Find the eigenvalues of the iteration matrix

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The damped Jacobi (or under-relaxed Jacobi) method is briefly described in Sections 7.2 and 7.6. Consider the system Ax = b, and let D be the diagonal part of A.

(a) Write down the corresponding splitting in terms of D, A, and ω.

(b) Suppose A is again the same block tridiagonal matrix arising from discretization of the two-dimensional Poisson equation.

i. Find the eigenvalues of the iteration matrix of the damped Jacobi scheme.

ii. Find the values of ω > 0 for which the scheme converges.

iii. Determine whether there are values ω ≠ 1 for which the performance is better than the performance of standard Jacobi, i.e., with ω = 1.

Reference no: EM131252595

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