MA3264 Mathematical Modelling Lecture 7

1. MA3264 Mathematical ModellingLecture 7 Review Chapters 1-6 (including dynamical systems, eigenvalues,cubic splines)

2. 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

3. 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

4. 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

5. 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

6. Discrete Dynamical Systems Example 4 Fibonacci Sequence Problem 1 page 290 n-th term of the Fibonacci sequence Linear Algebra Formulation

7. 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

8. Discrete Dynamical Systems Equilibrium is a vector that satisfies We observe that This gives us a closed formula for the n-th term !

9. 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 !

10. Eigenvalues Consider the following linear algebra equation set of complex numbers, please learn them ! where is an eigenvalue is an eigenvector with eigenvalue

11. Eigenvalues Eigenvalues are clearly useful ! Therefore the following questions are important. 1. When is an eigenvalue of a given matrix 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

12. 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 characteristicsCharakteristika {pl}; charakteristischeMerkmale; charakteristischeEigenschaften; 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.'

13. Characteristic Polynomial Example 1 Question What are the eigenvalues of Example 2

14. Characteristic Polynomial symmetric matrix Example 3 rotation matrix Example 4 Question When are the eigenvalues in Ex. 3,4 real ?

15. 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

16. 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.

17. 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/datafu13.html#17217 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

18. Tutorial 7 Due Week 13–17 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.