Introduction to Numerical Analysis
*
Hector D. Ceniceros
1
What is Numerical Analysis?
This is an introductory course of Numerical Analysis, which
comprises the
design, analysis, and implementation of constructive methods 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.
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. Con
*
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1
sider the problem of calculating a definite integral
I
[
f
] =
Z
b
a
f
(
x
)
dx.
(1)
In most cases we cannot find an exact value of
I
[
f
] and very often we only
know the integrand
f
at finite number of points in [
a, b
].
The problem is
then to produce an approximation to
I
[
f
] as accurate as we need and at a
reasonable computational cost.
2.1
An Approximation Principle and the Trapezoidal
Rule
One of the central ideas in Numerical Analysis is to approximate a given
function by simpler functions which we can analytically integrate, differenti
ate, etc. For example, we can approximate the integrand
f
(
x
) in [
a, b
] by the
segment of the straight line, a linear polynomial
P
1
(
x
), that passes through
(
a, f
(
a
)) and (
b, f
(
b
)). That is
f
(
x
)
≈
P
1
(
x
) =
f
(
a
) +
f
(
b
)

f
(
a
)
b

a
(
x

a
)
.
(2)
and
Z
b
a
f
(
x
)
dx
≈
Z
b
a
P
1
(
x
)
dx
=
f
(
a
)(
b

a
) +
1
2
[
f
(
b
)

f
(
a
)](
b

a
)
=
1
2
[
f
(
a
) +
f
(
b
)](
b

a
)
.
(3)
That is
Z
b
a
f
(
x
)
dx
≈
b

a
2
[
f
(
a
) +
f
(
b
)]
.
(4)
The right hand side is known as the
simple Trapezoidal Rule Quadrature
. A
quadrature is a method to approximate an integral.
How accurate is this
approximation?
Clearly, if
f
is a linear polynomial then the Trapezoidal
Rule would give us the exact value of the integral, i.e.
it would be exact.
The underlying question is how well does a linear polynomial
P
1
, satisfying
P
1
(
a
) =
f
(
a
)
,
(5)
P
1
(
b
) =
f
(
b
)
,
(6)
2
approximates
f
on the interval [
a, b
]? We can almost guess the answer. The
approximation is exact at
x
=
a
and
x
=
b
because of (5)(6) and it is
exact for all polynomial of degree
≤
1. This suggests that
f
(
x
)

P
1
(
x
) =
Cf
00
(
ξ
)(
x

a
)(
x

b
), where
C
is a constant. But where is
f
00
evaluated at?