MA3264 Mathematical Modelling Lecture 7

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MA3264 Mathematical ModellingLecture 7

• Review Chapters 1-6
• (including dynamical systems, eigenvalues,cubic
  splines)

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Discrete Dynamical Systems
Can be expressed recursively in the form
Initial State
set of d x 1 matrices
same as column vectors
Dynamics
where
set of d x d matrices
and
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Discrete Dynamical Systems
Example 1 A Car Rental Company pages 35-38
number of cars in Orlando at end of day n
number of cars in Tampa at end of day n
Linear Algebra Formulation
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Discrete Dynamical Systems
Example 2 The Battle of Trafalgar pages 38-41
number of British ships at stage n
number of French-Spanish ships at stage n
Linear Algebra Formulation
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Discrete Dynamical Systems
Example 3 Price Variation Problem 6 pages 49-50
price of product at year n
quantity of product at year n
Linear Algebra Formulation
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Discrete Dynamical Systems
Example 4 Fibonacci Sequence Problem 1 page 290
n-th term of the Fibonacci sequence
Linear Algebra Formulation
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Discrete Dynamical Systems
Example 5 Pollution in the Great Lakes pages
222-223
pollution in Lake A after n years
pollution in Lake B after n years
Linear Algebra Formulation
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Discrete Dynamical Systems
Equilibrium is a vector
that satisfies
We observe that
This gives us a closed formula for the n-th term !
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Discrete Dynamical Systems
Equilibria are clearly useful !
Therefore the following questions are important.
1. When do equilibria exist ?
Answer Iff
2. When do they exist and are unique ?
Answer Iff
3. When are they stable ? This means that x(n)
converges for every initial value x(0).
Answer Iff
Linear algebra and eigenvalues are very important
!
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Eigenvalues
Consider the following linear algebra equation
set of complex numbers, please learn them !
where
is an eigenvalue
is an eigenvector
with eigenvalue
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Eigenvalues
Eigenvalues are clearly useful !
Therefore the following questions are important.
an eigenvalue of a given matrix
1. When is
Answer Iff there exists a nonzero vector
or equivalently,
such that
2. What conditions on any matrix
determine the
existence of a nonzero vector
such that
Answer Iff the determinant of
vanishes.
This is expressed as
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Characteristic Polynomial
of a square (d x d) matrix
is defined by
Remark.
can and should be regarded as a function
defined by a monic degree d whose
roots are the eigenvalues of
characteristics Charakteristika pl
charakteristische Merkmale charakteristische
Eigenschaften Eigentümlichkeiten pl
The Man without Qualities (German original title
Der Mann ohne Eigenschaften) is a novel in three
books by the Austrian novelist and essayist
Robert Musil.
One of the great novels of the 20th century,
Musil's three-volume epic is now available in a
highly praised translation. It may look
intimidating, but in fact the story of Ulrich,
wealthy ex-soldier, seducer and scientist, the
'man without qualities', proceeds in short, pithy
chapters, each one abounding in wit and
intellectual energy. Lisa Jardine, the eminent
historian, wrote of it 'Musil's hero is a
scientist who finds his science inadequate to
help him understand the irrational and
unpredictable world of pre-World War I Austria.
The novel is perceptive and at times baroque
account of Ulrich's search for meaning and love
in a society hurtling towards political
catastrophe.'
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Characteristic Polynomial
Example 1
Question What are the eigenvalues of
Example 2
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Characteristic Polynomial
symmetric matrix
Example 3
rotation matrix
Example 4
Question When are the eigenvalues in Ex. 3,4 real
?
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Diving Boards
Remark. A diving board of length L bends to
minimize
Bending Energy
subject to the constraints at its ends. For small
deformations we use the approximation
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Cubic Spline Approximation
Therefore the shape of a diving board can be
approximately described by a function y
y(x), for x in the interval 0,L, that minimizes

subject to the constraints at its ends.
Theorem The condition above implies that
is a cubic polynomial. Furthermore, if
is unconstrained then
This is called a
natural, as opposed to a clamped, boundary
condition.
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Suggested Reading
Section 4.4 Cubic Spline Models pages 159-168.
Learn more about regression and its use in
statistics
http//en.wikipedia.org/wiki/Regression_analysis
file///C/MATLAB6p5/help/techdoc/math_anal/datafu
13.html17217
Experiment with the web based least squares
regression
http//www.scottsarra.org/math/courses/na/nc/polyR
egression.html
http//www.statsdirect.com/help/regression_and_cor
relation/poly.htm
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Tutorial 7 Due Week 1317 Oct
Problem 1. For each of the five examples of
discrete dynamical systems discussed in these
lectures, determine if (i) equilibria exist, (ii)
if they are unique, and (iii) are they stable.
Prove your answers by computing the appropriate
quantities (ranks and eigenvalues). Also write
and run a computer program to compute and plot
each component of x(n) for n 1,2,,40 where you
choose a reasonable starting value x(0).
Problem 2. Compute the coefficients of the cubic
polynomial y(x) that give the shape of a diving
board from these constraints
Problem 3. Write a program to generate the random
numbers
and use another program to fit a quadratic model
to this data. Explain the actual versus
expected sum of squared errors.