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AppSession_01

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Hello, and welcome to Application Session 1 Of this Introduction to Linear Algebra with Wolfram U. The topic for the session is polynomial interpolation. Let's begin with a brief overview of the session. Here is a pair of points, which are called data. The first point is {1, 3}, the next one is {3, 7}. And so you might guess That the polynomial 2<i>x</i> + 1 passes through these points. Here is a plot. It shows that you have The linear polynomial over here And the two data points over there. Now, this linear polynomial can be used to interpolate The values for points between the two given points. You could take the point over here, for example, Like 2.3, and work out the value over there based on The interpolating polynomial. This is a simple example of an interpolating polynomial. Here's a more formal discussion. So let's say given <i>n</i> points in the plane Then there's a polynomial of degree at most <i>n</i> - 1 Could be less than in some cases Which passes through these <i>n</i> points. This polynomial is called an interpolating polynomial For the given set of points. Let's say you have three points Then you have a polynomial of degree 2, etcetera. Now, the main point is that this polynomial Can be constructed by solving a system of linear equations. So although you have a non-linear problem You actually have a linear problem Sitting below it. That's a solving of a system of linear equations. Fortunately, we have a very nice function Called InterpolatingPolynomial Which can be used to compute this polynomial. I'll take a few examples of actually solving the system And then we'll use the built-in functions Straightaway. As a first example, here is, again, a set of two points. I want to construct a linear interpolating polynomial. I'll call it P[<i>x</i>] = <i>a x</i> + <i>b</i> The problem is to determine <i>a</i> and <i>b</i> So I'll set up a linear system for <i>a</i> and <i>b</i> And what we'll do is I set up the equations over here. So basically, I evaluate the polynomial at the <i>x</i> values And set them equal to the <i>y</i> values. OK, once I've done that I can solve the system for <i>a</i> and <i>b</i>. I can plug in the solution And so have that the polynomial is 2 <i>x</i> - 1 Which looks correct from here. So let's plot the polynomial and the data And you see that you actually have the correct fit over here. So onto the case of a quadratic interpolating polynomial. I've got three data points So I'm going to have a quadratic function Fitting them. I set up an equation system With three unknowns of <i>a</i>, <i>b</i> and <i>c</i> Three currently unknowns. I solve the system, plug it back, obtain the polynomial And when I plot over here Then again, there's a very nice fit. The next case will be a cubic interpolating polynomial. So now I've got four data points But this time I'm going to use the built-in function InterpolatingPolynomial To get the answer. Now, the point is this is in the so-called Horner form. It must be expanded to get a nice answer From a [indistinct] perspective, that's the expanded form. And again, if you plot over here You see you get a very nice plot Of the data points and the interpolating polynomial. As the first simple application Let's say I want to construct a quartic polynomial, degree 4 With these roots over here. So I set up a dataset with five points The first four points are just the roots. {2, 0}, {7, 0}, {5, 0}, {-3, 0} And the last one is more or less arbitrary But I just pick it so that I get a nice polynomial. So here's the interpolating polynomial. Clearly, that's going to pass through the required points. And then you can plot over here And you see that you have roots exactly at those points And there's a value at the fifth point. OK, let's take an application. I have a sine function A sine function is surely not a polynomial function But there's some data for it. So I construct an interpolating polynomial. It looks a bit messy. So I can use a Chop function which chops off very small parts To get a slightly more elegant expression. I can plot the data points and the polynomial over here. It looks like the fit is quite nice. So if I try to compute the value at Pi/4 It looks like the approximate and exact values Are quite close to each other But now if I go further away you see that actually The approximation gets quite poor. So the main point is that Interpolation is good where it is valid Namely between the points But once it’s three away from them You might get a pretty bad approximation. So let us be careful when doing interpolation Then using them for applications. That's the end of this session and the main point is that An interpolating polynomial of degree <i>n</i> - 1 Can be constructed for any set of <i>n</i> points in the plane And this polynomial, of course, passes through those points And it can be used to interpolate the values between the points. From the other perspective You actually would solve a linear system of equations To get the coefficients for that polynomial. We have a built-in function, InterpolatingPolynomial Which can be used to compute this polynomial For large sets of data Keeping in mind that as the digits become large The computation will become quite complicated. So I stop here. Thank you very much.

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Posted by: wolfram on Sep 30, 2020

AppSession_01

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