Numerical Mathematical Analysis
Outline
1
Introduction, matlab notes
2
Chapter 1: Taylor polynomials
3
Chapter 2: Error and Computer Arithmetic
4
Chapter 4: 4.1 - 4.3 Interpolation
5
Chapter 5: Numerical integration and differentiation
6
Chapter 3: Rootfinding
Numerical Mathematical Analysis
Math 1070
> Introduction
Numerical Analysis
Numerical Analysis: This refers to the analysis of mathematical
problems by numerical means, especially mathematical problems
arising from models based on calculus.
Effective numerical analysis requires several things:
An understanding of the computational tool being used, be it a
calculator or a computer.
An understanding of the problem to be solved.
Construction of an algorithm which will solve the given
mathematical problem to a given desired accuracy and
within the limits of the resources (time, memory, etc) that are
available.
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This is a complex undertaking. Numerous people make this their lifes
work, usually working on only a limited variety of mathematical
problems.
Within this course, we attempt to show the spirit of the subject.
Most of our time will be taken up with looking at algorithms for
solving basic problems such as rootfinding and numerical integration;
but we will also look at the structure of computers and the
implications of using them in numerical calculations.
We begin by looking at the relationship of numerical analysis to the
larger world of science and engineering.
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Traditionally, engineering and science had a two-sided approach to
understanding a subject: the theoretical and the experimental. More recently,
a third approach has become equally important: the computational.
Traditionally we would build an understanding by building theoretical
mathematical models, and we would solve these for special cases.
For example, we would study the flow of an incompressible irrotational fluid
past a sphere, obtaining some idea of the nature of fluid flow.
But more practical situations could seldom be handled by direct means,
because the needed equations were too difficult to solve.
Thus we also used the experimental approach to obtain better information
about the flow of practical fluids.
The theory would suggest ideas to be tried in the laboratory, and the
experimental results would often suggest directions for a further development
of theory.
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With the rapid advance in powerful computers, we now can augment
the study of fluid flow by directly solving the theoretical models of fluid
flow as applied to more practical situations; and this area is often
referred to as computational fluid dynamics.
At the heart of computational science is numerical analysis; and to
effectively carry out a computational science approach to studying a
physical problem, we must understand the numerical analysis being
used, especially if improvements are to be made to the computational
techniques being used.
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Mathematical Models
A mathematical model is a mathematical description of a physical
situation.
By means of studying the model, we hope to understand more about
the physical situation. Such a model might be very simple. For
example,
A = 4πRe2 ,
Re ≈ 6, 371 km
is a formula for the surface area of the earth. How accurate is it?
1
First, it assumes the earth is sphere, which is only an
approximation. At the equator, the radius is approximately 6,378
km; and at the poles, the radius is approximately 6,357 km.
2
Next, there is experimental error in determining the radius; and in
addition, the earth is not perfectly smooth.
Therefore, there are limits on the accuracy of this model for the
surface area of the earth.
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An infectious disease model
For rubella measles, we have the following model for the spread of the
infection in a population (subject to certain assumptions).
ds
= −asi
dt
di
= asi − bi
dt
dr
= bi
dt
In this, s, i, and r refer, respectively, to the proportions of a total
population that are susceptible, infectious, and removed (from the
susceptible and infectious pool of people). All variables are functions
of time t. The constants can be taken as
a=
1. Introduction, Matlab notes
6.8
,
11
b=
1
.
11
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Mathematical Models
The same model works for some other diseases (e.g. flu), with a
suitable change of the constants a and b. Again, this is an
approximation of reality (and a useful one).
But it has its limits. Solving a bad model will not give good results, no
matter how accurately it is solved; and the person solving this model
and using the results must know enough about the formation of
the model to be able to correctly interpret the numerical results.
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The logistic equation
This is the simplest model for population growth. Let N (t) denote the
number of individuals in a population (rabbits, people, bacteria, etc).
Then we model its growth by
N 0 (t) = cN (t),
t ≥ 0,
N (t0 ) = N0
