The methods of significance arithmetic are applied to the numerical solution of a nonlinear partial-differential equation. Our approach permits the use of initial values having imprecision considerably greater than that of rounding error; moreover, the intermediate and final quantities are monitored so that at any stage the precision of such quantities is available. An algorithm is found that represents faithfully the solution to a difference-equation approximation to Burgers' equation.

The methods of significance arithmetic are applied to the numerical solution of a nonlinear partial differential equation. Our approach permits the use of initial values having imprecision considerably greater than that of rounding error; moreover, the intermediate and final quantities are monitored so that at any stage the precision of such quantities is available. An algorithm is found that represents faithfully the solution to a difference equation approximation to Burgers' equation.

A single format for the representation of numbers in a computer is proposed to accommodate both exact and inexact quantities. A consistent set of rules is described for addition (subtraction), multiplication, and division of such quantities, both within their separate types, as well as in combination. Error correlation aside, the propagation of inherent errors is monitored in operations with at least one imprecise value. A definitive algorithm must, of course take into account any correlations of inherent errors; these correlations must be recognized and incorporated into the algorithm by the numerical analyst, not by the logical designer of the computer.

A single format for the representation of numbers in a computer is proposed to accommodate both exact and inexact quantities. A consistent set of rules is described for addition, multiplication and division of such quantities, both within their separate types, as well as in combination. Error correlation aside, the propagation of inherent errors is monitored in operations with at least one imprecise input. As a simple application, extension of the Forsythe algorithm for finding the roots of a quadratic equation is discussed.

A particular set of unnormalized arithmetic operations termed ``basic'' are described, in the context of the University of Chicago Maniac III Computer. Each basic operation involves three operand words and generates two result words, all in unnormalized floating point format. The use of these operations in the implementation of multi-precision arithmetic is explained; in particular, it is demonstrated that multiprecision division can be effected in a nontentative manner with their aid.

Algorithms for floating point computer arithmetic are described, in which fractional parts are not subject to the usual normalization convention. These algorithms give results in a form which furnishes some indication of their degree of precision. An analysis of one-stage error propagation is developed for each operation; a suggested statistical model for long-run error propagation is also set forth.

The usual floating point arithmetic makes error analysis difficult. This paper describes an alternative system which offers a means of analyzing floating point calculations more effectively and which also possesses certain advantages from an equipment standpoint.