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set13 - $ Set 13 General Projection Iterations Kyle A...

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a39 a38 a36 a37 Set 13: General Projection Iterations Kyle A. Gallivan Department of Mathematics Florida State University Numerical Linear Algebra 1 Fall 2010 1
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a39 a38 a36 a37 References Iterative methods for sparse linear systems, Yousef Saad, SIAM Press, Second Edition. Iterative Krylov methods for large linear systems, H. Van der Vorst, Cambridge Press Iterative methods for solving linear systems, A. Greenbaum, SIAM Press Matrix iterative analysis, R. Varga, Prentice Hall 2
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a39 a38 a36 a37 Basic Definitions < v 1 , v 2 > = v H 2 v 1 , A C n × n H = 1 2 ( A + A H ) , and S = 1 2 i ( A A H ) A = H + iS, H H = H and S H = S if Au i = u i λ i , λ i C , u i C n × n , < u i , u i > = 1 then λ i = < Au i , u i > = < Hu i , u i > + i < Su i , u i >, and ( λ i ) = < Hu i , u i >, ( λ i ) = < Su i , u i > 3
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a39 a38 a36 a37 Basic Definitions Definition 13.1. A C n × n is a positive definite matrix if any of the following equivalent conditions are true: 1. ( λ i ) > 0 for all eigenvalues of A . 2. < Hz, z >> 0 for all z C n . 3. < Ax, x >> 0 for all x R n . where the inequalities require that the inner products are real. 4
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a39 a38 a36 a37 Projections Choose two spaces of dimension 1 k n , K and L Determine a projector P such that for any vector x R n Px K x Px L You can think of L as defining a lens or a viewing angle through which you are drawing a vector at a right angle until you hit K . 5
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a39 a38 a36 a37 Simple Case k = 1 and n = 2 x - Px x L K Px 6
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a39 a38 a36 a37 Simple Case k = 1 and n = 2 x - Px x L K Px 7
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a39 a38 a36 a37 Algebraic Conditions Px K x Px L is equivalent to K L = { 0 } which is equivalent to K L = R n 8
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a39 a38 a36 a37 Projectors and Bases Theorem 13.1. Let
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