Exact Real Arithmetic
Real numbers can be represented elegantly with infinite precision by an infinite sequence of integer matrices
- PhD Thesis: Exact Real Arithmetic using Mobius Transformations
Develops a domain-theoretic, computationally feasible framework for exact real arithmetic. Unifies the redundant binary and continued-fraction representations into a general normal product built from Mobius transformations, specialised to "exact floating point", with algorithms for arithmetic and transcendental functions and a working implementation in Miranda.
- Algorithms for Exact Real Arithmetic using Mobius Transformations
Tutorial given at the Real Number Computation summer school (Indiana University, 1998). Introduces real numbers as sequences of nested closed intervals with endpoints restricted to countable subsets, building toward Mobius-transformation-based representations.
- Efficient on-line computation of real functions using exact floating point
Shows that incrementality and efficiency can be achieved simultaneously in exact real arithmetic, representing real functions as expression trees of Mobius transformations with efficient conversion and range-reduction algorithms.
- A New Representation for Exact Real Numbers
With Abbas Edalat. Develops the theoretical foundation for representing real numbers as infinite products of integer matrices, showing that a canonical choice of sign and digit matrices can represent any real number as a shrinking sequence of nested intervals.
- Exact Real Computer Arithmetic
With Abbas Edalat. Surveys the main frameworks for exact real arithmetic and introduces a new, feasible, incremental representation based on Mobius transformations with same-signed integer coefficients, implemented in C++, Java and Miranda.
- Semantics of Exact Real Arithmetic
With Abbas Edalat and Martin Escardo. Incorporates the Mobius-transformation representation of real numbers into PCF with products, presenting two models proven computationally adequate with respect to the operational semantics.
- Exact Real Arithmetic based on Linear Fractional Transformations
With Abbas Edalat. Introduces a feasible, incremental framework for exact real arithmetic based on linear fractional transformations with same-signed integer coefficients, including algorithms for basic arithmetic and elementary functions.