Computational Project # 2
Due February 28
Develop, debug, and test a MATLAB program that finds a root of a nonlinear equation
() 0fx
=
using (a) bisection method, (b) secant method, (c) Newton’s method, and (d) regularized
Newton’s method. The code should be able to solve for an arbitrary function f(x). Stop your
iterations, when 6
()10
n
fx −
<. Organize your code such a way that the code has either one or two
input parameters x1 and x2 (initial guess) and the sequence of guesses 123
(,,, ,)
n
x
xx x=x… as the
output parameter. For each method generate vectors
(
)
(
)() ()
(
)
123
,,,,
n
f
xfxfx fx=f… and
(
123
***
,,,,
n
)
*
x
xxxxx xxΔ= − − − −x…, plot them vs. the number of iteration using two
subplots within the same window. For both cases use semi-log scale (log scale in y direction and
linear scale in x). The figures you generate will be called convergence plot. Compare the
convergence of each method, i.e. compare how fast they converge (if the method converges).
For some functions f(x) the straightforward Newton’s method could diverge. In order to make
Newton’s method more robust, it is regularized the following way:
11
n
nn1n
x
xx
τ
+
++
=− Δ
where
(
)
()
1'
n
nn
f
x
x
f
x
+
Δ= and 1
1
2
2
n
nn
n
nn
n
xn
xx
τ
ττ
+
+
⎧
<
⎪
=⎨Δ≥
⎪Δ−Δ
⎩
.
for n=1,… and . Note that when
10.1
τ
=1
n
τ
=
the regularized Newton’s method reduces to the
original one.
Solve the following nonlinear equations with your own program using four methods:
a)
1/2
23
2cos 0
22
xx
π
⎛⎞
⎛⎞
−=
⎜⎟
⎜⎟
⎝⎠
⎝⎠ b)
(
)
12
tan 3 1 0x
−
−
=
Use the following initial guesses for both bisection and secant methods x1=0, x2=1 and x1=1 for
both Newton’s method.
Plot the convergence plots for each case. Discuss the results. Compare the roots you obtain with
the exact values
1/2
*2
3
x⎛⎞
=⎜⎟
⎝⎠ for problem (a) and 12
*3x−
= for problem (b).
P.S. Bring your solution to class. Email your code to [email protected]. The program
that you send should be a working program. All the codes will be checked whether they run or
not. If they are erroneous, but run, points will be taken for the errors. If the code does not run (it
has some syntax errors), an additional 25% will be taken off. The goal of this class is for you to
be comfortable solving engineering problems. Please take your time and learn how to trust the
computer.