Introduction to Numerical Math and Matlab ... - Ohio University

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Lab 13 Eigenvalues and Eigenvectors of Stochastic Matrices Finite Markov Process and Stochastic Matrices Imagine the following situation. Suppose that a system has n different states. Suppose that the system can change from one state to another state randomly, but with some restrictions. Namely, suppose that during a certain time interval, if the system is in some state i it has a fixed probabability of moving to another state j and this probability does not change over time. Denote this transition probability as t ij . If there are n states as supposed and there is probability of transition for each pair (i, j), then we can represent the system as the matrix T = (t ij ). Note that t ii would denote the probability of the system remaining in state i if it already is there. Given the basic law of probability that the sum of all probabilities must be one, we see that for any current state i, we have: n∑ t ij = 1. (13.1) j=1 A simple way of interpreting this equation is that no matter what state the system is in now it must be somewhere in the next step. A Stochastic or Markov matrix is an n × n matrix T for which 1. All entries t ij satisfy: 0 ≤ t ij ≤ 1. 2. Equation 13.1 holds. A 2 × 2 example of a Markov matrix is: T = ( .5 1/3 .5 2/3 We interpret this matrix in the following way: There are two possible states: A and B. If the system is in state A then there is a 50% chance of staying in state at the next time and a 50% chance of moving into state B. If the system is in State B, then there is an 2/3 chance it will remain there and a 1/3 chance it will move to state A at the next time step. You can imagine this concretely by marking two spots on the ground as A and B. Stand on A and roll a die. If 1, 2 or 3 is rolled stay where you are and if 4, 5 or 6 is rolled move to B. If you stay where you are then repeat. If you find yourself on B, roll a die. If 1 – 4 is rolled stay and if 5 or 6 is rolled move. 46 ) .
47 Likely States of a Markov Matrix If you actually carry out the above example you will find experimentally that you will spend more time on B than on A. Now suppose instead of one person there are many people playing this game. A natural question to ask that has very important implications in many applications is: “Typically, what portion of people will be on A and what portion will be on B?” We can investigate this question easily with the following mind experiment: If there are currently M 0 people on A at a certain time then at the next step move M 0 /2 of them to B and leave M 0 /2 at A. In there are N 0 people on B, then move N 0 /3 of them to A and leave 2N 0 /3 of them at B. It is easy to see that the new number of people at A and B are: M 1 = 1 2 M 0 + 1 3 N 0, N 1 = 1 2 M 0 + 2 3 N 0. and (13.2) It is not hard to see that this actually is an equation involving the matrix T, i.e.: ( ) ( ) ( ) M1 1/2 1/3 M0 = . N 1 1/2 2/3 N 0 Now if M and N are very large, than rather than count the numbers of people, we could simple keep track of the ratios of people on A and B. We can carry this out conveniently in Matlab. Make the matrix: > T = [ .5 1/3 ; .5 2/3] and make the intial vector supposing the half of the people are on each spot: > v = [ .5 ; .5 ] We can find the proportion at the next step by multiplying: > v = T*v If you repeat the above command (use the up arrow), it will produce the second step. If you continue to repeat it you will see very quickly that the vector v seems to be converging and after 20 steps it will have converged to ( .4 .6 If you continue repeating it will not change. We call this vector a Steady state of the Markov matrix, because ¯v = T ¯v. Implied by this equation is that ¯v is an eigenvector of T with eigenvalue 1. It is a mathematical fact that any transition matrix has 1 as an eigenvalue and that any eigenvector corresponding to 1 is a steady state of T. Thus to find the steady states of T all we really need to do is calculate the eigenvectors of T corresponding to eigenvalue 1. In Matlab find the eigenvalues and eigenvectors of T by the command: > [V E] = eig(T) You will see indeed that 1 is an entry of E in the second column. Thus the second column of V is the eigenvector corresponding to 1. Type: > v2 = V(;,2) to isolate this vector. You will see that it is not the vector ¯v = (.4, .6). However as we know eigenvectors are not unique and only their direction matters, thus with the command: ) .
Page 1 and 2: Introduction to Numerical Math and
Page 3 and 4: CONTENTS iii Lab 17. Integration: L
Page 5 and 6: Part I Matlab and Solving Equations
Page 7 and 8: 3 > plot(x,y,’*’) > plot(x,y,
Page 9 and 10: Lab 2 Matlab Programs - Input/Outpu
Page 11 and 12: 7 % mygraphs % plots the graphs of
Page 13 and 14: 9 The simplest way to accomplish th
Page 15 and 16: Lab 4 Controlling Error and while L
Page 17 and 18: 13 Exercises 4.1 In Calculus we lea
Page 19 and 20: 15 method and returns not only the
Page 21 and 22: 17 function x = mysecant(f,x0,x1,n)
Page 23 and 24: Lab 7 Symbolic Computations The foc
Page 25 and 26: 21 Other useful symbolic operations
Page 27 and 28: 23 Secant methods are better for ex
Page 29 and 30: Math 344 Sample Test Problems from
Page 31 and 32: Part II Linear Algebra c○Copyrigh
Page 33 and 34: 29 > A + A You can even add a numbe
Page 35 and 36: 31 Exercises 8.1 By hand, calculate
Page 37 and 38: 33 Figure 9.1: An equilateral truss
Page 39 and 40: 35 > b = [4 15 -10]’ > x = A\b Ne
Page 41 and 42: 37 On the other hand, a matrix that
Page 43 and 44: Lab 11 Accuracy, Condition Numbers
Page 45 and 46: 41 sensitivity is measured by the c
Page 47 and 48: Lab 12 Eigenvalues and Eigenvectors
Page 49: 45 Larger Matrices For a n × n mat
Page 53 and 54: Review of Part II Methods and Formu
Page 55 and 56: Part III Functions and Data c○Cop
Page 57 and 58: 53 When we tried a lower degree pol
Page 59 and 60: Lab 15 Spline Interpolation A chall
Page 61 and 62: Lab 16 Least Squares Interpolation:
Page 63 and 64: 59 Note that whenever you select a
Page 65 and 66: 61 ✻ ✟ ✟ ✟✟✟✟✟✟
Page 67 and 68: 63 efficient, first use x = linspac
Page 69 and 70: 65 ✻ ✟ ✉ ✟ ✟✟✟✟✟
Page 71 and 72: 67 In Matlab there is a built-in co
Page 73 and 74: 69 For another example of how meshg
Page 75 and 76: Review of Part III Methods and Form
Page 77 and 78: Part IV Appendices c○Copyright, T
Page 79 and 80: Appendix B Glossary of Matlab Comma
Page 81 and 82: 77 limit Finds two-sided limit, if
Page 83 and 84: 79 > 2*A ..........................
Page 85: Bibliography [1] Ayyup and McCuen,

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