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Chapter 1
Numerical Algorithms
This opening chapter introduces the basic concepts of numerical algorithms and scientiﬁc comput-ing.We begin with a general, brief introduction to the ﬁeld in Section 1.1. This is followed by themoresubstantialSections1.2and 1.3. Section1.2discusses the basic errorsthat maybe encounteredwhen applying numerical algorithms. Section 1.3 is concerned with essential properties of suchalgorithms and the appraisal of the results they produce.We get to the “meat” of the material in later chapters.
1.1 Scientiﬁc computing
Scientiﬁc computing is a discipline concerned with the development and study of
numerical al-gorithms
for solving mathematical problems that arise in various disciplines in science and engin-eering.Typically, the starting point is a given
mathematical model
which has been formulated inan attempt to explain and understand an observed phenomenon in biology, chemistry, physics, eco-nomics, or any other scientiﬁc or engineeringdiscipline. We will concentrate on those mathematicalmodels which are
continuous
(or
piecewise continuous
) and are difﬁcult or impossible to solve ana-lytically; this is usually the case in practice. Relevant application areas within computer science andrelatedengineeringﬁelds includegraphics, visionand motionanalysis, imageand 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 piecewisecontinuous problem is approximated by a discrete one. Functions are approximated by ﬁnite arraysof values. Algorithms are then sought which approximately solve the mathematical problem efﬁ-ciently, accurately, and reliably. This is the heart of scientiﬁc computing.
Numerical analysis
maybe 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 seemselementary and obvious, may actually pose rather difﬁcult and complex practical challenges. Themain purpose of this book is to equip you with basic methods and analysis tools for handling suchchallenges as they arise in future endeavors.
1
16.10. Additional notes 537
16.10 Additional notes
There are many books devoted to the numerical solution of initial value ODEs. The material in Sec-tions 16.2
–
16.7 is covered more thoroughly in Ascher and Petzold [5]. Deeper, more encyclopedicbooks are Hairer, Norsett, and Wanner [36] and Hairer and Wanner [37].Many iterative methods in optimization and linear algebra, including most of those describedin Chapters 3, 7, and 9, can be written as
y
i
+
1
=
y
i
+
h
i
f
(
y
i
),
i
=
0,1,
. . .
,where
h
i
is a scalar step size. This reminds one of Euler
’
s method for the ODE system
d
y
dt
=
f
(
y
).The independentvariable
t
is an
“
arti
ﬁ
cial time
”
variable. Much has been made of such connectionsrecently, and this simple observation does prove important in some instances. But caution should beexercised here: always ask yourself if the
“
discovery
”
of the arti
ﬁ
cial ODE actually adds somethingin your quest for better algorithms for your given problem.Much research on methods for stiff problems was carried out in the 1970s and 1980s. De-spite the simplicity of the test equation there is signi
ﬁ
cant general complication in stiff problems,essentially because some fast scales that are present in the given ODE are not approximated well incases where these scales don
’
t show up as a fast variation in the sought solution. This is fundamen-tally different from the usual nonstiff scenario, where the discretization typically approximates allscales well. Pioneering work was done by Gear [28]. The most exhaustive reference known to usremains [37].The problem described in Example 16.4 is one in a set of initial value ODE applications usedfor testing research codes and maintained by F. Mazzia and F. Iavernaro in http://pitagora.dm.uniba.it/
∼
testset/.A lot of attention has been devoted to numerical methods for
dynamical systems
; see Stuartand Humphries [66] and Strogatz [65].Signi
ﬁ
cant recent research work has been carried out in the context of
geometric integration
,and we refer the reader to Hairer, Lubich, and Wanner [35] and Leimkuhler and Reich [49] forcomprehensive accounts on this topic. There is a lighter version in Ascher [3].A relatively readable coverage of numerical methods for boundary value ODEs is [5]. Anearlier, pioneering work is Keller [46]. A deeper, more encyclopedic treatment can be found inAscher, Mattheij, and Russell [4].There is vast literature on numerical methods for PDEs. Two recent textbooks are LeVeque[50] and [3]. We mention further only Trefethen [69] for spectral methods, Elman, Silvester, andWathen[23]foranemphasisonlineariterativesolvers,andTrottenberg,Oosterlee,andSchuller[71]for multigrid methods. Let us repeat that our present treatment of this topic in Section 16.8 is meantto give just a taste, and other texts (and more advanced courses) are required to cover it properly.

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