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Hello, good morning, having introduced single
step methods, I kept this title as analysis
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of single step methods, because if you look
at the initial value problem y dash equals
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to f
of x y with given initial condition y of x
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0 equal x 0. So, what is y dash, you can think
as
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a slope now slope is equal to some function,
so we have to now find what is the
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corresponding y?
So, this definitely needs some approximation,
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as you see in the Taylor series method
which we have discussed. We were using various
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terms; second term, third term up to
fourth term, fifth term and then corresponding
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error etcetera, so that means definitely
there is some approximation involved. So,
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can we formulize this approximation? So, that
is the question, so this needs some analysis
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hence the title, so let us recall Taylor series
method.
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.
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So, this is Taylor series method, for example,
stops at third, and so then we write zeta
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n
where
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this interrupts, so the question is what happens?
If one uses more terms. So, the
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immediate answer is definitely the solution
may vary yeah of course you are true the
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.solution may vary, but what sense can we
say concretely if we use more number of
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terms. So, with general intuition the solution
will be better and things like that.
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So, we need
formulize this and one institution says number
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of terms controls the error, so
this may be formulized. So, if you look at
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this, this is first derivative, we have written
it
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as this is nothing but y dash second derivative
and so on. So, that means after some stage
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you can take for example, you can take out
h, but we need at least first approximation,
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so
suppose if we take out h then these terms
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can be controlled as some function and the
entire behavior may depend on this function.
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.
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So, suppose if we introduce this notation
for a particular h up to some terms, then
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what
kind of structure F may have. So, this depends
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on
the number of terms certainly, so this
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depends on number of terms, if this depends
on number of terms for sure we are
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approximating up to those many terms and started
neglecting from these. So, what is this
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then? So, this is residual or error, so what
do you mean by this if you compare with the
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exact solution up to these than the r neglecting
from here.
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So, that is called residual or error and we
introduce this notation because the minimum
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up to first. So, as I mentioned earlier, you
can pull out h and express the rest as tau
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and
plus 1, so what is then tau and plus 1? So,
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this is local truncation error why because
you
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are approximating up to some terms and you
are truncating therefore, this is local
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truncation error then we introduce tau h which
is max
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because in each interval if you
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.consider 0 to 1. I mean n 1 to 2, so we have
corresponding local truncation error
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involved, now we take the maximum and call
it tau. So, this is global
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truncation error,
now typically what kind of structure F would
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have.
.
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So, the structure F, so why this structure
because we have pulled out h, therefore f
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starts
from f of x and plus so on. So, when we start
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a competition, so when we start a
competition like this, so this would lead
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to say if we put down up to three terms third
term is like this. These are
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increment function, so these are called increment
function
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because you keep on adding more terms.
So, F behaves according to that, so we have
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to formulize the error and also we have to
formulize if ten terms are used what is the
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behavior and what do you call the method if
used twenty terms. So, this ten terms, twenty
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terms, forty terms, so this number also have
significance and same time whenever you have
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a h truncated up to something then
whatever you have thrown away, so that also
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has a significance we are trying to
formulize.
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..
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So, to this extent
how F behaves as h becomes small, so this
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is our first term then h over
2, then we can pull out, then we have some
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remaining terms. So, this would be plus, plus
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because the remaining will be of order h cube
right. So, this employs limit h goes to 0
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tau
and plus 1 h, so this goes to 0, so this goes
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to 0. So, this employs limit h goes to 0 tau
h
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goes to 0, so if this happen then we say
the single step method is consistent with
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the IVP.
So, why this is happening the way we have
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defined residual h times tau and plus 1,
therefore tau plus 1 goes to 0 because at
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least one h remain in these. Therefore, h
goes to
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0, this goes to 0, so if this happens then
we say single step method is consistent with
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the
IVP.
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..
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So, up to what terms we consider and then
how do we formulize, so we have to discuss
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that is called
order of a method. So, as your intuition says,
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the number of terms is related
to the word order, so a method is said to
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be of order p if every x a b the solution
y x of
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the IVP satisfy the condition. So, a method
is said to be of order p if for every x within
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in
this interval, where the corresponding IVP
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is defined, the solution y x of IVP satisfy
the
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condition.
