On Average Bit Complexity of Interval Arithmetic

In many practical situations, we know only the intervals which contain the actual (unknown) values of physical quantities. If we know the intervals x for a quantity x and y for another quantity y, then, for every arithmetic operation , the set of possible values of x y also forms an interval; the operations leading from x and y to this new interval are called interval arithmetic operations. For addition and subtraction, corresponding interval operations consist of two corresponding operations with real numbers, so there is no hope of making them faster. The best known algorithms for interval multiplication consists of 3 real-number multiplications and several comparisons. We describe a new algorithm for which the average time is equivalent to using only 2 multiplications of real numbers. What is interval arithmetic. Many computer algorithms for processing real numbers have been designed to process measurement results. Measurements are never 100% precise; therefore, when after measuring a physical quantity we get the measurement result e x, this does not mean that the actual value of this quantity is equal to e x: this actual value can take any value from the interval x = e x ? ; e x+ ], where is an upper bound on the measurement error (usually supplied by the manufacturer of the measuring instrument). If for two quantities x and y, we only know the intervals x = x; x] and y = y; y] of possible values, then, for every arithmetic operation (+; ?; ; =), we can only only conclude that x y 2 x y, where x y is deened as x y = fx y j x 2 x; y 2 yg: Thus deened arithmetic operations on the set of all intervals are called interval arithmetic. The results of interval arithmetic operations can be easily described in terms of lower and upper endpoints of the operand intervals: What is computational complexity of interval arithmetic? Formulation of the problem. When we process intervals corresponding to data uncertainty, our goal is to perform the corresponding interval arithmetic operations. The faster we perform them, the better, so the natural question is: what is the algebraic complexity of these operations? In other words, how many elementary operations with real numbers (actually, rational numbers) do we need to perform to implement one interval arithmetic operation? For addition, the answer is easy: addition of two intervals means two additions: …