First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 4. NUMERICAL QUADRATURE AND DIFFERENTIATION 160
and apply the quadrature rule again, using n 1 nodes, to get the approximation
∑n 2
j=1
This gives the two-dimensional rule
∫ b
a
(∫ d
c
(
n1
)
∑
w j w i f(x i ,y j ) .
i=1
)
f(x, y)dy dx ≈
∑n 2 n 1
j=1
∑
w i w j f(x i ,y j ).
For simplicity, we ignored the error term in the above derivation, however, its inclusion is
straightforward.
For an example, let’s derive the two-dimensional Gauss-Legendre rule for the integral
∫ 1 ∫ 1
0
0
i=1
f(x, y)dydx (4.5)
using two nodes for each axis. Note that each integral has to be transformed to (−1, 1).
Start with the inner integral ∫ 1
f(x, y)dy and use
0
t =2y − 1,dt=2dy
to transform it to
∫
1 1
(
f x, t +1 )
dt
2 −1 2
and apply Gauss-Legendre rule with two nodes to get the approximation
( (
) (
))
1
f x, −1/√ 3+1
+ f x, 1/√ 3+1
.
2
2
2
Substitute this approximation in (4.5) for the inner integral to get
∫ ( (
) (
))
1
1
f x, −1/√ 3+1
+ f x, 1/√ 3+1
dx.
2
2
2
0
Now transform this integral to the domain (−1, 1) using
s =2x − 1,ds=2dx