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# 10_iterative - Serial Iterative Methods Parallel Iterative...

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Serial Iterative Methods Parallel Iterative Methods Parallel Numerical Algorithms Chapter 10 – Iterative Methods for Linear Systems Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign CSE 512 / CS 554 Michael T. Heath Parallel Numerical Algorithms 1 / 43

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Serial Iterative Methods Parallel Iterative Methods Outline 1 Serial Iterative Methods Stationary Iterative Methods Krylov Subspace Methods 2 Parallel Iterative Methods Partitioning Ordering Chaotic Relaxation Michael T. Heath Parallel Numerical Algorithms 2 / 43
Serial Iterative Methods Parallel Iterative Methods Stationary Iterative Methods Krylov Subspace Methods Iterative Methods for Linear Systems Iterative methods for solving linear system Ax = b begin with initial guess for solution and successively improve it until solution is as accurate as desired In theory, inﬁnite number of iterations might be required to converge to exact solution In practice, iteration terminates when residual k b - Ax k , or some other measure of error, is as small as desired Iterative methods are especially useful when matrix A is sparse because, unlike direct methods, no ﬁll is incurred Michael T. Heath Parallel Numerical Algorithms 3 / 43

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Serial Iterative Methods Parallel Iterative Methods Stationary Iterative Methods Krylov Subspace Methods Jacobi Method Beginning with initial guess x (0) , Jacobi method computes next iterate by solving for each component of x in terms of others x ( k +1) i = ± b i - X j 6 = i a ij x ( k ) j ² /a ii , i = 1 ,...,n If D , L , and U are diagonal, strict lower triangular, and strict upper triangular portions of A , then Jacobi method can be written x ( k +1) = D - 1 ± b - ( L + U ) x ( k ) ² Michael T. Heath Parallel Numerical Algorithms 4 / 43
Serial Iterative Methods Parallel Iterative Methods Stationary Iterative Methods Krylov Subspace Methods Jacobi Method Jacobi method requires nonzero diagonal entries, which can usually be accomplished by permuting rows and columns if not already true Jacobi method requires duplicate storage for x , since no component can be overwritten until all new values have been computed Components of new iterate do not depend on each other, so they can be computed simultaneously Jacobi method does not always converge, but it is guaranteed to converge under conditions that are often satisﬁed (e.g., if matrix is strictly diagonally dominant), though convergence rate may be very slow Michael T. Heath Parallel Numerical Algorithms 5 / 43

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Serial Iterative Methods Parallel Iterative Methods Stationary Iterative Methods Krylov Subspace Methods Gauss-Seidel Method Faster convergence can be achieved by using each new component value as soon as it has been computed rather than waiting until next iteration This gives Gauss-Seidel method x ( k +1) i = ± b i - X j<i a ij x ( k +1) j - X j>i a ij x ( k ) j ² /a ii Using same notation as for Jacobi, Gauss-Seidel method can be written x ( k +1) = ( D + L ) - 1 ± b - Ux ( k ) ² Michael T. Heath Parallel Numerical Algorithms 6 / 43
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10_iterative - Serial Iterative Methods Parallel Iterative...

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