To solve the problem using the Jacobi iteration, the spectral radius of the matrix J must be less than one. Using the result of Problem show that the eigenvalues of J are where j are the...

To solve the problem

using the Jacobi iteration, the spectral radius of the matrix

_J
must be less than one.

Using the result of Problem
show that the eigenvalues of

_J
are

where

_j
are the eigenvalues of

Hint: Show that

_J

⁻¹

and let

be an eigenvector of

_J
with corresponding eigenvalue

Show that the spectral radius of

_J
is

and that

_J
. Thus, the Jacobi method converges. Hint: Use the half-angle formula

²

Using the McLaurin series for cos

show that