Numerical Analysis Assignment 1 — IEEE Arithmetic  
Question #1
Determine the maximum and minimum number that can be stored in:
(a)  A 5-byte unsigned integer
(b)  A 5-byte signed integer
08.20.16
Question #2
Consider a real number stored with 5 bytes. Bit #1 is reserved for the sign, while bits #2 to #10 are reserved for the biased exponent, and bits #11 to #40 are related to the significand. Do the following:
(a)  Find the minimum and maximum possible exponent $p$
(b)  Find the smallest possible positive number
(c)  Find the largest possible number
(d)  Find the smallest possible positive subnormal number
08.29.16
Question #3
Consider a real number stored with 6 bytes. Bit #1 is reserved for the sign, while bits #2 to #13 are reserved for the biased exponent, and bits #14 to #48 are related to the significand. Find the machine precision $\epsilon_{\rm mach}$.
Question #4
Say that $$ x=-g + \sqrt{g^2+1} $$ Say that both $x$ and $g$ are stored in memory using single precision numbers with a relative error due to machine accuracy of $\epsilon_{\rm mach}=2\times 10^{-10}$. Do the following:
(a)  Find the relative error on $x$ given $g=1000.0$
(b)  Recast the equation for $x$ in difference form to reduce its relative error
(c)  Find the relative error on $x$ given $g=1000.0$ for the recast equation outlined in (b)
08.30.16
Question #5
It is desired to minimize the number of bits that can store a certain range of numbers. The range lower limit is $3\times 10^{-65}$, and the range upper limit is $10^{32}$. Find the number of bits needed to store the exponent and the significand. Then find the total number of bits needed.
09.13.17
Question #6
(a)  Consider a number of real type. Knowing that the machine accuracy (non-denormal) is of $\epsilon_{\rm mach}=9.5367 \times 10^{-7}$ and that the maximum positive number must be at least as high as $10^{23}$, do the following:
(i) find the minimum number of bits for the exponent;
(ii) find the minimum number of bits for the significand.
(b)  Consider the number $9.5367\times 10^{-4}$ stored in memory as a real type. Knowing that the exponent of the real type has 4 bits what is the minimum number of bits that the significand should have if the relative error on the number is less than 0.01?
09.12.18
Answers
2.  255, $-254$, $3.454\times 10^{-77}$, $1.15792089 \times 10^{77}$, $3.217\times 10^{-86}$.
5.  9, 1, 11.
6.  19, 8, 11.
Due on Wednesday September 19th at 16:30. Do questions #2, #5, #6 only.
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