Page 271 - Applied Numerical Methods Using MATLAB

260 NUMERICAL DIFFERENTIATION/ INTEGRATION
(b) Consider the following double integral:

1 1 2
1 π
I = dx dy = (P5.14.2)
0 0 1 − xy 6
Noting that the integrand function is singular at (x, y) = (1, 1),use
the routine “int2s()” and the MATLAB built-in routine “dblquad()”
with the upper limit (d) of the integration interval along the y-axis d
= 0.999, d= 0.9999, d= 0.99999 and d= 0.999999 to compute this
double integral. Fill in Tables P5.14.2 and P5.14.3 with the results and
the times measured by using the commands tic/toc to be taken for
carrying out each computation.

Table P5.14.1 Results of Running ‘‘int2s()’’ and ‘‘dblquad()’’ for (P5.14.1)
int2s(), int2s(), int2s(),
M=N=20 M =N=100 M=N=200 dblquad()
|error| 2.1649 × 10 −8 1.3250 × 10 −8
time

Table P5.14.2 Results of Running ‘‘int2s()’’ and ‘‘dblquad()’’ for (P5.14.2)
a=0,b=1 a=0,b=1 a=0,b= 1 a = 0,b=1
c=0, c=0, c=0, c=0,
d = 1-10 −3 d = 1-10 −4 d =1-10 −5 d = 1-10 −6
int2s() |error| 0.0079 0.0024
M = 2000
N = 2000 time
dblquad |error| 0.0004 0.0006
time

Table P5.14.3 Results of Running the Double Integral Routine ‘‘int2s()’’ for
(P5.14.2)
M = 1000, M = 2000, M = 5000,
N = 1000 N = 2000 N = 5000
int2s() |error| 0.0003
a=0,b=1
c = 0, d = 1-10 −4 time

Based on the results listed in Tables P5.14.2 and P5.14.3, answer the
following questions.
(i) Can we say that the numerical error becomes smaller as we set the
upper limit (d) of the integration interval along the y-axis closer to
the true limit 1?