Core 3 Numerical Methods 3

1
Core 3 Numerical Methods (3)

• Learning Objectives
• Learn how to find the approximate area under a
  graph using Simpson's rule.

2
Previous methods

• Trapezium and mid-ordinate rules produce errors
  under curves because they are polygons (straight
  lines).

Mid-ordinate
trapezium
3
Introducing. SIMPSONS Rule

• Another numerical method for integration / area
  under graphs
• Based on quadratic functions
• Given three points you can fit a quadratic
• For example P(-1,1) Q(0,3) and R(1,2)

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SIMPSONS Rule

• Fit to a QUADRATIC of the form yax2 bx c
• Using P(-1,1)
• 1 a x (-1) 2 b x (-1) c
• So 1a-bc
• Now use Q(0,3) and R(1,2)
• Find values for a, b and c.

a -3/2
b 1/2
c 3
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SIMPSONS Rule

• The QUADRATIC is y-3/2x2 1/2x 3.
• The area under this curve is

• You can find this area by
• doing the definite integral

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SIMPSONS Rule

-1
0
1
Quadratics can give a give estimates in section,
even if we are dealing with a complex function
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SIMPSONS Rule

0
-1
1
Quadratics can give a give estimates in section,
even if we are dealing with a complex function

• Each strip can be estimated for 3 coordinates
• Read/work through idea development D on page
  140-141
• The gives you SIMPSONS RULE

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SIMPSONS Rule
The area of the nth strip is estimated to
h width of strip
y1
y2
y3
y6
y0
y5
y4
h
h
h
h
h
h
In this case you get
Area 1/3 h (y0 4y1 y2) 1/3 h (y2 4y3
y4) 1/3 h (y4 4y5 y6)
Area 1/3 h (y0 4y1 2y2 4y3 2y4 4y5
y6)
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SIMPSONS Rule
Our example
Area 1/3 h (y0 4y1 2y2 4y3 2y4 4y5
y6)
The end ones You count once
The odd ones.. You count 4 times
The remaining even ones.. You count 2 times
h width of strip
You have to have an even number of strips
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SIMPSONS Rule - Example
Divide into 8 strips
Width of strips (7-5)/8 0.25
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SIMPSONS Rule - Example
8 strips
Width (h) 0.25
Area 9.8639 (4 d.p)
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Comparison of Methods (1)
Trapezium Rule
Area ½ h (end ordinates twice sum of
interior ordinates)
Area 9.8172
Mid-ordinate Rule
Area width of strip x sum of mid-ordinates
Area 9.8870
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Integration by parts
Substitute in
Leads to
7
7
5
5
7 sin 7 cos 7 5 sin 5 cos 5 9.8637676
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Comparison of Methods (2)
Numerical methods all based on 8 strips.
Errors will change depending on the number of
strips used.
Simpsons Rule provides, by far, the most
accurate estimate
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Simpsons Rule Practice

• Page 142
• Core 3
• exercise D

in case you thought Id forgotten the obvious
picture