## Methods and applications of error-free computation |

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### Contents

Mapping Standard Residue Representations onto Integers | 17 |

The Forward Mapping and the Inverse Mapping | 31 |

MultipleModulus Residue Arithmetic with Rational Numbers | 49 |

Copyright | |

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### Common terms and phrases

adic expansion adic numbers Algorithm 6.26 arbitrary arithmetic operations base vector Cauchy sequence Chapter characteristic polynomial chemical equation chemical reaction coefficients column commutative ring components Consequently continued fraction convergence Corollary corresponding defined Definition divisor equivalent Example exists finite field finite-segment p-adic floating-point Hensel code g-inverse gcd(a gcd(c Gregory Hence implies integer elements integer in Dm integer solution inverse mapping iteration Krishnamurthy Lemma linear algebraic equations mantissa method metric space minimum-norm mixed-radix digits Moore-Penrose multiple-modulus residue arithmetic multiplicative inverse n-tuple non-zero nonsingular matrix nonsingular modulo normalized floating-point Hensel obtain operands order-N Farey fraction order-TV positive integer prime Proof Prove Theorem pseudo-overflow radix radix-5 rank rational numbers real numbers reflexive inverse Remark residue class residue number system residue representation result rounding errors Section seed matrix single-modulus residue arithmetic singular matrix Smith canonical form standard residue step subtraction system of linear unique write zero