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

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● 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

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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%!

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More on the Error

Now recall Simpson's 1/3 results:

Richardson's Extrapolation beats them both

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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+…

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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”