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MA3264 Mathematical ModellingLecture 7
Review Chapters 1-6
(including dynamical systems, eigenvalues,cubic splines)
Discrete Dynamical Systems
Can be expressed recursively in the form
dRx )0(Initial State
Dynamics
0,)()1( nvnxMnxwhere
ddRM set of d x d matrices
set of d x 1 matricessame as column vectors
dRvand
Discrete Dynamical SystemsExample 1 A Car Rental Company pages 35-38
nO number of cars in Orlando at end of day n
7.4.
3.6.,
0
0,)( Mv
T
Onx
n
n
Linear Algebra Formulation
nTnnn TOO 3.6.1
number of cars in Tampa at end of day n
nnn TOT 7.4.1
Discrete Dynamical SystemsExample 2 The Battle of Trafalgar pages 38-41
nB number of British ships at stage n
11.0
1.01,
0
0,)( Mv
F
Bnx
n
n
Linear Algebra Formulation
nF number of French-Spanish ships at stage n
nnn FBB 1.01 nnn BFF 1.01
Discrete Dynamical SystemsExample 3 Price Variation Problem 6 pages 49-50
nP price of product at year n
12.
1.1,
20
50,)( Mv
Q
Pnx
n
n
Linear Algebra Formulation
nQ)500(1.01 nnn QPP
quantity of product at year n
)100(2.01 nnn PQQ
Discrete Dynamical SystemsExample 4 Fibonacci Sequence Problem 1 page 290
nF n-th term of the Fibonacci sequence
01
11,
0
0,)( 1 Mv
F
Fnx
n
n
Linear Algebra Formulation
110 FF
0,12 nFFF nnn
Discrete Dynamical SystemsExample 5 Pollution in the Great Lakes pages 222-223
na pollution in Lake A after n years
9.65.
1.35.,
0
0,)( Mv
b
anx
n
n
Linear Algebra Formulation
nnn baa 1.35.1
nb pollution in Lake B after n years
nnn bab 9.65.1
Discrete Dynamical SystemsEquilibrium is a vector
dRx vxMx
We observe that
that satisfies
1,)1()( nxnxMxnx
xnxMxnx )2()( 2
xxMxnxM n )0()3(3
xxMxnx n )0()(
This gives us a closed formula for the n-th term !
Discrete Dynamical SystemsEquilibria are clearly useful !
1. When do equilibria exist ?
Therefore the following questions are important.
)(rank)(rank MIvMI
1||eigenvaluean M
Answer Iff
2. When do they exist and are unique ?
Answer Iff dMIMI )(rankinvertible is )(
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 !
EigenvaluesConsider the following linear algebra equation
where
vMv ddRM
C is an eigenvalue
0, vRv dis an eigenvectorwith eigenvalue
set of complex numbers, please learn them !
Eigenvalues
Therefore the following questions are important.
an eigenvalue of a given matrix
,vMv
?MvAnswer Iff there exists a nonzero vector
such that or equivalently,
1. When is
Eigenvalues are clearly useful !
.0)det( A
.0)( vMI2. What conditions on any matrix Adetermine the
existence of a nonzero vectorv such that ?0AvAnswer Iff the determinant of A vanishes.
This is expressed as
Characteristic Polynomialof a square (d x d) matrix is defined by M
Remark. can and should be regarded as a function.),det()( CzMzIzP
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.'
P
P,: CCP
roots are the eigenvalues of
defined by a monic degree d whose
.M
Characteristic PolynomialExample 1
Question What are the eigenvalues of
1][ RaM azzP )(?M
Example 22R
dc
baM
)()(det)( 2 bcadzdazdzc
bazzP
).det()(tr2 MzMz
2
4)d-(
2
)det(4))((tr)(trseigenvalue
22 bcadaMMM
Characteristic Polynomial
Example 32R
db
baM
2
4)d-(seigenvalue
22 bada
Example 4
symmetric matrix
cossin
sincosM rotation
matrix
1,sincosseigenvalue ii
Question When are the eigenvalues in Ex. 3,4 real ?
Diving Boards
Remark. A diving board of length L bends to minimize
)(y xdsd
xy
Bending Energy dsL
dsd
0
2subject to the constraints at its ends. For small deformations we use the approximation
,1)(0
2xdxxs
x
dxdy
Cubic Spline ApproximationTherefore 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
dxyEL
0
2subject to the constraints at its ends.
)(xyTheorem The condition above implies that
is a cubic polynomial. Furthermore, if )(Lyis unconstrained then .0)( Ly This is called a
natural, as opposed to a clamped, boundary condition.
Suggested Reading
Section 4.4 Cubic Spline Models pages 159-168.
Experiment with the web based least squares regression
http://www.scottsarra.org/math/courses/na/nc/polyRegression.html
http://www.statsdirect.com/help/regression_and_correlation/poly.htm
file:///C:/MATLAB6p5/help/techdoc/math_anal/datafu13.html#17217
http://en.wikipedia.org/wiki/Regression_analysis
Learn more about regression and its use in statistics
Tutorial 7 Due Week 13–17 OctProblem 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: .)(,0)0()0( dLyyy Problem 3. Write a program to generate the random numbers
10000,...,2,1,randn25.312)( 2 kkkky
and use another program to fit a quadratic model to this data. Explain the actual versus ‘expected’ sum of squared errors.
