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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
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St. John's University of Tanzania
MAT210 NUMERICAL ANALYSIS2013/14 Semester II
INTEGRATIONRichardson's Extrapolation & Romberg Integration
Kaw, Chapter 7.04http://nm.mathforcollege.com/topics/romberg_method.html
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
112h2 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=112h2 f ' '= 1
n2[(b−a)2 f ' '
12 ]⇒Et≈
Cn2
MAT210 2013/14 Sem II 4 of 17
Improved approximationI≈In+
Cn2
I≈I 2n+C
(2n)2⇒ 4 I≈4 I 2n+
Cn2
⇒ 3 I ≈ 4 I 2n−In
True Value, TV = I ≈ I 2n+I 2n−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%!
MAT210 2013/14 Sem II 6 of 17
More on the Error
Now recall Simpson's 1/3 results:
Richardson's Extrapolation beats them both
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=A1h2+A2h
4+A3h6+…
MAT210 2013/14 Sem II 8 of 17
Finding Recursion
The next term in the error expansion
Combining like before yields
MAT210 2013/14 Sem II 9 of 17
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
MAT210 2013/14 Sem II 10 of 17
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”