So, as h diminishes, your global truncation
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error is h power p, so then we say the method
is a fodder p. Let us say for example, when
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we use first three terms of Taylors method,
so
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what we would we have h, h square by 2, we
have y double plus y 3 at the time. So, if
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you use first three terms, then we are not
using fourth term, see one, two, three, fourth
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terms we are not using, this means we are
throwing it away.
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..
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Therefore, this would be h square by 6, so
remember we have defined xenon n plus 1 as
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h times. Now, n plus one, so there is h, so
I pulled out one h, therefore the coefficient
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is h
square. So, I made equality, you may correct
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it of this range, therefore tau h is max of
mod, so this behaves like this because we
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can estimate.
So, we can get a constant that together with
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the one over 6, we can get constant and
behaves like this. So, this employs, hence
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the method is second order, so this one hand
because the number of terms and then the how
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it behaves. We can construct ten terms,
twenty terms, what you call, that is nothing
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but a direct indication order of the method.
So, the next consent is if you consider twenty
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terms, five terms whether we really get
solution that is close enough, so which means
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convergence.
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..
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So, let us formulize what is convergence,
so we have this approximation, let this satisfy
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the scheme star in exact sense, then
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method is said to be convergent if
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you can easily.
Method is said to be convergence, if for every
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n this is exact of the scheme
approximating scheme. So, then y and what
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we obtain, so then the method is said to be
convergent.
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So, when we can obtain, we can see with the
corresponding error into consideration. So,
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then the method is said to be convergent if
the difference between the exact of the of
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the
approximating scheme actually obtained after
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considering the error is less than c of h
when we say c of h definitely this depends
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on h. Now, when do you say this is
convergent c h is an infinitesimal, that means
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this very small with respect to h. So, if
this
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is going to happen, then we say the method
is going to convergent, so what is the idea,
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so
you have approximated.
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So, what is the guarantee that this would
really give you solution with this converges
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this
scheme that means if you keep on complotting
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and if it blows up, and then it is beyond
out of control. So, to reach the solution,
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so when we say such an approximation is
convergent you consider without error than
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consider with error and the difference should
be diminution, it should be very of course
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depends on h. Now, correspondingly h the
number of terms also can be related, so this
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is convergence.
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..
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Further, the method is said to be convergent
with order p if there exists d great than
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0
such that
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the method is said to be convergent with order
p if their exist d great than 0 that
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c of h behaves like this. So, for example,
the Taylors up to three terms d times h cube
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where p is 2. So, this is convergent with
order 2, so what is the motivation, how do
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we
approximate, how many terms we consider? These
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are the quick questions we may come
across, so well we have assumed that F is
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smooth, then we try to expand and assuming
we have as many times as possible to differentiate.
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So, we propose Taylors series, but we started
compromising because we have restrict
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within the discrete notation. So, when you
have compromised the question are how many
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terms for this problem if I take this, many
terms what happens? So, these are the
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questions somebody would try to be a little
intelligent and then say I would consider
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h
pretty small, so small I mean I can start
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throwing from h square onwards.
So, if you if you argue like this yeah definitely
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you get some approximation because h
square onwards you are just throwing. Well,
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definitely what should be your h such that
your h square could be thrown it matters,
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but however if you assume that h square
onwards, you can throw definitely you are
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going to get some approximation, so let us
look at it, what is that?
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..
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So, this title Euler’s method, so what is
assumption, the assumption is step size h
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is very
small means order of h square can be neglected.
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Now, consider
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because that is our
assumption, so this is nothing but y 1 y 0
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because our IVP, now if we continue, the next
big point. So, we can formulize, so this approximation
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obviously
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is your Euler’s method.
Now, what is the corresponding error, the
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corresponding error we started throwing from
this now as we decided earlier we can estimate
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this.
Now, if error behaves like this, then that
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means Euler method is order 1, so Euler method
is first order method. So, you can probably
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guess because h is quite small started
throwing h square, so it is a first order
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method. So, definitely your solution you are
having big compromise in some sense, so we
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have to really think of better
approximations that where we are going to
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formulize better approximations.
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..
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So, before we go because in order to go better
approximations we should know what is
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the interpolation geometric views of this,
this would be given a hint for our
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generalization. So, say we have this x 0,
x 1, now this general Euler method, if we
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write
down the initial point
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and what is this? This is nothing but y dash
of x 0, so what is
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exactly happening, y dash of x 0 has been
approximated by this. So, use the tangent
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at x
0 y 0 as an approximation
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to the curve y x in this interval.
