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Unformatted text preview: Algebra C Numerical Linear Algebra — Sample Exam Problems Notation. Denote by V a finitedimensional Hilbert space with inner product ( · , · ) and corre sponding norm k·k . The abbreviation SPD is used for symmetric positive definite . If a linear operator A : V 7 → V is SPD, then ( · , · ) A := ( Ax,y ) is an inner product and the corresponding norm is denoted by k · k A . For a square matrix A , λ j ( A ) , 1 ≤ j ≤ n denote the eigenvalues of A . A generic linear iterative method towards the solution of Ax = b , A ∈ IR n × n is as follows: Given initial guess x , for k = 0,1,... define x k + 1 as x k + 1 = x k + B ( b Ax k ) . Different choices of B result in Jacobi, GaussSeidel, SOR, Richardson, etc methods. 1. Let A ∈ R n × n , and let A k = { a ij } k i,j = 1 , k = 1,2,...,n . a) Prove that if all of the A k are nonsingular, then there exists unique decom position A = LU , where L is a unit lower triangular matrix and U is an upper triangular matrix. b) Prove or disprove: if A is nonsingular (but nothing is known about the invertibility of A k ), then there exists a permutation P , and unique pair consisting of a unit lower triangular matrix L and an upper triangular matrix U such that PA = LU . 2. Consider the conjugate gradient method for the minimization of F ( u ) := 1 2 ( Au,u )  ( b,u ) , where A ∈ IR n × n is SPD. Thus starting with u = , r = b and p = r the successive approximations to the minimizer are computed by u k + 1 = u k + α k p k , r k + 1 = r k α k Ap k ; p k + 1 = r k + 1 β k p k , where α k = ( r k ,p k )...
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This note was uploaded on 01/27/2010 for the course EE 4343 taught by Professor Asdfasdsas during the Spring '10 term at Aarhus Universitet.
 Spring '10
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