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program4 - Program 4 Foundations of Computational Math 1...

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Program 4 Foundations of Computational Math 1 Fall 2010 Due date: via email before the Final Exam begins on Wednesday, 12/8/10 General Task Implement codes to use preconditioned steepest descent and preconditioned conjugate gra- dient to solve Ax = b where A R n × n is symmetric positive definite. You must demonstrate your code on multiple examples for three situations: 1. Take A = Λ = diag ( λ 1 , λ 2 , . . . , λ n ), i.e., a diagonal matrix and apply CG and steepest descent without preconditioning, i.e., set your preconditioner to I . You should choose various combinations of values for the λ i to demonstrate how the spectrum affects the convergence of the two methods. Make sure that you demonstrate for a range of n the key points made in the text and notes about the convergence of these methods. 2. Any symmetric positive definite matrix A can be written A = Q Λ Q T where Λ is a diagonal matrix with the eigenvalues of A (which are postive) forming the diagonal, and Q is an orthogonal matrix containing eigenvectors of A .
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