So, that means we have said function like
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this then this is y 0, so this is y 1 use
the
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tangent at x 0, y 0 at the approximation to
the curve y x in this interval. So, this is
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nothing but
according to this tangent which is by
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approximation to the function, so this
employs, so this we call h
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so this will give us y of x 1.
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..
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So, this give us that means you started x
0 x 1 x 2, so we started approximating so
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next
step if we approximate, so then we have correspondingly
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this difference will be y of x 2
minus y of x 1. So, we can march accordingly,
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so then
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in a general sense, we get this, so
this is a general idea. So, what exactly happens
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numerically, so this is geometrics so you
consider y dash of x equals to f of x y. So,
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then if you integrate this range we get, so
this
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what now this has to be we are having approximation
of this in different patterns leading
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to various method that is what is exactly
happening. So, in order this needs to be
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numerically integrated, so this needs to be
numerically integrated.
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.
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.Suppose, the simple situation
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is say we go for
is approximated by h times, so this is
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called rectangular rule. So, g of x between
x and x n plus 1 you are approximating by
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the
difference is h, so h times g of x n you are
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00:37:12,091 --> 00:37:18,010
approximating by rectangular rule. So, if
this
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00:37:18,010 --> 00:37:40,460
is the case, then we get from here
h times. So, this Euler instead of rectangular
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can be
generalized, so how do we generalize? So,
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I explain what is this, so we are generalizing
not using simply rectangular like this, you
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are giving some weightage 1 minus theta at
g
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of x n and theta g of x n plus 1. So, restrict
theta in this range, now this if we
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use it in A,
so we get corresponding approximation for
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00:38:58,040 --> 00:39:03,030
f then we may get different method.
.
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00:39:03,030 --> 00:39:38,240
So, we get
plus h remember this is at x n plus 1, so
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with this more general if theta equal
to 0 employs definitely Euler’s method theta
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equal to 0. So, this sometimes called
implicit Euler method, suppose theta equals
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to half, so both terms survive we get, so
this
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00:40:49,150 --> 00:41:13,680
is trapezium rule method. So, we discuss geometrically
another observation we see to
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compute y n plus 1, right hand side is also
demanding y n plus 1. So, this is
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then I am
sorry this Euler method theta equal 0 this
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explicit. So, this is not a implicit explicit
and
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this is implicit, now say theta equals to
1, so this goes away so we get y n plus h
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because
theta 1 goes away this survives.
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00:42:06,869 --> 00:42:26,170
So, this looks like Euler method, but this
is implicit Euler method, now the idea is
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if you
keep on changing, then we are getting different
201
00:42:31,750 --> 00:42:41,930
approximations. So, we have to really
generalize what are we doing geometrically
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we started an interval. Within in the interval,
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.we approximate the slope with the corresponding
function value at the initial point the
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00:42:53,190 --> 00:42:59,320
beginning if the interval is x k, x k plus
1 then y dash of x k, we have approximated
205
00:42:59,320 --> 00:43:02,670
by
the function value at x k.
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00:43:02,670 --> 00:43:10,870
So, definitely this is giving up to first
order, so then we started splitting and giving
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00:43:10,870 --> 00:43:14,350
it 1
minus theta by it to left and theta to right
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00:43:14,350 --> 00:43:24,900
and we tried to estimate depending on theta
value we get various methods. So, what is
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00:43:24,900 --> 00:43:33,720
generality in this, it is nothing approximating
with various slopes, so you can now think
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00:43:33,720 --> 00:43:42,890
of how to generalize that means we can
approximate using two, three, four, five slopes
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00:43:42,890 --> 00:43:44,050
etcetera.
.
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So, let us more carefully, so this generalization
to single step methods, so generalization
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of single step methods
h times
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some kind of average of slopes
at intermediate points. So,
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this is nothing but
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I put some kind definitely the method would
be different depending on
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what kind average. So, what is general idea
approximate y dash of zeta n as a weighted
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average of slopes at intermediate points x
n plus say x 0 plus n h. So, you have say
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00:46:18,380 --> 00:46:31,900
x 0 x
1, so we approximate. So, if you want to generalize
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00:46:31,900 --> 00:46:56,450
this so may be this is x n and x n plus
a 2 h and then x n plus a 3 h, then somewhere
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00:46:56,450 --> 00:47:04,270
we get x n plus 1 that means we have
changed slightly.
