Book contents
- Frontmatter
- Contents
- List of statistical and mathematical tables
- Preface
- PART I BASIC NUMERICAL TECHNIQUES
- 1 Introduction
- 2 Errors, mistakes and the arrangement of work
- 3 The real roots of non-linear equations
- 4 Simple methods for smoothing crude data
- 5 The area under a curve
- 6 Finite differences, interpolation and numerical differentiation
- 7 Some other numerical techniques
- PART II BASIC STATISTICAL TECHNIQUES
- PART III THE METHOD OF LEAST SQUARES
- Appendix
- References
- Author index
- Subject index
- Frontmatter
- Contents
- List of statistical and mathematical tables
- Preface
- PART I BASIC NUMERICAL TECHNIQUES
- 1 Introduction
- 2 Errors, mistakes and the arrangement of work
- 3 The real roots of non-linear equations
- 4 Simple methods for smoothing crude data
- 5 The area under a curve
- 6 Finite differences, interpolation and numerical differentiation
- 7 Some other numerical techniques
- PART II BASIC STATISTICAL TECHNIQUES
- PART III THE METHOD OF LEAST SQUARES
- Appendix
- References
- Author index
- Subject index
Summary
Summary In this chapter we summarise some important mathematical results which are frequently required to solve numerical and statistical problems arising in scientific work. First, we discuss some Taylor series expansions for functions of one variable; we continue with the notion of functions of two or more variables and of partial differentiation. Lastly, we outline the concept of a matrix and the basic matrix operations: addition, subtraction, multiplication and the derivation of an inverse.
The level of mathematics required
The level of mathematics necessary for understanding the techniques described in this book is quite modest. Yet, using these techniques we are able to solve many difficult numerical and statistical problems. We are able to solve non-linear equations in one or more unknowns, integrate and differentiate a given function (including empirical functions), smooth crude data, fit curves and interpolate. Some of the techniques we develop can be used to solve other complex problems. For example, the method of finite differences (chapter 6) is used extensively for solving differential equations.
Readers will be familiar with the mathematical concepts of differentiation (obtaining the slope of a curve) and integration (finding the area under a curve). There are certain other basic concepts and formulae which bear repeating and these are summarised in the remainder of this chapter.
The Taylor series expansion
This formula uses the value of a function f(x) and its derivatives at a particular point x to produce the value of that function at a neighbouring point x + h.
- Type
- Chapter
- Information
- A Handbook of Numerical and Statistical TechniquesWith Examples Mainly from the Life Sciences, pp. 3 - 13Publisher: Cambridge University PressPrint publication year: 1977