Download Intro to Numerical Methods

Intro to Numerical Methods
APAM E4300 (1)
ASSIGNMENT 2 – SOLUTIONS (ROUNDOFF AND TRUNCATION ERRORS)
Due: Tuesday, February 26, 2013
Problem 1 [9 points]:
1. Write down the IEEE double precision representation for the following decimal
numbers:
a. 1.5, using round up.
b. 5.1, using round to nearest.
c. – 5.1, using round towards zero.
d. – 5.1, using round down.
2. Write down the IEEE double precision representation for the decimal number
50.2, using round to nearest.
3. What is the gap between 2 and the next larger double precision number?
Solution:
Problem 2 [9 points]:
Describe an algorithm to compare two double precision floating point numbers a and b
to determine whether a < b, a = b, or a > b, by comparing each of their bits from left to
right, stopping as soon as a differing bit is encountered.
Solution:
Problem 3 [8 points]:
Consider a very limited system in which significands are only of the form 1. and
the only exponents are 0, 1, and – 1. What is the machine precision for this system?
Assuming that subnormal numbers are not used, what is the smallest positive number
that can be represented in this system, and what is the largest number that can be
represented? Express your answers in decimal form.
Solution:
Problem 4 [9 points]:
Using Taylor series, derive the error term for the approximation
1
3 4 2.
2
Solution:
Problem 5 [10 points]:
Consider a forward difference approximation for the second derivative of the form
2.
Use Taylor’s theorem to determine the coefficients A, B, and C that give the maximal
order of accuracy and determine what this order is.
Solution: