Lecture slides introducing Romberg Integration based on Chapter 7.04 of Prof. Anton Kaw's Numerical Methods textbook. Some parts of this presentation are based on resources at http://nm.MathForCollege.com, primarily http://mathforcollege.com/nm/topics/romberg_method.html

1. St. John's University of Tanzania
MAT210 NUMERICAL ANALYSIS
2013/14 Semester II
INTEGRATION
Richardson's Extrapolation & Romberg Integration
Kaw, Chapter 7.04
http://nm.mathforcollege.com/topics/romberg_method.html

2. MAT210 2013/14 Sem II 2 of 17
● Approximating Error was not exact
●
Error for multi-segment Trapezoidal was
known to be
● What if we tried to say that the next
approximation for the integral was
● In the example the “predicted” error was 51
and the “actual” error was 48, but at least we
jump from error of ~50 to one of ~3
Recall
1
12
h
2
f ''
In+1
=In
+predicted error

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Richardson's Extrapolation
● That is the idea behind a technique known
as Richardson's Extrapolation
●
Jump to a new approximation using an
approximation of the error
●
It works out in this integration case because
the error is nearly a function of n alone
Et
=
1
12
h
2
f ''=
1
n
2 [(b−a)2
f ''
12 ]
⇒Et
≈
C
n
2

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Improved approximation
I≈In
+
C
n
2
I≈I2n
+
C
(2n)2
⇒ 4 I≈4 I2n
+
C
n2
⇒ 3 I ≈ 4 I2n
−In
True Value, TV = I ≈ I2n
+
I2n
−In
3
Richardson's
Extrapolation
for True Value

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Apply it to the Example
Exact is 11061m, so error is only 0.00904%!

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More on the Error
Now recall Simpson's 1/3 results:
Richardson's Extrapolation beats them both

7. MAT210 2013/14 Sem II 7 of 17
Romberg Integration
● Romberg Integration takes Richardson's
Extrapolation and builds a recursive
algorithm around it
●
It uses the added fact that
to create the recursion formula
Et
=A1
h2
+A2
h4
+A3
h6
+…

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Finding Recursion
The next term in the error expansion
Combining like
before yields

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Recursion Formula and Error
● The process can repeated to produce:
● j = Level of accuracy
● k = Order of the extrapolation
●
k = 1 – Trapezoidal rule, O(h²) error
●
k = 2 – 1st Level of Romberg, O(h4) error
● k = 3 – 2nd level of Romberg, O(h6) error

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Applied to the Example
●
Results from the 1,2,4 & 8
segment Trapezoidal Rule
●
This is the hard work
●
The rest is easy & recursive

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Pictorial View

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Spreadheet View

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Not just known functions!

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Not just known functions!

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Not just known functions!

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Not just known functions!

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Good enough to be “standard”