternatives these $K$-times compensated algorithms are competitive for $K$ up to 4, i.e., up to 212 mantissa bits.

This paper is concerned with a robust geometric predicate for the 2D orientation problem. Recently, a fast and accurate floating-point summation algorithm is investigated by Rump, Ogita and Oishi, which provably outputs a result faithfully rounded from the exact value of the summation of floating-point numbers. We optimize their algorithm for applying it to the 2D orientation problem which requires only a correct sign of a determinant of a $3\times 3$ matrix. Numerical results illustrate that our algorithm works fairly faster than the state-of-the-art algorithm in various cases.

A numerical verification method of steady state solutions for a system of reaction-diffusion equations is described. Using a decoupling technique, the system is reduced to a single nonlinear equation and a computer-assisted method for second-order elliptic boundary value problems based on the infinite dimensional fixed-point theorem can be applied. Some numerical examples confirm the effectiveness of the method.

Let an $n\times n$ matrix $A$ of floating-point numbers in some format be given. Denote the relative rounding error unit of the given format by $\mathtt{eps}$. Assume $A$ to be extremely ill-conditioned, that is $\cond(A)\gg\mathtt{eps}^{-1}$. In about 1984 I developed an algorithm to calculate an approximate inverse of $A$ solely using the given floating-point format. The key is a multiplicative correction rather than a Newton-type additive correction. I did not publish it because of lack of analysis. Recently, in [9] a modification of the algorithm was analyzed. The present paper has two purposes. The first is to present reasoning how and why the original algorithm works. The second is to discuss a quite unexpected feature of floating-point computations, namely, that an approximate inverse of an extraordinary ill-conditioned matrix still contains a lot of useful information. We will demonstrate this by inverting a matrix with condition number beyond $10^{300}$ solely using double precision. This is a workout of the invited talk at the SCAN meeting 2006 in Duisburg.

We give a survey on direct methods for interval linear systems. We also consider various kinds of solution sets and show how the interval hull can be computed.

After the introduction basic properties of interval arithmetic are discussed and different approaches are repeated by which one can compute verified numerical approximations for a solution of a nonlinear equation.

The construction of spherical $t$-designs with $(t+1)^2$ points on the unit sphere $S^2$ in $\mathbb{R}^3$ can be reformulated as an underdetermined system of nonlinear equations. This system is highly nonlinear and involves the evaluation of a degree $t$ polynomial in $(t+1)^4$ arguments. This paper reviews numerical verification methods using the Brouwer fixed point theorem and Krawczyk interval operator for solutions of the underdetermined system of nonlinear equations. Moreover, numerical verification methods for proving that a solution of the system is a spherical $t$-design are discussed.

In this paper, we are concerned with the analysis of two-transistor circuits. Applying technique for the numerical verification, we prove rigorously the existence of five solutions in a two-transistor circuit. The system of equations for a transistor circuit is obtained as nonlinear equations, therefore Krawczyk's method is applied for proving the existence of a solution.

This survey contains recent developments for computing verified results of convex constrained optimization problems, with emphasis on applications. Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analysis, for linear programming, and for semidefinite programming. A discussion of important problem transformations to special types of convex problems and convex relaxations is included. The latter are important for handling and for reliability issues in global robust and combinatorial optimization. Some remarks on numerical experiences, including also large-scale and ill-posed problems, and software for verified computations concludes this survey.

The authors give short survey on validated computaion of initial value problems for ODEs especially Taylor model methods. Then they propose an application of Taylor models to the Nakao method which has been developed for numerical verification methods on PDEs and apply it to initial value problems for ODEs with some numerical experiments.

We highlight selected results of recent development in the area of rigorous computations which use interval arithmetic to analyse dynamical systems. We describe general ideas and selected details of different ways of approach and we provide specific sample applications to illustrate the effectiveness of these methods. The emphasis is put on a topological approach, which combined with rigorous calculations provides a broad range of new methods that yield mathematically reliable results.

For second-order semilinear elliptic boundary value problems on bounded or unbounded domains, a general computer-assisted method for proving the existence of a solution in a ``close'' and explicit neighborhood of an approximate solution, computed by numerical means, is proposed. To achieve such an existence and enclosure result, we apply Banach's fixed-point theorem to an equivalent problem for the error, i.e., the difference between exact and approximate solution. The verification of the conditions posed for the fixed-point argument requires various analytical and numerical techniques, for example the computation of eigenvalue bounds for the linearization at the approximate solution. The method is used to prove existence and multiplicity results for some specific examples.

We first summarize the general concept of our verification method of solutions for elliptic equations. Next, as an application of our method, a survey and future works on the numerical verification method of solutions for heat convection problems known as Rayleigh--Bénard problem are described. We will give a method to verify the existence of bifurcating solutions of the two-dimensional problem and the bifurcation point itself. Finally, an extension to the three-dimensional case and future works will be described.

This paper treats a linear equation \begin{equation*} Av=b, \end{equation*} where $A \in \mathbb{F}^{n\times n}$ and $b \in \mathbb{F}^n$. Here, $\mathbb{F}$ is a set of floating point numbers. Let $\mathbf{u}$ be the unit round-off of the working precision and $\kappa(A)=\|A\|_{\infty}\|A^{-1}\|_{\infty}$ be the condition number of the problem. In this paper, ill-conditioned problems with \begin{equation*} 1 < \mathbf{u}\kappa(A) < \infty \end{equation*} are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.

We consider an eigenvalue problem for differential operators, and show how guaranteed bounds for eigenvalues (together with eigenvectors) are obtained and how non-existence of eigenvalues in a concrete region can be assured. Some examples for several types of operators will be presented.

In solving elliptic problems by the finite element method in a bounded domain which has a re-entrant corner, the rate of convergence can be improved by adding a singular function to the usual interpolating basis. When the domain is enclosed by line segments which form a corner of $\pi/2$ or $3\pi/2$, we have obtained an explicit a priori $H^{1}_{0}$ error estimation of $O(h)$ and an $L^{2}$ error estimation of $O(h^{2})$ for such a finite element solution of the Poisson equation. Particularly, we emphasize that all constants in our error estimates are numerically determined, which plays an essential role in the numerical verification of solutions to non-linear elliptic problems.

Today simulation technologies (based on numerical computation) are definitely vital in many fields of science and engineering. The accuracy of numerical simulations grows in importance as simulation technologies develop and prevail, and many researches have been carried out for establishing numerically stable algorithms. In recent years, combining computer algebra and other guaranteed accuracy approaches draws much attention as one of promising directions for developing guaranteed accuracy algorithms for a wider class of problems. This paper illustrates several typical usages of symbolic and algebraic methods for guaranteed accuracy computation, highlighting some of the recent applications in control problems.