AMSC/CMSC 460 Dr. Wolfe ASSIGNMENT #4 Due July 10, 2009
1. Write a MATLAB program to evaluate I=Rb
af(x)dx using the trapezoidal rule
with nsubdivisions, calling the result In. Use the program to calculate the following
integrals with n= 2,4,8,16,...,512.
(a) Z1
0p16 + x2dx (b) Z1
0
x3/8dx
The exact value of the integral in (a) is 4.04128450518694.
Analyze emperically the rate of convergence of Into Iby calculating the ratios
Rn=I2n−In
I4n−I2n
and pn=log(Rn)
log(2)
In part (b) compute the extrapolated approximation to I,
I˜=I4n−(I4n−I2n)2
(I4n−I2n)−(I2n−In)
for n= 128.
2. Repeat problem 1 using Simpson’s rule.
3. Apply the corrected trapezoidal rule to the integral in problem 1(a). Compare the
results with those of problem 2 for Simpson’s rule.
4. Use Gauss-Legendre integration with n= 2,4,8 nodes to the integrals of problem 1.
Compare the results with those for the trapezoidal and Simpson methods.
5. Find approximate values of the integral in problem 1(a) by computing the Romberg
integral I(5)
32 where I(0)
nis the n-panel trapezoid approximation and
I(k)
n=4kI(k−1)
n−I(k−1)
n/2
4k−1
for ndivisible by 2k.
6. Use the MATLAB function QUADL to find approximate values of the integrals 1(a)
and 1(b).
7. Ex. 6.13 p.181 Numerical Computing with MATLAB.
8. The 11 point Newton-Cotes integration rule on [0,1] is
Z1
0
f(x)dx ≈
10
X
i=0
wif(i
10)
