1. Show that the floating point multiplication of two numbers is backward stable. 2. Prove that a small residual for the solution of  implies backward stability. 3. Prove that if  and  are in n  the...

1.
Show that the floating point multiplication of two numbers is backward stable.

2.
Prove that a small residual for the solution of implies backward stability.

3.
Prove that if
 and
 are in

ⁿ
 the
matrix

^T
has rank

4.
If A and B are matrices and α is a floating point number, prove the following forward error results.

a. fl (αA) = αA + E, |E| ≤ eps |αA| .

.

5.
Show that the roots of the polynomial

<sup>3

2</sup>
 are ill-conditioned and explain why

6.
Let
 ln

 Show that the condition number of f at
 is

 Using the above result, show that ln
 is ill-conditioned near

7.
Show that computing
 for
 is well-conditioned.

8.
What is the condition number for
at
 Where is it ill-conditioned