# MA580/CSC580 Homework 1 due 22 January 2014 in class

Read chapter 4 in the book by Higham & Higham, as well as the IEEE Standard for Binary Floating Point Arithmetic.

Try out Cleve Moler's Matlab program floatgui.m.

The program shows you the spacing of floating point numbers. The parameter t specifies the number of bits in the fraction, and the parameters emin and emax specify the range of the exponent. (This is for self-study only. You do not need to hand in anything.)

1. [10 points]   Let A be a mxn matrix, and e the nx1 vector of all ones. What does the matrix vector product Ae do?

2. [10 points]   Characterize the class of unitary, Hermitian positive definite matrices. Prove your answer.

3. [10 points]   The expressions 2^(-1024) and 1/(2^1024) are equal in exact arithmetic. Explain what happens when they are evaluated in IEEE double precision arithmetic, and why.

4. [10 points]   Determine the exact values of x and xp1, in terms of powers of two, at the termination of the Matlab algorithm below. Explain the purpose of the algorithm.
```x = 1;
xp1 = x + 1;
while xp1 > 1
x = x/2;
xp1 = x + 1;
end
x = 2*x;
```