NUMERICAL INTEGRATION: ANOTHER APPROACH
NUMERICAL INTEGRATION: ANOTHER APPROACH
We look for numerical integration formulas
Z1
X n
-1 f (x) dx j=1 wjf (xj)
which are to be exact for polynomials of as large a
degree as possnibleo. There are no renstricotions placed on the nodes xj nor the weights wj in working
towards that goal. The motivation is that if it is exact
for high degree polynomials, then perhaps it will be
very accurate when integrating functions that are well
approximated by polynomials.
There is no guarantee that such an approach will work.
In fact,nit oturns out to be a bad idea when the node points xj are required to be evenly spaced over the innteorval of integration. But without this restriction on
xj we are able to develop a very accurate set of quadrature formulas.
The case n = 1. We want a formula
Z1
w1f (x1)
f (x) dx
-1
The weight w1 and the node x1 are to be so chosen that the formula is exact for polynomials of as large a
degree as possible.
To do this we substitute f (x) = 1 and f (x) = x. The
first choice leads to
Z1
w1 ? 1 =
1 dx
-1
w1 = 2
The choice f (x) = x leads to
Z1
w1x1 =
x dx = 0
-1
x1 = 0
The desired formula is
Z1
f (x) dx 2f (0)
-1
It is called the midpoint rule and was introduced in
the problems of Section 5.1.
The case n = 2. We want a formula
Z1
w1f (x1) + w2f (x2)
f (x) dx
-1
The weights w1, w2 and the nodes x1, x2 are to be so
chosen that the formula is exact for polynomials of as
large a degree as possible. We substitute and force equality for
f (x) = 1, x, x2, x3
This leads to the system
Z1
w1 + w2
=
1 dx = 2
Z-11
w1x1 + w2x2 w1x21 + w2x22 w1x31 + w2x32
= = =
x dx = 0
Z-11 x2 dx = 2
Z-11
x3
dx
=
3 0
-1
The solution is given by
w1 = w2 = 1,
x1
=
-1 sqrt(3)
,
x2
=
1 sqrt(3)
This yields the formula
Z1 -1
f
(x)
dx
f
?
?
-1
sqrt(3)
+
f
?
?
1
sqrt(3)
(1)
We say it has degree of precision equal to 3 since it
integrates exactly all polynomials of degree 3. We
can verify directly that it does not integrate exactly f (x) = x4.
?
Z
1 x4 dx =
-?1 ?
2 5
?
f
-1 sqrt(3)
+f
1 sqrt(3)
=
2 9
Thus (1) has degree of precision exactly 3.
EXAMPLE Integrate Z 1 dx = log 2 =. 0.69314718 -1 3 + x
The formula (1) yields
1
1
+
= 0.69230769
3 + x1 3 + x2
Error = .000839
THE GENERAL CASE
We want to find the weights {wi} and nodes {xi} so
as to have
Z1
X n
f (x) dx wjf (xj)
-1
j=1
be exact for a polynomials f (x) of as large a degree as possible. As unknowns, there are n weights wi and n nodes xi. Thus it makes sense to initially impose 2n conditions so as to obtain 2n equations for the 2n
unknowns. We require the quadrature formula to be
exact for the cases
f (x) = xi, i = 0, 1, 2, ..., 2n - 1
Then we obtain the system of equations
w1xi1
+
w2xi2
+
?
?
?
+
wnxin
=
Z1 -1
xi
dx
for i = 0, 1, 2, ..., 2n - 1. For the right sides,
Z 1 xi dx
-1
=
2, i+1
0,
i = 0, 2, ..., 2n - 2 i = 1, 3, ..., 2n - 1
................
................
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