Richardson Extrapolation

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ME6758, Dr. Ferri
Richardson Extrapolation
Start with some integration rule, N(h), and some
general trend of the error as a function of h
(1)
Now, divide h by 2 (double the number of
intervals)
(2)
To eliminate the lowest order error term, take
2Eq2 Eq1
Call this N2(h)
2
weighted average
Now,
(3)
Halve h again (double the number of intervals)
(4)
Eliminate h2 terms by taking 4Eq4 Eq3
Call this N3(h)
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This is accurate to order O(h3)
Now,
The O(h4) approximation is given by
The O(h5) approximation is given by
This is accurate to order O(hj)
In general
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Construct a Table
O(h) O(h2) O(h3) O(h4)
N2(h) N2(h/2) N2(h/4)
N3(h) N3(h/2)
N4(h)
N1(h) N1(h/2) N1(h/4) N1(h/8)
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Romberg Integration
Start with composite trapezoidal rule with
m-intervals, (m1) points
a lt m lt b
h (b-a)/m
xj a jh
Perform n trapezoidal integrations using m1 1
interval, m2 2 intervals, m3 4 intervals,
mn 2n-1 intervals. In each case hk (b-a)/mk
(b-a)/ 2k-1
Call this Rk,1
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h3
h2
h1
R1,1
R2,1
R3,1
h1/2
just R1,1/2
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h3
h2
h1
R1,1
R2,1
R3,1
h2/2
just R2,1/2
odd multiples of hk
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R1,1
new point
R2,1
new point
new point
R3,1
h4
R4,1
h5
R5,1
R6,1
2 endpoints, 2k-1-1 interior points, 2k-2
additional points
Rk,1
odd multiples of hk
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Example
Converges, but rather slowly
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Subtract top equation from 4 times bottom
equation. After algebra, get
Re-write as
Now, extrapolate again to eliminate O(hk4) terms
to obtain an O(hk6) result, etc
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Define
Error O(hk2j)
Generalize to
R1,1 R2,1 R3,1 R4,1 Rn,1
R2,2 R3,2 R4,2 Rn,2
R3,3 R4,3 Rn,3
R4,4 Rn,4
Rn,n
Construct Romberg Table
Obtain from a single composite trapezoidal integra
tion
IRn,n O(hn2n)

Obtain by simple averaging
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Example
0.00000000          
1.57079633 2.09439511        
1.89611890 2.00455976 1.99857073      
1.97423160 2.00026917 1.99998313 2.00000555    
1.99357034 2.00001659 1.99999975 2.00000001 1.99999999  
1.99839336 2.00000103 2.00000000 2.00000000 2.00000000 2.00000000
Error in R6,6 is only 6.61026789e-011 !!!!
Accurate to O(h612)
h6 p/32 9.817e-002, h612 8.017e-013