Algebracprob - Algebra C Numerical Linear Algebra — Sample Exam Problems Notation Denote by V a finite-dimensional Hilbert space with inner

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Algebra C Numerical Linear Algebra — Sample Exam Problems Notation. Denote by V a finite-dimensional 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, Gauss-Seidel, 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 )...
View Full Document

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.

Page1 / 5

Algebracprob - Algebra C Numerical Linear Algebra — Sample Exam Problems Notation Denote by V a finite-dimensional Hilbert space with inner

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online