Composite Numerical Integration

The accuracy of the Quadrature formulae is a function of a fraction of the length of the integration interval. For integration over a larger interval, they tend to give rather large errors. For example:

> f:=x->cos(ln(x)):Int(f(x),x=1..11);exact:=evalf(value(%));

For Simpson's Rule:

> h:=evalf(10./2.):intn2:=h*(f(1.)+4*f(6.)+f(11.))/3.;err2:=abs(intn2-exact);

Even for n=4:

> h:=evalf(10./4.):intn4:=2*h*(7*f(1.)+32*f(7./2.)+12*f(6.)+32*f(17./2.)+7*f(11.))/45.;err4:=abs(intn4-exact);

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One can, of course, split the integral in the sum of a number of integrals and apply these methods on each one of them separately. For example, let us split this interval in 4. Then

> Int(f(x),x=1..11)=Int(f(x),x=1..7/2)+Int(f(x),x=7/2..6)+Int(f(x),x=6..17/2)+Int(f(x),x=17/2..11);

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Applying Simpson's Rule on these integrals then gives:

> intcn2:=0:h:=5/4.:for i from 0 to 3 do a:=1.+i*5/2.:b:=7/2.+i*5/2.:intcn2:=intcn2+h*(f(a)+4*f((a+b)/2.)+f(b))/3.: od:print(intcn2,abs(intcn2-exact));

Using the 5-point method yields:

> intcn4:=0:h:=5/8.:for i from 0 to 3 do a:=1.+i*5/2.:b:=7/2.+i*5/2.:intcn4:=intcn4+2*h*(7*f(a)+32*f(a+h)+12*f(a+2*h)+32*f(a+3*h)+7*f(b))/45.: od:print(intcn4,abs(intcn4-exact));

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Compare this with the earlier results:

> print(`n=`.2,abs(intn2-exact),abs(intcn2-exact));

> print(`n=`.4,abs(intn4-exact),abs(intcn4-exact));

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The idea of splitting the integration interval and appying a quadrature formula on each of the subintervals is called composite numerical integration.

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Theorem 14 (Composite Simpson's Method)

Theorem 15 (Composite Trapezoidal Rule)

Theorem 16 (Composite Midpoint Rule)