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set7 - $ Set 7 Iterative Methods for Solving Equations Part...

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a39 a38 a36 a37 Set 7: Iterative Methods for Solving Equations: Part 1 Kyle A. Gallivan Department of Mathematics Florida State University Foundations of Computational Math 1 Fall 2010 1
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a39 a38 a36 a37 Overview The second half of the course deals with the application of the ideas of convergent iteration and optimization of a scalar cost function to solve: 1. linear sytems of equations 2. nonlinear equations 3. systems of nonlinear equations 4. unconstrained optimization problems 2
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a39 a38 a36 a37 Overview Two parts: 1. Iterations that are contraction mappings for solving: linear sytems of equations nonlinear equations systems of nonlinear equations 2. Iterations that minimize scalar cost functions for solving: linear sytems of equations unconstrained optimization problems 3
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a39 a38 a36 a37 Overview Iteration x k = F ( x 0 , . . . , x k 1 ) in general particular form of F depends on problem and other constraints such as efficiency x 0 , x 1 , x 2 , . . . must converge to a solution of the problem We will consider: 1. construction of the iteration 2. necessary and sufficient conditions for convergence to the desired solution 3. efficiency of the iteration 4
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a39 a38 a36 a37 Iterative Methods for Linear Systems Additonal References and Source Material: Iterative Methods for Sparse Linear Systems, Yousef Saad, SIAM Press, Second Edition. Matrix Iterative Analysis, Richard Varga, Prentice Hall. Applied Iterative Methods, L. A. Hageman and D. M. Young, Academic Press. Iterative Solution Methods, O. Axelsson, Cambridge University Press. Analysis of Numerical Methods, E. Isaacson and H. Keller, Wiley. 5
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a39 a38 a36 a37 Outline for Iterative Methods for Linear Systems Motivation Linear Stationary Methods Examples Convergence Analysis Convergence Behavior Implementation Issues 6
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a39 a38 a36 a37 Motivation Iterative methods produce a series of approximations to the solution of Ax = b , i.e., x 0 , x 1 , x 2 , . . . x k , . . . such that x k x = A 1 b A is an n × n and n very large. A is a sparse matrix and the fill-in in the factorization is unacceptably large. A good guess at x is avaiable and we wish to improve it. High accuracy is not required and we want to save computations. We want to improve the accuracy of a direct method that was degraded due to saving computations. 7
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a39 a38 a36 a37 Motivation The matrix A is not available finite element discretization of a continuous domain analysis of a discrete network of devices – circuit simulation action of Av z is the sum of actions of elements or devices on pieces of v Assume that there is structure that makes computation of Av z O ( n ) or O ( n log n ) typically, e.g., sparse or Toeplitz. If A is available then it is stored in an efficient data structure.
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