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Unformatted text preview: Comment on Program 2 Foundations of Computational Math 1 Fall 2009 1 The Spectral Radii From homework we have that A = T =  1 ... ... ... 1  1 ... ... 1  1 ... . . . . . . . . . . . . . . . ... 1  1 ... ... 1  1 ... ... ... 1 j =  2cos j, = n + 1 q j = ( sin( j ) , sin(2 j ) ,..., sin( nj ) ) T and that G J =  1 T . We therefore have ( G J ) =  2cos n  = 2cos 1 and since the matrix is tridiagonal ( G gs ) = 2 ( G J ) The number of expected steps for Jacobi given an initial error norm of e and a desired error norm of e d is then d = (log 10 e d log 10 e ) log 10 ( G J ) and half of that for GaussSeidel. We have the following spectral radii for the values of and n of interest. 1 n cos 1 ( G J ) ( G gs ) 2 100 0.99951628 0.9995163 0.9990328 2 1000 0.99999508 0.9999951 0.9999902 2 2000 0.99999877 0.9999988 0.9999975 2 10000 0.99999995 1.0000000 0.9999999 3 100 0.99951628 0.6663442 0.4440146 3 1000 0.99999508 0.6666634 0.4444401 3 2000 0.99999877 0.6666658 0.4444433 3 10000 0.99999995 0.6666666 0.4444444 4 100 0.99951628 0.4997581 0.2497582 4 1000 0.99999508 0.4999975 0.2499975 4 2000 0.99999877 0.4999994 0.2499994 4 10000 0.99999995 0.5000000 0.2500000 Note that for = 2 the methods converge very slowly if at all in practice. Also note that is more important than n in determining the radii. So we would not expect significant variation in the convergence as a function of n for a given . For Symmetric GaussSeidel we have for a general symmetric positive definite matrix, A A = D L L T M = ( D L ) D 1 ( D L T ) = A + LD 1 L T G = I M 1 A = ( D L T ) 1 L ( D L ) 1 L T If A is symmetric positive definite then so is D and M . M therefore has a Cholesky factor ization M = CC T , C = ( D L ) D 1 / 2 The iteration matrix G is not necessarily symmetric but it is similar to a symmetric positive definite matrix. It therefore has real positive eigenvalues and is positive definite in this more general sense. The eigenvalues of G can also be shown to satisfy a generalized symmetric definite eigenvalue problem. It can also be seen from this that ( G ) < 1 for A symmetric positive definite. All of these facts can be derived as follows....
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This note was uploaded on 07/25/2011 for the course MAD 5403 taught by Professor Gallivan during the Spring '11 term at University of Florida.
 Spring '11
 Gallivan

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