Concise Guide to Numerical Algorithmics. The Foundations and Spirit of Scientific Computing

Preface
	Acknowledgements
	References
Contents
1 The Big Picture: Mathematics, Science and Engineering
	1.1 Introduction
	1.2 Arenas and Their Interfaces
	1.3 Modi Operandi
	1.4 Summary
	References
2 Number: The Language of Science
	2.1 Introduction
	2.2 Symbolism in Human Languages
	2.3 From Words to Numbers
	2.4 History and Background
	2.5 The Symbolism of Mathematics
	2.6 From Numbers to Number Systems
		2.6.1 Positive Numbers
		2.6.2 Negative Numbers
		2.6.3 Other Number Systems and Beyond
	References
3 Algorithmics: The Spirit of Computing
	3.1 Introduction: What is an Algorithm?
	3.2 Theoretical Foundations: The Grand Unified Theory of Computation
	3.3 A Hierarchy of Theoretical Models
		3.3.1 Knuth’s Realistic Model: The MMIX Computer
		3.3.2 Register, or Random-Access, Computational Models
		3.3.3 The Computational Models of Turing and Post
		3.3.4 Computational Equivalence and the Halting Problem
	3.4 Practical Foundations: Floating-Point Model
		3.4.1 Finite-Precision Representation: Decimal Base
		3.4.2 Finite-Precision Representation: Arbitrary Base
		3.4.3 Axiomatic Floating-Point Arithmetic and an FP-RAM
	3.5 Algorithmics
	3.6 Computational Complexity
	3.7 Algorithmic Systems and Beyond
	References
4 A Taxonomy of Numerical Problems
	4.1 Introduction
	4.2 A “Cubist” Portrait
	4.3 Classes of Numerical Problems
		4.3.1 A-F: Deterministic Parameters
		4.3.2 G-L: Stochastic Parameters
	4.4 Discussion
	References
5 Numerical Analysis: Algorithm Under the Rubric of Number
	5.1 Introduction
	5.2 Theoretical Foundations of Numerical Analysis
		5.2.1 The BCSS Model of Computation
	5.3 Practical Foundations of Numerical Analysis
		5.3.1 The IEEE 754 Floating-Point Standard
		5.3.2 The Chebfun Model
	References
6 Numerical Algorithmics: Number Under the Rubric of Algorithm
	6.1 Introduction
	6.2 Definition and Rationale for Numerical Algorithmics
	6.3 Theoretical Foundations
		6.3.1 Real-Number Models in the Turing Tradition
	6.4 Practical Foundations
		6.4.1 Floats
		6.4.2 Unums and Posits
	6.5 NAS&E: Content and Organization
	6.6 Conclusion
	References