Computers manipulate approximations of real numbers, called floating-point numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floating-point operations lose so much precision that they quickly become irrelevant. In this article, we review some of the problems one can encounter, focussing on the IEEE754-1985 norm. We give a (sketch of a) semantics of its basic operations then abstract them (in the sense of abstract interpretation) to extract information about the possible loss of precision. The expected application is abstract debugging of software ranging from simple on-board systems (which use more and more on-the-shelf micro-processors with floating-point units) to scientific codes. The abstract analysis is demonstrated on simple examples and compared with related work.

Static Analyses of the Precision of Floating-Point Operations

An algorithm of Schur for characterizing power series that are bounded in the unit circle is shown to have applications to a variety of problems in science and engineering. These include speech analysis and synthesis, inverse scattering, decoding of error-correcting codes, synthesis of digital filters, modeling of random signals, Pade approximation for linear systems, and zero location of polynomials.

A Theorem of I. Schur and Its Impact on Modern Signal Processing

An Introduction to Numerical Analysis for Electrical and Computer Engineers: Zarowski/Intro to Numerical Analysis

Preface.1 Functional Analysis Ideas.1.1 Introduction.1.2 Some Sets.1.3 Some Special Mappings: Metrics, Norms, and Inner Products.1.3.1 Metrics and Metric Spaces.1.3.2 Norms and Normed Spaces.1.3.3 Inner Products and Inner Product Spaces.1.4 The Discrete Fourier Series (DFS).Appendix 1.A Complex Arithmetic.Appendix 1.B Elementary Logic.References.Problems.2 Number Representations.2.1 Introduction.2.2 Fixed-Point Representations.2.3 Floating-Point Representations.2.4 Rounding Effects in Dot Product Computation.2.5 Machine Epsilon.Appendix 2.A Review of Binary Number Codes.References.Problems.3 Sequences and Series.3.1 Introduction.3.2 Cauchy Sequences and Complete Spaces.3.3 Pointwise Convergence and Uniform Convergence.3.4 Fourier Series.3.5 Taylor Series.3.6 Asymptotic Series.3.7 More on the Dirichlet Kernel.3.8 Final Remarks.Appendix 3.A COordinate Rotation DIgital Computing (CORDIC).3.A.1 Introduction.3.A.2 The Concept of a Discrete Basis.3.A.3 Rotating Vectors in the Plane.3.A.4 Computing Arctangents.3.A.5 Final Remarks.Appendix 3.B Mathematical Induction.Appendix 3.C Catastrophic Cancellation.References.Problems.4 Linear Systems of Equations.4.1 Introduction.4.2 Least-Squares Approximation and Linear Systems.4.3 Least-Squares Approximation and Ill-Conditioned Linear Systems.4.4 Condition Numbers.4.5 LU Decomposition.4.6 Least-Squares Problems and QR Decomposition.4.7 Iterative Methods for Linear Systems.4.8 Final Remarks.Appendix 4.A Hilbert Matrix Inverses.Appendix 4.B SVD and Least Squares.References.Problems.5 Orthogonal Polynomials.5.1 Introduction.5.2 General Properties of Orthogonal Polynomials.5.3 Chebyshev Polynomials.5.4 Hermite Polynomials.5.5 Legendre Polynomials.5.6 An Example of Orthogonal Polynomial Least-Squares Approximation.5.7 Uniform Approximation.References.Problems.6 Interpolation.6.1 Introduction.6.2 Lagrange Interpolation.6.3 Newton Interpolation.6.4 Hermite Interpolation.6.5 Spline Interpolation.References.Problems.7 Nonlinear Systems of Equations.7.1 Introduction.7.2 Bisection Method.7.3 Fixed-Point Method.7.4 Newton-Raphson Method.7.4.1 The Method.7.4.2 Rate of Convergence Analysis.7.4.3 Breakdown Phenomena.7.5 Systems of Nonlinear Equations.7.5.1 Fixed-Point Method.7.5.2 Newton-Raphson Method.7.6 Chaotic Phenomena and a Cryptography Application.References.Problems.8 Unconstrained Optimization.8.1 Introduction.8.2 Problem Statement and Preliminaries.8.3 Line Searches.8.4 Newton's Method.8.5 Equality Constraints and Lagrange Multipliers.Appendix 8.A MATLAB Code for Golden Section Search.References.Problems.9 Numerical Integration and Differentiation.9.1 Introduction.9.2 Trapezoidal Rule.9.3 Simpson's Rule.9.4 Gaussian Quadrature.9.5 Romberg Integration.9.6 Numerical Differentiation.References.Problems.10 Numerical Solution of Ordinary Differential Equations.10.1 Introduction.10.2 First-Order ODEs.10.3 Systems of First-Order ODEs.10.4 Multistep Methods for ODEs.10.4.1 Adams-Bashforth Methods.10.4.2 Adams-Moulton Methods.10.4.3 Comments on the Adams Families.10.5 Variable-Step-Size (Adaptive) Methods for ODEs.10.6 Stiff Systems.10.7 Final Remarks.Appendix 10.A MATLAB Code for Example 10.8.Appendix 10.B MATLAB Code for Example 10.13.References.Problems.11 Numerical Methods for Eigenproblems.11.1 Introduction.11.2 Review of Eigenvalues and Eigenvectors.11.3 The Matrix Exponential.11.4 The Power Methods.11.5 QR Iterations.References.Problems.12 Numerical Solution of Partial Differential Equations.12.1 Introduction.12.2 A Brief Overview of Partial Differential Equations.12.3 Applications of Hyperbolic PDEs.12.3.1 The Vibrating String.12.3.2 Plane Electromagnetic Waves.12.4 The Finite-Difference (FD) Method.12.5 The Finite-Difference Time-Domain (FDTD) Method.Appendix 12.A MATLAB Code for Example 12.5.References.Problems.13 An Introduction to MATLAB.13.1 Introduction.13.2 Startup.13.3 Some Basic Operators, Operations, and Functions.13.4 Working with Polynomials.13.5 Loops.13.6 Plotting and M-Files.References.Index.

An Introduction to Numerical Analysis for Electrical and Computer Engineers

CORDIC processors are powerful computing systems for applications involving large amount of rotation operations or for evaluating several elementary functions. In this paper, a unified view of the different architectures proposed in literatures is presented. The goal is to provide a wide spectrum of architectures, a coordinated and comprehensive design methodology, and the main figures of merit characterizing architectures' performance and complexity. This provides the basic guidelines to the designer for an optimal choice of the architectural approach with respect the specific requirements and constraints of the specific application.

A unified view of CORDIC processor design

(1983). Calculator Function Approximation. The American Mathematical Monthly: Vol. 90, No. 5, pp. 317-325.

Charles W. Schelin

Papers

Calculator Function Approximation

Multicomponent vapor-liquid equilibrium: Direct integration of the Gibbs-Duhem equation