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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