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1 Project 2 Discussion Robb T. Koether Hampden-Sydney College Fri, Feb 8, 2013 Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
2 1 Introduction 2 Markov Processes 3 The Matrix Class Matrix Constructors Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
3 Outline 1 Introduction 2 Markov Processes 3 The Matrix Class Matrix Constructors Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
4 Three Vats Imagine we have three vats of water. Vat A contains 25 gallons. Vat B contains 50 gallons. Vat C contains 75 gallons. 25 gal Vat A 50 gal Vat B 75 gal Vat C Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
5 Three Vats We decide to transfer 10% of the water in A from A to B. 20% of the water in A from A to C. 20% of the water in B from B to A. 20% of the water in B from B to C. 30% of the water in C from C to A. 10% of the water in C from C to B. Then how much is in each vat? Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
6 Three Vats 50 gal Vat B 25 gal Vat A 75 gal Vat C Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
7 Three Vats 2.5 gal 50 gal Vat B 5.0 gal 25 gal 75 gal Vat A 10% from Vat A to Vat B 20% from Vat A to Vat C Vat C Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
8 Three Vats 2.5 gal 50 gal Vat B 10.0 gal 10.0 gal 5.0 gal 25 gal 75 gal Vat A 20% from Vat B to Vat A 20% from Vat B to Vat C Vat C Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
9 Three Vats 2.5 gal 50 gal Vat B 7.5 gal 10.0 gal 10.0 gal 25 gal 22.5 gal 5.0 gal 75 gal Vat A 30% from Vat C to Vat A 10% from Vat C to Vat B Vat C Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
10 Three Vats 2.5 gal 40 gal Vat B 7.5 gal 10.0 gal 10.0 gal 50 gal Vat A 22.5 gal 5.0 gal 60 gal Vat C Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
11 Three Vats 50 gal Vat A 40 gal Vat B 60 gal Vat C Now Vat A contains 50 gallons. Vat B contains 40 gallons. Vat C contains 60 gallons. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
12 Three Vats 5.0 gal 35 gal Vat B 6.0 gal 8.0 gal 8.0 gal 61 gal Vat A 18.0 gal 10.0 gal Now do it again 54 gal Vat C Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
13 Three Vats 5.0 gal 35 gal Vat B 6.0 gal 8.0 gal 8.0 gal 61 gal Vat A 18.0 gal 5.0 gal Now do it again 54 gal Vat C Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
14 Three Vats 61 gal Vat A 35 gal Vat B 54 gal Vat C Now Vat A contains 61 gallons. Vat B contains 35 gallons. Vat C contains 54 gallons. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
15 Some Questions Where is this headed? If we perform this transfer a total 50 times, how much water will be in each vat? What if we had started with 50 gallons in Vat A, 75 gallons in Vat B, and 25 gallons in Vat C? What if we had started with all 150 gallons in Vat A? Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
16 Outline 1 Introduction 2 Markov Processes 3 The Matrix Class Matrix Constructors Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
17 Markov Processes This situation is an example of a Markov process. The process may be represented by a matrix, called the transition matrix: The initial distribution may be represented by a vector: Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
18 Markov Processes To A B C From A B C Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
19 Markov Processes If we multiply the matrix by the vector, we get the resulting distribution after one transition = Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
20 Markov Processes If we multiply the matrix by the vector, we get the resulting distribution after one transition = Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
21 Markov Processes If we multiply the matrix by the vector, we get the resulting distribution after one transition = Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
22 Markov Processes If we multiply the matrix by the vector, we get the resulting distribution after one transition = Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
23 Markov Processes To get the result after a second transition, multiply again = Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
24 Markov Processes Let M be the transition matrix and let v 0 be the distribution vector. Then, after one transition, we have the distribution v 1 = Mv 0. After two transitions, we have the distribution v 2 = Mv 1 = M(Mv 0 ) = M 2 v 0. It is easy to see that after n transitions, the distribution will be v n = M n v 0. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
25 Outline 1 Introduction 2 Markov Processes 3 The Matrix Class Matrix Constructors Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
26 The Matrix Class What information (i.e., data) is relevant to the matrix? Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
27 The Matrix Class What information (i.e., data) is relevant to the matrix? The dimensions (rows, columns). Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
28 The Matrix Class What information (i.e., data) is relevant to the matrix? The dimensions (rows, columns). The matrix elements. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
29 The Matrix Class What actions do we want to take with the matrix? Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
30 The Matrix Class What actions do we want to take with the matrix? Constructors. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
31 The Matrix Class What actions do we want to take with the matrix? Constructors. Destructor. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
32 The Matrix Class What actions do we want to take with the matrix? Constructors. Destructor. Inspectors. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
33 The Matrix Class What actions do we want to take with the matrix? Constructors. Destructor. Inspectors. Mutators. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
34 The Matrix Class What actions do we want to take with the matrix? Constructors. Destructor. Inspectors. Mutators. Operators. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
35 The Matrix Class What actions do we want to take with the matrix? Constructors. Destructor. Inspectors. Mutators. Operators. Facilitators. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
36 The Matrix Class What actions do we want to take with the matrix? Constructors. Destructor. Inspectors. Mutators. Operators. Facilitators. Other functions. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
37 Outline 1 Introduction 2 Markov Processes 3 The Matrix Class Matrix Constructors Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
38 title How would we construct a matrix of doubles with 3 rows and 4 columns? We allocate each row as an array of 4 doubles. For each of these rows, we have a pointer-to-double that points to it. To hold these pointers, we allocate an array of 3 pointers-to-double. Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
39 title A 3 4 Matrix double Row 1 Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
40 title A 3 4 Matrix double double Row 1 Row 2 Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
41 title A 3 4 Matrix double double double Row 1 Row 2 Row 3 Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
42 title A 3 4 Matrix double** double* double double double Row 1 Row 2 Row 3 Robb T. Koether (Hampden-Sydney College) Project 2 Discussion Fri, Feb 8, / 25
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