Math 551 - Introduction to Scientific Computing
Chapter 15. Numerical Integration
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Outline
15.1 Basic quadrature algorithms
15.2 Composite numerical integration
Goals
To develop some classical methods for approximating definite integrals;
to concentrate on issues that arise also in more involved and complicated circumstances, such as high order method development and adaptivity;
to consider some challenging problems in multidimensional integration.
Numerical integration (quadrature)
The need to integrate arises very frequently in numerical computations. Instance: finite element methods for differential equations use basis functions, in combination with integrals computed over tiny pieces of the computational domain.
As opposed to differentiation, which is local in nature, integration is a global operation.
Note that while the derivative of f (x) is typically rougher than f , the integral of f (x) is smoother. Consequently, no special roundoff error difficulties are expected here.
Many-dimensional integration often arises in statistical applications.
15.1 Basic quadrature algorithms
Mathematical question we are interested in numerically answering
How do we evaluate
b
I = f (x)dx?
a
Calculus tells us that if F (x) is the antiderivative of a function
f (x) on the interval [a, b], then
b b
I = f (x)dx = F (x)
a
= F (b) − F (a).
Practically, most integrals cannot be evaluated using this approach. For example,
1 1
1 + x5 dx,
1
e−x2
0
dx,
xπ sin(√
0
x)dx,
have complicated antiderivatives and it is easier to adopt a numerical method to approximate this integral.
Numerical Integration: A General Framework
If you cannot solve a problem, then replace it with a “near-by” problem that you can solve!
Our problem: Evaluate
b
I = f (x)dx.
a
To do so, many of the numerical schemes are based on replacing
f (x) with some approximate function f˜(x) so that
∫ b ˜ ˜
|
Example: f˜(x) could be an easy to integrate function approximating f (x).
Then, the approximation error in this case is
|
|
˜ ∫ b ˜ ˜
Question: How to choose f˜(x)?
