Linear Algebra with Applications, 10th Edition

Chapter 6 Exercises

MATLAB Exercises

Critical Loads for a Beam

1. Consider the application relating to critical loads for a beam from Section 6.1. For simplicity, we will assume that the beam has length 1 and that its flexural rigidity is also 1. Following the method described in the application, if the interval [0, 1] is partitioned into n subintervals, then the problem can be translated into a matrix equation $A\mathbf{y}=\lambda \mathbf{y}$. The critical load for the beam can be approximated by setting $P=s{n}^2$, where s is the smallest eigenvalue of A. For $n=100,200,400$, form the coefficient matrix by setting

   $$\begin{array}{cc}\hfill D& =\mathsf{\text{diag}}\left(\mathsf{\text{ones}}\right(n-1,1),1);\hfill \\ \hfill A& =2*\mathsf{\text{eye}}\left(n\right)-D-D\prime ;\hfill \end{array}$$

   In each case, determine the smallest eigenvalue of A by setting

   $$s=\mathsf{\text{min}}\left(\mathsf{\text{eig}}\right(A\left)\right)$$

   and then compute the corresponding approximation ...