The goal of this exercise is to observe the behavior of n as goes to where n is the Hilbert matrix, defined by nij In MATLAB, generate n using the command hilb( Let vary from to and make a plot...

The goal of this exercise is to observe the behavior of

_n

as

goes to
where

_n
is the

Hilbert matrix, defined by

<sub>n

ij</sub>

In MATLAB, generate

_n
using the command

hilb(
Let

vary from

to

and make a plot of

versus log
_n

Draw conclusions about the experimental behavior. Hint: Use the MATLAB plotting function semilogy.

Problems 10.26 and 10.27 deal with non-square systems.

The technique of linear least-squares solves a system
by minimizing the residual

₂. Chapter 12 introduces the topic and Chapter 16 discusses it in depth. Least-squares problems usually involve dealing with

systems,
Since solutions are obtained using floating point arithmetic, they are subject to errors, and it certainly is reasonable to ask the questions

Is there a definition of ill-conditioning for non-square systems

Can perturbations in entries of an
system,

cause large fluctuations in the solution

The answer to both of these questions is

Define the condition number of an

matrix as

where

is the largest and

the smallest nonzero singular values of A. Of course if

is square, this is

As we have noted, never compute singular values by finding the eigenvalues of

^T
. The MATLAB command

returns a vector

containing the singular values of