Methods and Applications of Error-Free Computation

This book is written as an introduction to the theory of error-free computation. In addition, we include several chapters that illustrate how error-free com­ putation can be applied in practice. The book is intended for seniors and first­ year graduate students in fields of study involving scientific computation using digital computers, and for researchers (in those same fields) who wish to obtain an introduction to the subject. We are motivated by the fact that there are large classes of ill-conditioned problems, and there are numerically unstable algorithms, and in either or both of these situations we cannot tolerate rounding errors during the numerical computations involved in obtaining solutions to the problems. Thus, it is important to study finite number systems for digital computers which have the property that computation can be performed free of rounding errors. In Chapter I we discuss single-modulus and multiple-modulus residue number systems and arithmetic in these systems, where the operands may be either integers or rational numbers. In Chapter II we discuss finite-segment p-adic number systems and their relationship to the p-adic numbers of Hensel [1908]. Each rational number in a certain finite set is assigned a unique Hensel code and arithmetic operations using Hensel codes as operands is mathe­ matically equivalent to those same arithmetic operations using the cor­ responding rational numbers as operands. Finite-segment p-adic arithmetic shares with residue arithmetic the property that it is free of rounding errors.

Table of Contents:
I Residue or Modular Arithmetic.- 1. Introduction.- 2. Single-Modulus Residue Arithmetic.- 3. Multiple-Modulus Residue Arithmetic.- 4. Mapping Standard Residue Representations onto Integers.- 5. Single-Modulus Residue Arithmetic with Rational Numbers.- 6. The Forward Mapping and the Inverse Mapping.- 7. Multiple-Modulus Residue Arithmetic with Rational Numbers.- II Finite-Segment p-adic Arithmetic.- 1. Introduction.- 2. The Field of p-adic Numbers.- 3. Arithmetic in Qp.- 4. A Finite-Segment p-adic Number System.- 5. Arithmetic Operations on Hensel Codes.- 6. Removing a Leading Zero from a Hensel Code.- 7. Mapping a Hensel Code onto a Unique Order-N Farey Fraction.- III Exact Computation of Generalized Inverses.- 1. Introduction.- 2. Properties of g-inverses.- 3. Applications of g-inverses.- 4. Exact Computation of A+ if A Is a Rational Matrix.- 5. Failures of Residue Arithmetic and Precautionary Measures.- IV Integer Solutions to Linear Equations.- 1. Introduction.- 2. Theoretical Background.- 3. The Matrix Formulation of Chemical Equations.- 4. Solving the Homogeneous System.- 5. Solving a Non-Homogeneous System.- 6. Solving Interval Linear Programming Problems.- 7. The Solution of Systems of Mixed-Integer Linear Equations.- V Iterative Matrix Inversion and the Iterative Solution of Linear Equations.- 1. Introduction.- 2. The Newton-Schultz Method for the Matrix Inverse.- 3. Iterative Solution of a Linear System.- 4. Iterative Computation of g-inverses.- VI The Exact Computation of the Characteristic Polynomial of a Matrix.- 1. Introduction.- 2. The Algorithm Applied to Lower Hessenberg Matrices.