The constant c is the growth constant, and it usually must be
determined empirically.
Over short periods of time, this is often an accurate model for
population growth. For example, it accurately models the growth of
US population over the period of 1790 to 1860, with c = 0.2975.
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The predator-prey model
Let F (t) denote the number of foxes at time t; and let R(t) denote the
number of rabbits at time t. A simple model for these populations is called
the Lotka-Volterra predator-prey model:
dR
= a[1 − bF (t)]R(t)
dt
dF
= c[−1 + dR(t)]F (t)
dt
with a, b, c, d positive constants.
If one looks carefully at this, then one can see how it is built from the logistic
equation. In some cases, this is a very useful model and agrees with physical
experiments.
Of course, we can substitute other interpretations, replacing foxes and rabbits
with other predator and prey. The model will fail, however, when there are
other populations that affect the first two populations in a significant way.
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1. Introduction, Matlab notes
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Newton’s second law
Newtons second law states that the force acting on an object is directly
proportional to the product of its mass and acceleration,
F ∝ ma
With a suitable choice of physical units, we usually write this in its scalar
form as
F = ma
Newtons law of gravitation for a two-body situation, say the earth and an
object moving about the earth is then
m
d2 r(t)
Gm me r(t)
=−
·
2
dt
|r(t)|2 |r(t)|
with r(t) the vector from the center of the earth to the center of the object
moving about the earth. The constant G is the gravitational constant, not
dependent on the earth; and m and me are the masses, respectively of the
object and the earth.
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This is an accurate model for many purposes. But what are some
physical situations under which it will fail? When the object is very
close to the surface of the earth and does not move far from one spot,
we take |r(t)| to be the radius of the earth. We obtain the new model
m
d2 r(t)
= −mgk
dt2
with k the unit vector directly upward from the earths surface at the
location of the object. The gravitational constant
.
g = 9.8 meters/second2
Again this is a model; it is not physical reality.
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Matlab notes
Matlab designed for numerical computing.
Strongly oriented towards use of arrays, one and two dimensional.
Excellent graphics that are easy to use.
Powerful interactive facilities; and programs can also be written in
it.
It is a procedural language, not an object-oriented language.
It has facilities for working with both Fortran and C language
programs.
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At the prompt in Unix or Linux, type Matlab.
Or click the Red Hat, then DIVMS, then Mathematics, then Matlab.
Run the demo program (simply type demo). Then select one of the
many available demos.
To seek help on any command, simply type
help command
or use the online Help command. To seek information on Matlab
commands that involve a given word in their description, type
lookfor word
Look at the various online manuals available thru the help page.
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MATLAB is an interactive computer language.
For example, to evaluate
y = 6 − 4x + 7x2 − 3x5 +
3
x+2
use
y = 6 − 4 ∗ x + 7 ∗ x ∗ x − 3 ∗ x5 + 3/(x + 2);
There are many built-in functions, e.g.
√
exp(x), cos(x), x, log(x)
The default arithmetic used in MATLAB is double precision (16
decimal digits and magnitude range 10−308 : 10+308 ) and real.
However, complex arithmetic appears automatically when needed.
sqrt(-4) results in an answer of 2i.
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The default output to the screen is to have 4 digits to the right of the
decimal point.
To control the formatting of output to the screen, use the command
format. The default formatting is obtained using
format short
To obtain the full accuracy available in a number, you can use
format long
The commands
format short e
format long e
will use ‘scientific notation’ for the output. Other format options are
also available.
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SEE plot trig.m
MATLAB works very efficiently with arrays, and many tasks are best done
with arrays. For example, plot sin x and cos x on the interval 0 ≤ x ≤ 10.
t = 0 : .1 : 10;
x = cos(t); y = sin(t);
plot(t, x, t, y, ’LineWidth’, 4)
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
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1
2
3
4
5
6
7
8
9
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The statement
t = a : h : b;
with h > 0 creates a row vector of the form
t = [a, a + h, a + 2h, . . .]
giving all values a + jh that are less than b.
When h is omitted, it is assumed to be 1. Thus
n=1:5
creates the row vector
n = [1, 2, 3, 4, 5]
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Arrays
b = [1, 2, 3]
creates a row vector of length 3.
A = [1 2 3; 4 5 6; 7 8 9]
creates the square matrix

