MATH 609-602: Numerical Methods
Lecturer: Prof. Wolfgang Bangerth
Blocker Bldg., Room 507D
(979) 845 6393
Teaching Assistant: Seungil Kim
Blocker Bldg., Room 507A
(979) 862 3259
Homework assignment 1
All problems are discussed in the lab on Wednesday. You do not need to hand
anything in.
Problem 1 (Continuous vs. discrete). Functions f(x) are usually defined
over an entire domain x∈I= (a,b)⊂Rand – if interesting – take values in an
image f(I)⊂R. Both domain and image are sets with infinitely many elements.
On the other hand, computers can only represent numbers using a finite number
of bits, most often as 32-bit (float, or REAL*4) or 64-bit (double, or REAL*8)
IEEE floating point numbers, which store numbers in the form ±m2e, where
0≤m < 1 is the mantissa
m=b12−1+b22−2+b32−3+· · · +bM2−M(1)
and eis the exponent and has the form
e=±(u121+u222+u323+· · · +uE2E).(2)
The coefficients bi, eiare single-bit numbers, i.e. either 0 or 1. In the bi-
nary system, floating point numbers can therefore be written as ±0.b1b2b3. . . ×
2±u1u2u3...uE. The total number of bits needed for the representation are M
bits for the mantissa, E+ 1 bits for the exponent, and 2 bits for the two signs.
Obviously, not all elements of Iand f(I) can be represented. Write a short
program to find
a) an approximation to the smallest and largest positive numbers that can
be represented in float and double precision;
b) the smallest float and double floating point number you can add to 1 such
that the result is different from 1.
c) In exact arithmetic, the system of linear equations
x1+x2= 2,
x1+ 1020x2= 1 + 1020
1
