Floating-point Interval Arithmetic Research Articles

An interval version of the Kuntzmann-Butcher method for solving the initial value problem

The Kutzmann-Butcher method is the unique implicit four-stage Runge-Kutta method of order 8. In many problems in ordinary differential equations this method realized in floating-point arithmetic gives quite good approximations to the exact solutions, but the results obtained do not contain any information on rounding errors, representation errors and the error of the method. Thus, we describe an interval version of this method, which realized in floating-point interval arithmetic gives approximations (enclosures in the form of interval) containing all these errors. The described method can also include data uncertainties in the intervals obtained.

On interval predictor-corrector methods

One can approximate numerically the solution of the initial value problem using single or multistep methods. Linear multistep methods are used very often, especially combinations of explicit and implicit methods. In floating-point arithmetic from an explicit method (a predictor), we can get the first approximation to the solution obtained from an implicit method (a corrector). We can do the same with interval multistep methods. Realizing such interval methods in floating-point interval arithmetic, we compute solutions in the form of intervals which contain all possible errors. In this paper, we propose interval predictor-corrector methods based on conventional Adams-Bashforth-Moulton and Nyström-Milne-Simpson methods. In numerical examples, these methods are compared with interval methods of Runge-Kutta type and methods based on high-order Taylor series. It appears that the presented methods yield comparable approximations to the solutions.

Calculation of Rational Numbers in an Interval Whose Denominator is the Smallest by using FP Interval Arithmetic

The continued fraction expansion method is a fast solver to find a rational number in a given real interval whose denominator is the smallest. A simple implementation of the CF expansion method which uses floating point numbers as real numbers has a possibility to give a wrong answer by the effect of numerical round-off errors. In this paper, we show a modification of the algorithm of the CF expansion method so that it uses floating point (FP) intervals as replacements of real numbers. By this modified algorithm, the answer is obtained only when its correctness is guaranteed and the possibility to give a wrong answer is eliminated.

Interval Versions of Central-Difference Method for Solving the Poisson Equation in Proper and Directed Interval Arithmetic

Abstract To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h2 +k2), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained.

Solving the Poisson Equation by an Interval Difference Method of the Second Order

The paper deals with an interval difference method for solving the Poisson equation based on the conventional central-difference method. We present the interval method in full details. The method is constructed in such a way that the exact solution is included in the interval solution obtained. Some numerical results obtained in floating-point interval arithmetic are also presented.

AN INTERVAL FINITE DIFFERENCE METHOD FOR THE BIOHEAT TRANSFER PROBLEM DESCRIBED BY THE PENNES EQUATION WITH UNCERTAIN PARAMETERS

In this paper the transient bioheat transfer problem given by the one-dimensional Pennes equation with mixed boundary conditions is considered. The model assumes the heat transfer between the skin and its surroundings in the case of a natural and forced convection. For computations the interval finite difference method of Crank- -Nicolson type together with the floating-point interval arithmetic is used. In this way, uncertain geometric and thermophysical parameters can be represented in the form of intervals as well as the resultant temperature distribution over time.