This opening chapter introduces the basic concepts of numerical algorithms and scientific comput-
We begin with a general, brief introduction to the field in Section 1.1. This is followed by the
more substantial Sections 1.2 and 1.3. Section 1.2 discusses the basic errors that may be encountered
when applying numerical algorithms. Section 1.3 is concerned with essential properties of such
algorithms and the appraisal of the results they produce.
We get to the “meat” of the material in later chapters.
Scientific computing is a discipline concerned with the development and study of
for solving mathematical problems that arise in various disciplines in science and engin-
Typically, the starting point is a given
which has been formulated in
an attempt to explain and understand an observed phenomenon in biology, chemistry, physics, eco-
nomics, or any other scientific or engineering discipline. We will concentrate on those mathematical
models which are
) and are difficult or impossible to solve ana-
lytically; this is usually the case in practice. Relevant application areas within computer science and
related engineering fields include graphics, vision and motion analysis, image and signal processing,
search engines and data mining, machine learning, and hybrid and embedded systems.
In order to solve such a model approximately on a computer, the continuous or piecewise
continuous problem is approximated by a discrete one. Functions are approximated by finite arrays
of values. Algorithms are then sought which approximately solve the mathematical problem effi-
ciently, accurately, and reliably. This is the heart of scientific computing.
be viewed as the theory behind such algorithms.
The next step after devising suitable algorithms is their implementation. This leads to ques-
tions involving programming languages, data structures, computing architectures, etc. The big pic-
ture is depicted in Figure 1.1.
The set of requirements that good scientific computing algorithms must satisfy, which seems
elementary and obvious, may actually pose rather difficult and complex practical challenges. The
main purpose of this book is to equip you with basic methods and analysis tools for handling such
challenges as they arise in future endeavors.
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