Graphics Programs Reference

In-Depth Information

Using the extrapolation formulas in Eqs. (6.13), wecan now construct the following

table:

⎡

⎣

⎤

⎦
=

⎡

⎣

⎤

⎦

R
1
,
1

R
2
,
1
R
2
,
2

R
3
,
1
R
3
.
2
R
3
,
3

R
4
,
1
R
4
,
2
R
4
,
3
R
4
,
4

0

1

.

57082

.

0944

.

.

.

1

8961 2

00461

9986

1

.

97422

.

00032

.

0000 2

.

0000

It appearsthat the procedurehasconverged. Therefore,
0

sin
x dx

=

R
4
,
4
=

2

.

0000,

which is, of course, the correct result.

EXAMPLE 6.7

Use Romberg integration to evaluate
√
π

0

2
x
2
cos
x
2
dx
and compare the results with

Example 6.4.

Solution

>> format long

>> [Integral,numEval] = romberg(@fex6

_

7,0,sqrt(pi))

Integral =

-0.89483146948416

numEval =

257

>>

Here the M-file defining the function to be integratedis

functiony=fex6

7(x)

% Function used in Example 6.7

y = 2*(xˆ2)*cos(xˆ2);

_

It is clear that Romberg integrationisconsiderablymoreefficientthan the trape-

zoidal rule. It required 257 function evaluations ascompared to 4097 evaluations with

the composite trapezoidal rule in Example 6.4.

PROBLEM SET 6.1

Use the recursivetrapezoidal rule to evaluate
π/
4

0

1.

ln(1

+

tan
x
)
dx

.

Explain the

results.

2.

The table shows the power
P
supplied to the driving wheels of a car as a function

of the speed
v
. If the mass of the car is
m

=

t
it

takes for the car to accelerate from1m/s to 6m/s. Use the trapezoidal rule for

2000 kg, determine the time

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