(a). Show that the eigenvalue problem for the Gauss-Seidel iteration matrix may be expressed in the...

 

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(a). Show that the eigenvalue problem for the Gauss-Seidel iteration matrix may be expressed in the form RGSW = Aw Uw (DL)w, = where D, -L, -U is the usual diagonal lower triangular and upper triangular matrices in the splitting of matrix A. (b). Assuming that A is the finite difference discretization of -u" (x) = f(x), x = (0,1) u(0) = u(1) = 0 What equation must this eigenvalue problem satisfy at an interior node of the grid? Taking the ansatz that the eigenvector has the form w; = r, determine the eigenvalues. (c). Determine the corresponding eigenvectors. (a). Show that the eigenvalue problem for the Gauss-Seidel iteration matrix may be expressed in the form RGSW = Aw Uw (DL)w, = where D, -L, -U is the usual diagonal lower triangular and upper triangular matrices in the splitting of matrix A. (b). Assuming that A is the finite difference discretization of -u" (x) = f(x), x = (0,1) u(0) = u(1) = 0 What equation must this eigenvalue problem satisfy at an interior node of the grid? Taking the ansatz that the eigenvector has the form w; = r, determine the eigenvalues. (c). Determine the corresponding eigenvectors.