WEBVTT
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Using Newton's method. We will approximate the fourth root
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of 75, Correent to eight decimal places. So
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remember Newton's method is a method with which we calculate
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are approximately route of no linear equation of the forum
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F of X equals zero. That is a general
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nonlinear expression equated to zero. And this method help
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us a person made solutions to this equation in the
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idea behind the method is that we start with some
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initial gas. It's not. And if we have
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something like this, this is a function. This
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route we want to approximate. And we started with
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, let's say some initial guess it's not relatively close
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to the root in some way. So we take
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the tangent line to the curve at that initial guess
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if he takes you take the tunnel line at this
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point, let's say a tangent line really got the
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X axis at some point which we which will be
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the next iterate. And we repeat the process will
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take tiny line at that point. And we continue
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that way. If we can see in this small
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sketch, we're going to be hopefully we will be
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approximated to the word or brushing the root of long
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nonlinear equation. That's equivalent to saying that we're calculating
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the zero of a function. So here we are
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asked to present the numbers of the question is which
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is the integration to apply Newton's method. But we're
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gonna say we gotta do before applying this method is
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uh state to following. We have the value X
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is the number 4th Root of 75. It means
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that X to the fourth is equal to 75.
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And then Eggs to the 4th 75 is equal to
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zero. And now we have an equation and only
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near creation where we can apply this in medicine.
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And what happens if we found an X. Which
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satisfies this equation? Well we do the reverse process
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, we have this equation and then this employed this
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. We can take in this step here. From
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here. We can take the fourth word because 75
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is positive number. And so they're these equations are
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equivalent are the same. That is we can go
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from one to the other. So it means that
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if we solve this nonlinear equation we will have a
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value X. Which must be equal to the 4th
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root of 75. Sorry, just put this year's
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. Okay. Okay. So we at the souls
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the non linear equation we find to value Mhm.
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Force Group of 75. And the previous year the
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two equations that are equivalent. Well that's it.
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Now we know that our function F. of X
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. X. to the 4th-75. It's derivative
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is for X cube and derivative is needed because this
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idea have explained here. We made the calculations of
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the tender line at the root or point of intersection
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of the detainee in line with the X axis.
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We get something like this. The new eatery is
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equal to the previous one minus the function the that
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previous iterate over to the relative at that trade.
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So this is a way of advancing the illiterates using
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the previous one. We use this formula where we
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use the function and its derivative and doing the calculations
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, we get the next citrate and we repeat the
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process as we said in the graph. And that
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for any greater than or equal to zero, it
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means we can have an initial gas for the rule
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we can obtain the guests maybe in the licenses equation
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or the function or having a graph of the function
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in some way. So for our case here as
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we are calculating The 4th Root of 75. And
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the idea behind Newton's method is that the initial gas
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be close to the root in some way. What
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we can see here is that we can think about
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an exact false route. The number Which is close
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to 75 for example, 81, is exactly three
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to the fourth. That is The 4th Root of
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81. He's three and 81 is not Far from
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75 6 units away. It means that if we
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start with this valuable, we should be close to
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the four food of 75 sell the good initial guests
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for a problem is three. And now you applied
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Newton's method in this case and some to Medicare to
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be programmed some language to the calculations the iterations are
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Yeah, updating the traits using this formula. That's
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all. But there are some things we got to
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consider that is when the method is going to stop
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generally we used to um conditions to stop the iterations
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. One of these conditions can be two consecutive illiterates
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are separate from small distance. That is these conditions
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here. This number is small in some way which
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we get to precise in the code. Another way
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is that because we are solving the nonlinear equation F
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of x equals zero. We can shake also this
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if the function evaluated at an illiterate is also small
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but we can combine both of those conditions and with
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these conditions we stopped the iterations and here for the
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quantity to be small is where we decide how many
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decimals would get and putting a small number as the
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accuracy here we get more correct decimals. The other
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things we got to check in our code is that
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this denominator here is not no. In the sense
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of numerical note that is it is a small number
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in terms of the calculations in the computer. Now
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, not as a mathematical zero. Well if we
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It started three with small, Curious E that is
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put for example, 10- 1913 or 12.
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We'll get Almost 15 decim of security. And when
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newton smith converged converges. Okay, quadratic lee that
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is, it's a rapid convergence and here with and
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a heresy of let's say 10 to the-14.
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That is we you said number to compare these quantities
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and initial guess three. We Need four Iterations.
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In fact converges very quickly for alterations to obtain an
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approximation of four through 75 with 13 correct decimals.
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It is more much more than was asked if you
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want And the value is approximately equal to 2.9 four
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28 28 30 95 638. Okay, 27 12
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In four iterations that is a very fast convergence of
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this method For these function X to the 4th mine
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and 75 An initial guess of three with an accuracy
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for double precision of 10-2014 we have here we have
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double precision meaning that we have around 14 13 14
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15 not fisting my women between 13 and 14,
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correct. In fact I have checked with calculator different
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ways and we have all 15 correct decimals here.
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And the important fact is that we have calculated that
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you've seen a very simple function in this case a
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polynomial function X to the fourth minus 75 with very
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clear need derivative for X cube and uh it means
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that it's a very effective way of calculating numbers and
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uh of course in the case where the method converges
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because in that case it converges quite dramatically that he
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is very fast