AMS210-02Spring09PracticeTest2 - stable distribution p Ap=p...

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AMS 210.03 David Fried Test 2 B Name_________________________________ No calculators are permitted. Show ALL work. 1. Let A= 2 3 7 4 7 3 2 3 3 ± a) Use elimination by pivoting (Gauss-Jordan elimination) to find the inverse of the matrix A. (10 points) b) Use your answer in (a) to find the solutions to Ax=b, where b= 100 100 100 ± . (5 points) c) If b 3 (the third entry in b) is increased by 10, what will the change in the solutions be (DO NOT answer by first solving Ax=b with the new b). (5 points) d) Give the LU decomposition of A. (5 points) e) Find the determinant of A. (5 points) 2. a) Set up the following system of equations in the form x=Dx+c for solution by Jacobi iteration. Show why it will converge: (10 points) 6x -y+2z=30 x+4y -z=12 -x+2y+4z=20 b) Let x (k-1) , x (k) , and x (k+1) be the (k-1) th , k th , and (k+1) th iterates, respectively. Prove that: |x (k+1) -x (k) |<|x (k) -x (k-1) | (for any norm). (10 points) 3. For the Markov chain transition probability matrix A below, solve the system of equations for finding the
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Unformatted text preview: stable distribution p: Ap=p or (A-I)p=0 (*) First find the general family of solutions to (*), and then find the solutions that is a probability vector. A= ² ³ ³ ³ ³ ´ 1/3 2/3 2/3 1/6 2/3 1/6 1/6 1/6 1/6 1/6 1/6 2/3 1/6 2/3 2/3 1/3 µ ¶ ¶ ¶ ¶ · (10 points) 4. Consider the following growth model for elephants and mice: E’ = E - M M’=2E+4M a) Determine the eigenvalues of the coefficient matrix. (10 points) b) Determine the associated eigenvectors. (10 points) c) Write the coefficient matrix in the diagonalized form UD λ U-1 . (5 points) 5. a) Find the condition number (with respect to the sum norm) of the coefficient matrix for the system of equations: (10 points) x- y= 1 2x+4y=-1 b) Give a bound on the relative error (percentage error) in the solution if the coefficient 4 in the second equation is erroneously entered as 3. (5 points)...
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This note was uploaded on 08/08/2010 for the course AMS 210 taught by Professor Fried during the Spring '08 term at SUNY Stony Brook.

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