(Thomas) Simpson’s rule
A great mathematician
Riley Wang
Mr. Sidanycz
Block 2nd
Thomas Simpson
• 1. Thomas Simpson, born on August 20, 1710,
in Market Bosworth, Leicestershire, England,
was the son of a self-taught weaver.
• 2. Simpson’s father naturally expected his son
to take up the same profession as his “ol’
man”. However, with the occurrence of a
solar eclipse in 1724, Thomas Simpson turned
to “mathematical interests”, changing his life
forever.
Thomas Simpson
• 3. By 1735, he was able to solve puzzles
concerning infinitesimal calculus.
• 4. Not only did Simpson work on
mathematics, but he also delved heavily into
probability theory and the concept of
approximation and error.
• 5. Simpson died on May 14, 1761, he is usually
remembered for his contribution to numerical
integration: “Simpson’s Rule”.
.
Simpson’s rule
• 1.Discussed here at the Holistic Numerical
Methods Institute, Simpson’s Rule “has
become a popular and useful special case of
the Newton-Cotes formula for approximating
an integral”.
• Simpson's rule is a method for numerical
integration, the numerical approximation of
definite integrals. Specifically, it is the
following approximation:
Simpson’s rule
Simpson’s Quadrati interpolation
Error of Simpson's rule
Simpson's rule Example
Example 1: Approximating the graph of y = f(x) with parabolic arcs across
successive pairs of intervals to obtain Simpson's Rule
Since only one quadratic function can interpolate any three (non-colinear)
points, we see that the approximating function must be unique for each
interval . Note that the following quadratic function interpolates the three
points
Example of Simpson's rule
Then you get
• Finally we get
To nine decimal places, we get better
approximation
Probability Theory
This part of mathematics is concerned with the analysis
of random phenomena that much of
Simpson’s life was dedicated to.
Simpson found this to be true by deriving
several equations with an end result of
p =(1 + i)ax/ax + 1
He used several proofs and De Moivre’s
work to compile to this answer. This
’branch’ of Simpson’s
work in mathematics deals with things such
as mortality rates and life insurance.
How much it is better than trapezoid
rule
Table 7.2 Composite Trapezoidal Rule for
f (x) = 2 + sin(2√x) over [1, 6]
M h T ( f, h) ET ( f, h) = O(h2)
10 0.5 8.19385457 −0.01037540
20 0.25 8.18604926 −0.00257006
40 0.125 8.18412019 −0.00064098
80 0.0625 8.18363936 −0.00016015
160 0.03125 8.18351924 −0.00004003
Table 7.3 Composite Simpson Rule for
f (x) = 2 + sin(2√x) over [1, 6]
M h S( f, h) ES( f, h) = O(h4)
5 0.5 8.18301549 0.00046371
10 0.25 8.18344750 0.00003171
20 0.125 8.18347717 0.00000204
40 0.0625 8.18347908 0.00000013
80 0.03125 8.18347920 0.00000001
Now we get how Simpson's rule works and why it has
better approximation