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In this segment, we're going to talk about bisection
method,
and look at the advantages and drawbacks
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of the bisection method.|Now, let's go ahead and enumerate the advantages
to begin with.|The advantage, first advantage
of the bisection method is that it is always convergent.
And the reason why it's always convergent
is . . . is evident from the approach itself, if you look at it from a graphical
point of view is that we have . . . we start with some bracketing
of the root.|You start with a bracketing of the root of the equation f of x equal to
0, and if you look at this as the lower . . . lower guess, this as the upper guess,
what you are finding out here is that as you keep on
decreasing the interval size by choosing a new guess as the midpoint,
what is happening is that it is . . . the interval length keeps on diminishing,
it keeps on halving as you keep on going from one iteration to another,
so it always is going to find the interval where the function is changing sign, so it's
always going to be convergent, there's no way for it to diverge.
So that's . . . that's a big advantage of the bisection method.|The second one is that the
error can be controlled.
What I mean by that is that we already know that if the root
is between, let's suppose, this point and this point, then we know that the maximum error which you're going to have
in the root will be that width, because if the . . . if the root
is right here, or something like that, then we know that the, in the starting bracket, this is the
maximum error which you're going to have.|Once you have halved the bracket you got right here, then you know
the root . . . that the root is between this point and this point, then you know that the true error can be
maximum of that.|So, you can see that as you keep on applying one
iteration after another that the amount of error which you're going to have is getting halved,
the maximum amount of error which you're going to have keeps on getting halved as you keep on
going from one iteration to another, so your error can be controlled, so you can, in the beginning itself,
know that how much error you're going to have, as you go from one iteration to another.
Now, let's go ahead and look at some of the drawbacks of bisection method.
One of the
main drawbacks of bisection method is that convergence is slow.
Although convergence is guaranteed, convergence is
generally slow.|And the reason why it is slow is because
every time your interval is getting halved, and that's the max . . . that's the amount of error which is
getting . . . getting it halved by as you go from one iteration to another, so the
convergence is slow.|If you take different examples of the bisection method and run them through,
you'll find out that to be the case.|The second one here
is that . . . that you can . . .
you can get a lot of . . . you may have to do a lot of iterations if you choose one of
the guesses to be very close to the . . . to the root.|So, let's suppose
if we have a function like this, and you choose this to be
the lower guess, and you choose this to be the upper guess.
So you can very well see that as you are halving the step
size, or halving the interval, that you're going to . . . it's going to take a lot
of iterations to keep on zeroing in on the root here, because
you chose one of the initial guesses to be very close to the root itself, so
that is . . . that is a drawback, choosing . . . so let me write this down, choosing
a guess close to the root
may result
in needing
many iterations
to converge.
It is going to converge, but if you're going to choose . . .
choose one of the guesses, one of the initial guesses to be close to the root,
then the number of iterations which you'll need for the same order of convergence
will be quite . . . quite a few.|Let's look at some other
drawbacks.|Another drawback can be something like this, you can have . . .
I can have a function like
this, so let's suppose this function is x squared.
So if I have function is equal to x squared, so the
equation is x squared equal to 0, let's suppose, then we know that, hey, x equal to 0 is the root of that particular
equation.|But you won't be able to find out a lower guess
and upper guess where the function changes sign, because the
root is non-negative everywhere,
so, cannot find
roots of some equations.
Again, it's not necessarily directly the fault of bisection method that you
cannot find the root of that equation because of the fact that bisection method requires
you to find the lower guess and the upper guess to be such that the function
changes sign.|So, if it is not able to do that, it's just that you cannot apply this method
to find the root of an equation like that one.|Another one can be something like this,
may seek
a singularity point
as root.|What is an example of
that is that if your equation is f of x equal to 1 by x equal to 0.
So if we go ahead and plot the function f of x
for this case, you're going to get something like this.
So, what you can have is that you can have
this as your lower limit and this as your upper limit,
or, I shouldn't say limits, but lower guess and upper guess,
so if you have that as your lower guess and your upper guess of the bracket, the function
is changing sign, it is positive here, it is negative here,
and what's going to happen is that it is going to start converging to the singular
point, which is x equal to 0.|Again, this is not the fault of the bisection method
theorem which we use, because the bisection method says that the function has to be
real and continuous between xl and xu.|This function,
1 by x, is continuous 0 onwards on the
right side of the y-axis, or on the left . . . left side of the y-axis, but
if you are crossing the y-axis, it is not continuous, because you're getting discontinuity at x equal to 0,
at x equal to 0 getting discontinuity.|So, but if you were doing this
without paying attention that whether the function is continuous between the lower and upper limit,
you would get this phenomenon where you might converge on a singular point, assuming
that you have the lower guess and the upper guess where the function is changing
sign.|So those are the advantages and drawbacks of the bisection
method.|And that's the end of this segment.
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