Graphics Programs Reference

In-Depth Information

EXAMPLE 6.10

Evaluate as accuratelyas possible

∞

x

3

√
x

+

e
−
x
dx

F

=

0

Solution
Inits presentform, the integral is not suited to any of the Gaussian quadra-

tures listedinthis article.But using the transformation

t
2

x

=

dx

=

2
tdt

wehave

2
∞

0

∞

3)
e
−
t
2
dt

3)
e
−
t
2
dt

(
t
2

(
t
2

F

=

+

=

+

−∞

which can beevaluated exactlywith Gauss-Hermite formulausing only two nodes

(
n

=

2). Thus

A
1
(
t
1

A
2
(
t
2
+

=

+

+

F

3)

3)

886 227
(0

3
+

886 227
(

3

707107)
2

707107)
2

=

0

.

.

+

0

.

−

0

.

+

=

6

.

20359

EXAMPLE 6.11

Determinehowmany nodes are required to evaluate

π

sin
x

x

2

dx

0

with Gauss-Legendrequadraturetosix decimal places. The exact integral, rounded

to six places, is 1

.

418 15.

Solution
The integrand is a smooth function; hence it issuited for Gauss-Legendre

integration. There is anindeterminacyat
x

0,butthisdoesnotbother the quadrature

since the integrand is never evaluatedatthat point.We used the following program

thatcomputes the quadraturewith 2

=

,

3

,...

nodes until the desiredaccuracyis reached:

% Example 6.11 (Gauss-Legendre quadrature)

a=0;b=pi;Iexact=1.41815;

forn=2:12

I = gaussQuad(@fex6

_

11,a,b,n);

if abs(I - Iexact) < 0.00001

I

n

break

end

end

Search WWH ::

Custom Search