Numerical Integration - Numerical Methods in Engineering with MATLAB - page 214

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