A precision of computation in the projective space

Abstract

Precision of computation and stability are the key issues in all computational methods. There are a lot of problems that lead to a “nearly singular” formulation and if standard approaches are taken wrong results are usually obtained. The projective formulation of many computational problems seems to be very appealing as the division operation is not needed if result(s) can remain in the projective representation. This paper focuses on computational precision using the projective space representation. Properties of this approach are demonstrated on an inversion of the Hilbert matrix, as the inverse is known analytically and determinant converges to zero. Also, we will compare the proposed approach with the standard method for solving linear systems of equations – the comparison is based on pivoted Gaussian method and its projective variant, using the previously developed library PLib for the .NET environment. The paper proves that elimination of the division operation is entirely possible while preserving the precision of the calculation and simplicity of code. This could even lead to a significant performance boost with appropriate hardware support.