miércoles, 22 de mayo de 2024

Demostración de Integración de Romberg

Demostración de Integración de Romberg 


https://personal.math.ubc.ca/~CLP/CLP2/clp_2_ic/ap_Romberg.html

Romberg Integration

The formulae (E4a,b) for 𝐾 and 𝐴 are, of course, only 1  approximate since they are based on (E2), which is an approximation to (E1). Let's repeat the derivation that leads to (E4), but using the full (E1),
“Only” is a bit strong. Don't underestimate the power of a good approximation (pun intended).
𝐴=𝐴()+𝐾𝑘+𝐾1𝑘+1+𝐾2𝑘+2+
Once again, suppose that we have chosen some  and that we have evaluated 𝐴() and 𝐴(/2). They obey
(E5a)𝐴=𝐴()+𝐾𝑘+𝐾1𝑘+1+𝐾2𝑘+2+(E5b)𝐴=𝐴(/2)+𝐾(2)𝑘+𝐾1(2)𝑘+1+𝐾2(2)𝑘+2+
Now, as we did in the derivation of (E4b), multiply (E5b) by 2𝑘 and then subtract (E5a). This gives
(2𝑘1)𝐴=2𝑘𝐴(/2)𝐴()12𝐾1𝑘+134𝐾2𝑘+1+
and then, dividing across by (2𝑘1),
𝐴=2𝑘𝐴(/2)𝐴()2𝑘11/22𝑘1𝐾1𝑘+13/42𝑘1𝐾2𝑘+2+
Hence if we define our “new improved approximation”
(E6)𝐵()=2𝑘𝐴(/2)𝐴()2𝑘1 and 𝐾~=1/22𝑘1𝐾1 and 𝐾~1=3/42𝑘1𝐾2
we have
𝐴=𝐵()+𝐾~𝑘+1+𝐾~1𝑘+2+
which says that 𝐵() is an approximation to 𝐴 whose error is of order 𝑘+1, one better 2  than 𝐴()'s.
If 𝐴() has been computed for three values of , we can generate 𝐵() for two values of  and repeat the above procedure with a new value of 𝑘. And so on. One widely used numerical integration algorithm, called Romberg integration 3 , applies this procedure repeatedly to the trapezoidal rule. It is known that the trapezoidal rule approximation 𝑇() to an integral 𝐼 has error behaviour (assuming that the integrand 𝑓(𝑥) is smooth)
𝐼=𝑇()+𝐾12+𝐾24+𝐾36+
Only even powers of  appear. Hence
𝑇() has error of order 2

so that, using (E6) with 𝑘=2,

𝑇1()=4𝑇(/2)𝑇()3 has error of order 4

so that, using (E6) with 𝑘=4,

𝑇2()=16𝑇1(/2)𝑇1()15 has error of order 6

so that, using (E6) with 𝑘=6,

𝑇3()=64𝑇2(/2)𝑇2()63 has error of order 8
and so on. We know another method which produces an error of order 4 — Simpson's rule. In fact, 𝑇1() is exactly Simpson's rule (for step size 2).

Example C.2.2. Finding 𝜋 by Romberg integration. 

The following table 4  illustrates Romberg integration by applying it to the area 𝐴 of the integral 𝐴=0141+𝑥2d𝑥. The exact value of this integral is 𝜋 which is 3.14159265358979, to fourteen decimal places.
𝑇()𝑇1()𝑇2()𝑇3()
1/43.131183.141592502463.141592661143.14159265359003
1/83.138993.1415926512253.141592653708
1/163.140943.141592653553
1/323.14143
This computation required the evaluation of 𝑓(𝑥)=41+𝑥2 only for 𝑥=𝑛32 with 0𝑛32 — that is, a total of 33 evaluations of 𝑓. Those 33 evaluations gave us 12 correct decimal places. By way of comparison, 𝑇(132) used the same 33 evaluations of 𝑓, but only gave us 3 correct decimal places.
As we have seen, Richardson extrapolation can be used to choose the step size so as to achieve some desired degree of accuracy. We are next going to consider a family of algorithms that extend this idea to use small step sizes in the part of the domain of integration where it is hard to get good accuracy and large step sizes in the part of the domain of integration where it is easy to get good accuracy. We will illustrate the ideas by applying them to the integral 01𝑥 d𝑥. The integrand 𝑥 changes very quickly when 𝑥 is small and changes slowly when 𝑥 is large. So we will make the step size small near 𝑥=0 and make the step size large near 𝑥=1.