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MACM 316 – Assignment 4

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MACM 316 – Assignment 4
A. Computing Assignment – Finding zeros of Bessel functions
The Bessel function J0(x) of the first kind is the solution of Bessel’s differential equation
x
2 d
2y
dx
2
+ x
dy
dx
+ x
2
y = 0,
that is finite at the origin x = 0 (the other solution, the Bessel function of the second kind Y0(x) is
singular at the origin). In Matlab you can evaluate J0(x) using the command besselj(0,x). A plot
of J0(x) is given on the next page.
A number of important physical applications require explicit calculation of the zeros of J0(x). It is
known that J0(x) has infinitely-many zeros in the range 0 < x < ∞. We shall write these zeros as
0 < x1 < x2 < x3 < . . . < ∞.
In this assignment you will use the bisection method to compute the first M of these zeros. It is up
to you to choose appropriate parameters a, b and TOL for the bisection method. To aid in choosing
a and b, I strongly recommend you plot J0(x) first.
Once you have computed the first M zeros, your goal is to determine the asymptotic behaviour of
the Mth zero xM as M → ∞. You should find that xM obeys an approximate linear relationship
for large M. That is to say
xM = α(M + β) + O (1/M), M → ∞,
for certain constants α and β. Using the roots you have calculated, find approximate values of α
and β. Make sure to take M large enough to get good approximations for α and β.
Your conclusions should be explained in a one-page report. Make sure to include the following:
• Justification for the values of a, b, TOL and M you chose in terms of the three key concepts
of the course: accuracy, efficiency and robustness.
• The values of α and β you computed, and an explanation as to how you found them.
• A hypothesis, based on the values you computed, as to the exact values of α and β.
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