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Fundamentals of Computing

Davoud Mirzaei

TL;DR

This lecture surveys the fundamentals of numerical computing, focusing on how numerical representations, rounding, and error sources affect the accuracy of computations on continuous problems. It develops a framework for analyzing error through notions of conditioning and stability, and applies it to simple algorithms (multiplication, summation, inner products, and linear systems) to illustrate forward and backward error concepts. Key topics include fixed-point and floating-point representations (IEEE 754), machine epsilon, subnormals, rounding modes, and the dangers of ill-conditioned problems such as those arising from Hilbert-type matrices and polynomial roots. The discussion emphasizes the practical impact of rounding and discretization on algorithmic reliability, and provides guidelines for avoiding numerical pitfalls through modeling choices, error analysis, and algorithm design. The material connects theoretical conditioning with concrete computational strategies, illustrating how to assess and improve robustness in scientific computing tasks.

Abstract

This lecture addresses some general ideas behind numerical computations ranging from representation of numbers in computers to stability and accuracy of standard algorithms for some simple mathematical problems.

Fundamentals of Computing

TL;DR

This lecture surveys the fundamentals of numerical computing, focusing on how numerical representations, rounding, and error sources affect the accuracy of computations on continuous problems. It develops a framework for analyzing error through notions of conditioning and stability, and applies it to simple algorithms (multiplication, summation, inner products, and linear systems) to illustrate forward and backward error concepts. Key topics include fixed-point and floating-point representations (IEEE 754), machine epsilon, subnormals, rounding modes, and the dangers of ill-conditioned problems such as those arising from Hilbert-type matrices and polynomial roots. The discussion emphasizes the practical impact of rounding and discretization on algorithmic reliability, and provides guidelines for avoiding numerical pitfalls through modeling choices, error analysis, and algorithm design. The material connects theoretical conditioning with concrete computational strategies, illustrating how to assess and improve robustness in scientific computing tasks.

Abstract

This lecture addresses some general ideas behind numerical computations ranging from representation of numbers in computers to stability and accuracy of standard algorithms for some simple mathematical problems.
Paper Structure (22 sections, 4 theorems, 179 equations, 10 figures, 6 tables)

This paper contains 22 sections, 4 theorems, 179 equations, 10 figures, 6 tables.

Table of Contents

  1. What is scientific computing?
  2. Computational simulation
  3. Sources of approximation
  4. Absolute and relative errors
  5. Computer representation of numbers
  6. Fixed-point representation of numbers
  7. Floating-point representation of numbers
  8. Rounding modes
  9. Subnormal numbers
  10. IEEE standard for floating point arithmetic
  11. Disasters caused by inappropriate use of floating point arithmetic
  12. Algebraic properties of floating point arithmetic
  13. Floating-point arithmetic models
  14. Conditioning of problems and stability of algorithms
  15. Conditioning of a mathematical problem
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let $f$ be continuously differentiable up to order $n+1$ on the interval $[a, b]$ and $x_0 \in [a, b]$. Then, for every $x \in [a, b]$, there exists a point $\xi$ between $x_0$ and $x$ such that where and

Figures (10)

  • Figure 1: The land of scientific computing (SC)
  • Figure 2: Problem solving steps: birds on the wire!
  • Figure 3: Simulation of the velocity filed and pressure of the air surrounding a car (image from www.vias3d.com)
  • Figure 4: Positive normalized floating-point numbers in $\mathbb{F}(2,3,-2,1)$
  • Figure 5: Non-negative normal and subnormal floating-point numbers in $\mathbb{F}(2,3,-2,1)$
  • ...and 5 more figures

Theorems & Definitions (38)

  • Example 2.1
  • Theorem 2.1
  • Example 2.2
  • Example 3.1
  • Remark 3.1
  • Definition 3.3
  • Example 3.2
  • Remark 3.2
  • Theorem 3.4
  • Theorem 3.5
  • ...and 28 more