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00:47:04,270 --> 00:47:05,270
..
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So, because within an interval x n to x n
plus 1, we have introduced an a i h where
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00:47:21,160 --> 00:47:36,490
a i is 0
then a i equals to a i depends, so it will
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00:47:36,490 --> 00:47:50,410
be i equal to 2, 3, l. So, within interval
x n plus a
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00:47:50,410 --> 00:48:05,130
2 h x n plus a 3 h x n plus a l h. So, that
means within interval, so first time you recall,
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so
this is y dash of x n, we have approximated
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00:48:11,110 --> 00:48:19,470
only using this slope at y dash of x n. Now,
within one interval slopes act several intermediate
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points say in this case l slopes hope
you get this is generalization.
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00:48:27,840 --> 00:48:55,260
So, let us see a simple case in this namely
modified Euler’s, so this is x n, then you
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pick
up x n plus h by 2. So, where you are getting
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00:49:03,210 --> 00:49:19,230
x n plus h by 2, so this look at this notion,
so I picked as half, so it is not equal, so
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00:49:19,230 --> 00:49:25,990
8 x value depending upon i, now a i, so here
so a
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00:49:25,990 --> 00:49:44,930
i is zero, so a 2 is chosen as half you can
see, so we are using midpoint. So, we are
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00:49:44,930 --> 00:49:58,220
step
half way across
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00:49:58,220 --> 00:50:16,560
this point, goes step half way across this
then evaluate F at midpoint,
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00:50:16,560 --> 00:50:37,610
then use the
corresponding slope to move further, so which
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00:50:37,610 --> 00:50:53,650
means y n plus 1 is y n plus h
by dash of x n plus h by 2.
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00:50:53,650 --> 00:50:54,650
..
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00:50:54,650 --> 00:51:08,990
So, this is nothing but f of x n plus n by
2 y n plus h by 2, so this is equal to f of
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00:51:08,990 --> 00:51:21,770
x n plus
h by 2, y n plus h by 2 times. Therefore,
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00:51:21,770 --> 00:51:41,030
we get the method, so this is modified if
we use
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00:51:41,030 --> 00:51:48,150
midpoint, now there is no necessity that we
should use midpoint.
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00:51:48,150 --> 00:51:49,150
.
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00:51:49,150 --> 00:52:22,400
So, let us say step across x n, x n plus 1
at x n plus k, so then we get y dash of n,
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00:52:22,400 --> 00:52:29,010
x n plus
k now depends on what kind of approximation
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00:52:29,010 --> 00:52:38,220
we go for. Suppose, somebody goes for
approximation
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00:52:38,220 --> 00:52:59,510
x n plus h, so we get
suppose we go for this approximation by average
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00:52:59,510 --> 00:53:00,510
at
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00:53:00,510 --> 00:53:30,720
.x n, so at x n plus 1. So, then we get approximated
like this then we get, so this is your
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00:53:30,720 --> 00:53:34,890
trapezoidal rule.
.
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00:53:34,890 --> 00:53:58,840
So, we are coming to generalization, so what
is generalization evaluate f of x y at several
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00:53:58,840 --> 00:54:39,930
intermediate points within. Then take a linear
combination of these values and add it to
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00:54:39,930 --> 00:54:45,940
y
n, so you have seen how considered initially
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00:54:45,940 --> 00:54:53,099
the weights on theta. So, similarly instead
of
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00:54:53,099 --> 00:54:59,300
considering only the end points, you can step
across several intermediate points within
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00:54:59,300 --> 00:55:05,320
the small interval x n, x n plus 1.
I am not talking about Euler’s method from
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00:55:05,320 --> 00:55:08,930
x 0 to x 1, x 1 to x 2, x 2 to x 3 and then
we
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00:55:08,930 --> 00:55:14,750
got the generalized method I am talking within
the interval x n and x n plus 1. You go for
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00:55:14,750 --> 00:55:21,320
several intermediate points and then you start
approximating tau, those slopes and take a
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00:55:21,320 --> 00:55:28,070
linear combination, so that it would give
better approximation. So, this is this
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00:55:28,070 --> 00:55:35,300
generalization is leading to something called
arcade methods, which we discuss in the
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00:55:35,300 --> 00:55:38,110
next lecture until then, bye.
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00:55:38,110 --> 00:55:38,110
.