123
A= 456 
789

Spaces or commas can be used as delimiters in giving the components of an
array; and a semicolon will separate the various rows of a matrix.
For a column vector,
b = [1 3 − 6]0


1
results in the column vector  3 
−6
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Array Operations
Addition: Do componentwise addition.
A = [1, 2; 3, −2; −6, 1];
B = [2, 3; −3, 2; 2, −2];
C = A + B;
results in the answer


3
5
0 
C= 0
−4 −1
Multiplication by a constant: Multiply the constant times each component
of the array.
D = 2 ∗ A;
results in the answer
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

2
4
6 −4 
D=
−12
2
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Array Operations
Matrix multiplication: This has the standard meaning.
E = [1, −2; 2, −1; −3, 2]
F = [2, −1, 3; −1, 2, 3];
G = E ∗ F;
results in the answer




4 −5 −3
1 −2 2
−1
3
3 
=  5 −4
G =  2 −1 
−1
2 3
−8
7 −3
−3
2
A nonstandard notation:
H = 3 + F;
results in the computation
1 1 1
2 −1 3
5 2 6
H=3
+
=
1 1 1
−1
2 3
2 5 6
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Componentwise operations
Matlab also has component-wise operations for multiplication, division and
exponentiation. These three operations are denoted by using a period to
precede the usual symbol for the operation. With
a = [1 2 3];
b = [2 −, 1 4];
we have
a.∗b = [2 − 2 12]
a./b = [.5 − 2.0 0.75]
a.ˆ3 = [1 8 27]
2.ˆa = [2 4 8]
b.ˆa = [2 1 64]
The expression
3
x+2
can be evaluated at all of the elements of an array x using the command
y = 6 − 4x + 7x2 − 3x5 +
y = 6 − 4 ∗ x + 7 ∗ x.∗x − 3 ∗ x.ˆ5 + 3./(x + 2);
The output y is then an array of the same size as x.
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Special arrays
A = zeros(2, 3)
produces an array with 2 rows and 3 columns, with all components set
to zero,
0 0 0
0 0 0
B = ones(2, 3)
produces an array with 2 rows and 3 columns, with all components set
to 1,
1 1 1
1 1 1
eye(3) results in the 3 × 3 identity matrix,


1 0 0
 0 1 0 
0 0 1
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Array functions
There are many MATLAB commands that operate on arrays, we
include only a very few here.
For a vector x, row or column, of length n, we have the following
functions.
max(x) = maximum component of x
min(x) = minimum component of x
abs(x) = vector of absolute values of components of x
sum(x) = sum
p of the components of x
norm(x)= |x1 |2 + · · · + |xn |2
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Script files
A list of interactive commands can be stored as a script file.
For example, store
t = 0 : .1 : 10;
x = cos(t); y = sin(t);
plot(t, x, t, y)
with the file name plot trig.m. Then to run the program, give the
command
plot trig
The variables used in the script file will be stored locally, and
parameters given locally are available for use by the script file.
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Functions
To create a function, we proceed similarly, but now there are input and
output parameters. Consider a function for evaluating the polynomial
p(x) = a1 + a2 x + a3 x2 + · · · + an xn−1
MATLAB does not allow zero subscripts for arrays.
The following function would be stored under the name polyeval.m.
The coefficients {aj } are given to the function in the array named
coeff, and the polynomial is to be evaluated at all of the components
of the array x.
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Functions
function value = polyeval(x,coeff);
%
% function value = polyeval(x,coeff)
%
% Evaluate a polynomial at the points given in x.
% The coefficients are to be given in coeff.
% The constant term in the polynomial is coeff(1).
n = length(coeff)
value = coeff(n)*ones(size(x));
for i = n-1:-1:1
value = coeff(i) + x.*value;
end
>> polyeval(3,[1,2])
yields
n=2
ans = 7
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