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Unformatted text preview: Iterative Methods for Large Linear Systems Section 11.5 Not in Custom Edition – use these slides • Suppose that linear system Ax = b has 1 million equations and unknowns • Gaussian elimination requires O(n 3 ) arithmetic operations • 10 18 work is prohibitive • Iterative methods for solving Ax=b can be more economical • But they require and infinite number of iterations to give an exact solution • Thus we terminate them when desired accuracy is reached • Need an initial estimate of the solution General Iterative Method Idea: suppose that an estimate x of the solution is known. Compute the new estimate x 1 as: In general: A=L+D+U Strictly lower triangular diagonal strictly upper triangular Jacobi’s method Jacobi’s method Choose the matrix splitting as: Lower, diagonal, upper scalar form A= L+D+U Matrix form Jacobi Process one equation at a time: i=1,…n Diagonal must be nonzero How fast does it converge? How fast does it converge?...
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 Fall '11
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 Economics, Gauss–Seidel method, Jacobi method, Iterative method, Jacobi

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