Solution of linear systems with verified accuracy

Abstract

In the past several years new methods have been derived for solving algebraic problems with high accuracy and with automatic verification of the correctness of the result. The introduction of the new methods consists of four parts: (1) TO obtain a result on a computer the correctness of which is verified to be correct, a precisely defined computer arithmetic is indispensable. Moreover, a computer arithmetic with maximum accurate results for an> single operation would be desirable. The definition of maximum accuracy for a computed result is easy, the theoretical and practical foundation is a new theory developed by Kulisch and Miranker [2] to be described in the following. A computer result is an approximation of the precisely defined real result by a floating-point number. Although the real (infinite precise) result is in general not computable, it may serve to define the term maximum accuracy. As long as no overflow or underflow occurs, there are essentially two cases. First, the precise result may be exactly a machine (floating-point) number. Second, there are two floating-point neighbours left and right to the precise real result. In the first case, the maximum accurate floating-point result is obviously the exact result which happens to be a machine number. In the second case, both floating-point neighbours of the precise result are of maximum accuracy and it depends on the rounding mode which result will be delivered. There are four essential rounding modes, namely rounding to nearest, rounding downwards, rounding upwards and rounding towards zero. For rounding to the nearest floating-point number there is the special case that the precise result is exactly the midpoint of the two floating-point neighbours. In this case the result is usually the floating-point number of larger magnitude. The definition of maximum accurate floating-point operations is clear, but the theoretical bases and implementation is not trivial. One reason for this is that the exact, infinite precise result is in general not known or difficult to compute. However, in numerical analysis not only bare numbers as approximations to real quantities occur but also complex numbers and, more general, vectors and matrices over real and complex numbers. It is desirable to provide the well-defined operations for vectors and matrices also with maximum accuracy. Obviously for that purpose a dot product with maximum accuracy is necessary. The implementation of a dot product with maximum accuracy is possible as has been shown by Bohlender [l]. It turns out that the maximum accurate dot product is essentially sufficient to provide all operations for real or complex scalars, vectors and matrices with maximum accuracy 121. (2) The ultimate goal is the implementation of algorithms delivering result with automatic verification of the correctness. A verification can be provided by delivering sets including the solution. The correctness, i.e. the fact that the delivered set actually contains the solution to the