1. The biharmonic matrix is ill-conditioned. Verify this for matrices with over the interval 2. We will investigate the one-dimensional version of the biharmonic equation: Assuming the boundary...

1.

The biharmonic matrix is ill-conditioned. Verify this for matrices with

over the interval

2.

We will investigate the one-dimensional version of the biharmonic equation:

Assuming the boundary conditions

and using a five-point central finite-difference approximation over

with uniform step size

gives rise to the following pentadiagonal

matrix

A is symmetric positive definite and ill-conditioned, so the preconditioned

method should be used.

Write a function [x, y, u, residual, iter] = biharmonic1D(n, f, tol,maxiter) that uses preconditioned CG with n subintervals, specified tolerance and maximum number of iterations to approximate a solution to Equation 21.33 and graph it.

Consider the specific problem