Math 5660 review questions

1. reformulate an expression to avoid loss of accuracy

2. show the bit pattern in IEEE airthmetic that represents the floating 
point number largest, smallest, smallest nonzero in magnitude, smallest in magnitude
normalized

3. represent the sum of three numbers, product of three numbers, inner product 
of vectors dimension 3, as exact operations on perturbed data

4. write Newton's method as functional iteration on x=F(x) and show that F'=0
at the solution

5. compute the condition number of the evaluation of a given scalar expression at
a given point

6. compute the condition number of a 2x2 matrix 

7. given function F(x) and iterval [a,b] show that there is exactly one fixed point
in the interval and fixed point iterations converge to it (apply Banach theorem)

7a. given a function F(x) and interval I the iterations on x=F(x) do not converge 
to a fixed point. Show which assumption of the Banach contraction theorem is violated.

8. Given nxn matrix such that ||A||<1 show iterations on x=Ax+b converge to the
unique solution; be sure to verify all assumptions of the Banach theorem

9.  Given function F compute the order of functional iterations

10. Use Horner's rule to evaluate given p(x)

11. Use Horner's rule to evaluate p'(x)

12. Given p(x) and root r, p(r)=0 , use Horner's rule to divide p(x)/(x-r)

13. Given p(x), write companion matrix, and compute its characteristic polynomial
by expansion of the determinant

14. Use Gauss elimination to compute LU decomposition of a give 3x3 matrix

15. Find LU decomposition of a 3x3 matrix by the recursive outer product method