Chapter 1
Introduction
1.1
What is Numerical Analysis?
This is an introductory course of Numerical Analysis, which
comprises the
design, analysis, and implementation of constructive methods and algorithms
for the solution of mathematical problems
.
Numerical Analysis has vast applications both in Mathematics and in
modern Science and Technology. In the areas of the Physical and Life Sci-
ences, Numerical Analysis plays the role of a virtual laboratory by providing
accurate solutions to the mathematical models representing a given physical
or biological system in which the system’s parameters can be varied at will, in
a controlled way. The applications of Numerical Analysis also extend to more
modern areas such as data analysis, web search engines, social networks, and
basically anything where computation is involved.
1.2
An Illustrative Example: Approximating
an Integral
The main principles and objectives of Numerical Analysis are better illus-
trated with concrete examples and this is the purpose of this chapter.
Consider the problem of calculating a definite integral
I
[
f
] =
Z
b
a
f
(
x
)
dx.
(1.1)
3

4
1.2.1An Approximation PrincipleOne of the central ideas in Numerical Analysis is to approximate a givenfunction or data by simpler functions which we can analytically evaluate,integrate, di↵erentiate, etc. For example, we can approximate the integrandfin [a, b] by the segment of the straight line, a linear polynomialp1(x), thatpasses through (a, f(a)) and (b, f(b)). That isf(x)⇡p1(x) =f
(1.4)The right hand side is known as thesimple Trapezoidal Rule Quadrature. Aquadrature is a method to approximate an integral.How accurate is thisapproximation? Clearly, if