Question

Prove tha how I=epsilon/R+r

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Answer 1

Answer:

(a) Find the eigenvalues and eigenvectors of A. Hint: A is an example of a TST matrix.

(b) Suppose A = D+E+ET, where A is the diagonal parta of A and E is the strict upper triangular part. Find the eigenvalues and eigenvectors of J=1-D-¹A₁ the iteration matrix for the Jacobi method.

(c) The damped Jacobi method has the iteration matrix

J(w) = -WD ¹(E+E) + (1-w)l. 0<w <2,

so that J(1) = J. What is the corresponding linear fixed point iteration function T? Recall that

T(z)=(1-B¹A)z + B-¹b.

in general.

(d) Find the eigenvalues and eigenvectors of J(w) € Cnxn (e) Let u(*)(w) be the kth eigenvalue of J(w), with the ordering

4(2) <μ(2) <...<(").

Assume that n + 1 is even. Prove that the quantity

S(w): max |u(k) (w)|

is minimized by w = wo = 3, and

for all t≤k≤n.

Explanation:

i