Hwa Chong Institution
HCI Projects’ Day 2014
Mathematics Project Summary
Newton’s Road to Chaos
Group 9-28
Group Members:
Yew Tze Yong (4A3-28) Leader
Tin Jun Hao (4A2-29)
Yip Seng Yeun (4A3-29)
Yuan Khai Yeep (4A3-30)
Introduction
1
Newton’s Method is an iterative method which aids in the construction of a sequence of
approximations to a root of an equation (Dugdale, 1998). Newton’s Method can be applied to
xn +1
finding square roots, with the formula: =
1
k
(x n + ) where x = k (Simpsons, 2008).
2
xn
Chaos Theory is a branch of study in applied mathematics in which systems described by the
theory are seemingly disordered; but the theory seeks to find the underlying order in the
seemingly random data. Dugdale (1998) stated that the tweaking of the iterative method & the
results obtained or the changing of the initial value can be extended naturally into Chaos Theory.
Theorems such as Newton’s Method provide a solid understanding for Chaos Theory, and help
to explain the mathematical concepts behind a seemingly complex phenomenon.
2
Objectives
●
To investigate on how the initial value and initial estimate affects the expansion of the
graph of the square root estimates against the number of iterations
●
To investigate how the initial estimate can affect the chaotic property of the graph
●
To explore on Newton’s Method’s relevance on cube roots
●
To create our own formula for deriving 4th root value of a real number, and apply it into
Chaos Theory
3
Methodology
1. Conduct research on Newton’s Method and Chaos Theory through magazines, articles,
journal and online resources.
2. Build a computing model through Excel to simulate the Newton’s Method for square
root by plotting the graph of square root estimates against the number of iterations.
3. Test the graph obtained by changing the initial value, k and the initial estimate, n.
4. Find the relevance of the graph of square root estimates against the number of iterations
to the Chaos Theory
5. Repeat the same process by finding the cube root and expand into finding the nth root of
a number.
6. Find the relevance of the graph of cube root and nth root estimates against the number of
iterations to the Chaos Theory
4
Findings and conclusion
Newton’s method for square root works well as an iterative method for square root when
positive initial value is input. However, when negative initial value is used, the method fails to
function as expected because square root value of a number is complex. As a result, the graph
will display irregularities, which leads us to discovering “seemingly disordered patterns” as
described by Chaos Theory.
Newton’s Method for cube root leads us nowhere near Chaos Theory because cube root
value of a negative number is not complex and can be derived accurately through the iterative
method. Nonetheless, we came up with our own formula for deriving cube root values xn +1
=
k .
1
( xn + 2
)
xn
3
After a few rounds of testing, we found out that this formula has an even higher
efficiency in deriving cube root values as compared to the existing methods, and can effectively
negate the flaws of existing iterative method. The only limitation of our method is that it does not
work when a negative initial value or estimate is used.
Based on the above finding, we decided to create our own formula for deriving 4th root
values, as we thought that it might shed some light to the mysterious Chaos Theory. We started
off by applying the Arithmetic Mean-Geometric Mean inequality concept and come up with 3
3
th
k
1 3
trial formulae. The first trial,=
xn +1
( x + 3( ) , was unsuccessful in both deriving 4 root values
4
xn
1
k
as well as displaying chaotic patterns. The second trial, x=
( x + 3( 3 ) was partially successful
n +1
4
x
as it can derive 4th root value of positive number, but cannot display chaotic pattern when
negative value is applied. The third trial, xn +1= 1 ( x + x + 2k ) was also partially successful as it is
4
k
x
able to display chaotic patterns, but unable to derive the 4th root value of a number. Therefore,
the extension of this project is to come up with a formula that can both derive the 4th root value
of a number as well as applicable to Chaos Theory.