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NUMERICAL METHODS

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1
Basis of Bisection Method
Theorem
x
f(x) x
u
x
An equation f(x)=0, where f(x) is a real continuous function, has at least one root between x
l
and x
u
if f(x
l
) f(x
u
) < 0.
Figure 1
At least one root exists between the two points if the function is real, continuous, and changes sign.
x
f(x) x
u
x
2
Basis of Bisection Method
Figure 2
If function does not change sign between two points, roots of the equation may still exist between the two points.
x f
0
x f
x
f(x) x
u
x
3
Basis of Bisection Method
Figure 3
If the function does not change sign between two points, there may not be any roots for the equation between the two points.
x
f(x) x
u
x
x f
0
x f
x
f(x) x
u
x
4
Basis of Bisection Method
Figure 4
If the function changes sign between two points, more than one root for the equation may exist between the two points.
x f
0
x f

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