2. 2
• Easy
• But, not easy
• How about these?
a
acbb
xcbxax
2
4
0
2
2
?02345
xfexdxcxbxax
Roots of Equations
?0)3sin()10cos(
?0sin
xxx
xxx
3. Graphical Approach
• Make a plot of the
function f(x) and
observe where it
crosses the x-axis,
i.e. f(x) = 0
• Not very practical
but can be used to
obtain rough
estimates for roots
• These estimates can
be used as initial
guesses for numerical
methods that we’ll
study here.
Using MATLAB, plot f(x)=sin(10x)+cos(3x)
4. 4
Why Called Bracketing Methods?
They require two initial guesses which “bracket”
either side of the root.
5. 5
Example1: the parachutist problem
Given m = 68.1 kg, v = 40 m/s, t = 10 s, g = 9.8 m/s2, find the corresponding c
Graphical Methods
401
38667
401
16889
1 146843010168
)(
.
)(
).(.
)()( .)./()/( cctmc
e
c
e
c
ve
c
gm
cf
)( )/( tmc
e
c
gm
v
1
Make a plot of the function and observe where it crosses the x axis
Solution:
c f(c)
4 34.115
8 17.653
12 6.067
16 -2.269
20 -8.401
Root is between 12 and 16
6. 6
Estimate Properties of Roots By Graphical Methods
From (a) and (c):
if both f(xl) and f(xu) have the same sign, there must be 0 or even number of roots
From (b) and (d):
if f(xl) and f(xu) have different signs, there must be 1 or odd number of roots
Exceptions:
multiple roots
f(x) = (x-2)2(x-4)
discontinuous f(x)
7. 7
Find x such that f(x) = 0:
Step 1: Choose lower xl and upper xu such that f(xl)f(xu) < 0
Step 2: an estimate of the root
Step 3: Make the evaluations to determine in which interval the root lies:
(a) if f(xl)f(xr) < 0, the root lies in the lower interval; set xu = xr and
return to Step 2.
(b) if f(xl)f(xr) > 0, the root lies in the upper interval; set xl = xr and
return to Step 2.
(c) if f(xl)f(xr) = 0, the root = xr, stop.
Bisection Method
2
ul
r
xx
x
15. 15
Bisection Method
Pros
• Easy
• Always finds a root
• Number of iterations
required to attain an
absolute error.
Cons
• Slow
• Need to find initial
guesses for xl and xu
• No account is taken
of the fact that if f(xl)
is closer to zero, it is
likely that root is
closer to xl .
17. 17
Incremental Search and Determining Initial Guesses
• For every interval xi – xi+1, apply bisection or false-
position method to find the roots in the interval
• Only roots in x4-x5 will be found