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More concretely, in what situations are the predictions made by Lagrange's Interpolation accurate? Say, if we have 4 known values of some country's populations, would the unknown value predicted by interpolating the points be accurate? I doubt it would, because population doesn't have to increase/decrease to fit an equation predicted by Lagrange. In what cases would the prediction be accurate?

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    This is not clear. Are your four populations just listed in a table? But nobody thinks the population of a country is a function of its order in a table you drew up at random. You need the function values to be functions of the index variables. But, if you have that, then the method is helpful. Why not? People who model financial curves of various sorts use all sorts of interpolation schemes, and polynomial fitting is certainly useful. – 2017-02-12
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    The reference about country's population doesn't seem very relevant. Lagrange's Interpolation gives a polynomial of degree $n$ when given $n$ points in $\mathbb{R}^2$. (Like $y=x^2-1$ or whatever). – 2017-02-12
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    The reference was used to illustrate the point of my question. I do understand what and how interpolation is used, but I'm wondering in what cases are the predictions of missing values are accurate. As with reference, I'm asking the following: say, we had Russia's population in 1970, 1971, 1972 and 1973; would P(1974) be more or less relevant prediction? (P stands for population function). Also, I understand it's not that simple to predict a country's population in that manner, but I'm questioning if there are cases in which interpolating would be sufficient to predict missing values. – 2017-02-12
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    It's really not clear that you understand interpolation, given the example you used (in which nothing is being interpolated). But if you want a general rule then I'd argue that polynomial interpolation, much like linear interpolation, works best locally. For global fitting....well, unless you have an a priori belief that the system should follow a polynomial...it's not so great. And it is easy to misuse. Also, for fitting noisy data it is often better to use some error minimizing model rather than demand an entirely spurious precision.... – 2017-02-12
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    ...But these are technical points. I use Lagrange constantly in designing smooth models for curves. Specifically, I use the technique to fit nice polynomials locally, which are then smoothed globally by a different method. – 2017-02-12
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    Let me ask the following then: are there such systems that follow a polynomial? – 2017-02-12
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    Globally or locally? I'd have faith even in your population model if you asked about what happened in between 1972 and 1973....I'd have no faith at all if you used it to predict the population in 2050 or in 1630. Simple trajectory models are quadratic, but I'd tend to agree that few interesting systems are globally modeled by polynomials, nor by any other elementary functions. – 2017-02-12
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    The question about population was just an example. I understand it'd be incorrect to assume P(2050) would be somewhat close to reality. However, does a system exist in which predicting by modeled polynomials is helpful? Maybe a relationship between # of sheep and # of wolves, or anything? And, if such systems exist, what are they? – 2017-02-12
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    I don't know how else to answer that question beyond what I've already said. Real functions, functions arising from the real world, are messy. Simple interpolating methods, as here, are good at local approximations and they are good for designing smooth models. They are not especially good globally (and indeed I've never seen them used globally except as exercises). – 2017-02-12
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    Just to repeat: even if the physical system was known to be governed by a polynomial, I wouldn't use Lagrange globally. That is because data gathering is intrinsically noisy and insisting on exactness at certain points tends to mean modeling that noise. I'd use the model to derive a polynomial shape and then use some error minimizing regression to fit the parameters. But that does not in any way reduce the value of Lagrange! – 2017-02-12
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    That's exactly what I was interested in. Thanks. – 2017-02-12

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The basic Lagrange interpolation with many points is primarily of theoretical interest.

Polynomials are pretty well-behaved and well-understood functions, and it is sometimes useful to know while conducting a proof that there exists a well-behaved function that goes through any finite number of points you care to specify.

In practice, however, Lagrange interpolation on real-world data sets with many points fails horribly. Even tiny rounding or measurement errors in the middle of the sequence will blow up into insane oscillations of the interpolated function at the ends.

With a few points given -- no more than about 4, I'd say -- Lagrange interpolation can be useful for producing an approximation of the underlying function. It is probably misleading to think of it as a prediction, though. But if you already know that the underlying curve is nice and smooth throughout the interval you're looking at, then a polynomial interpolation can give you an arithmetic expression for something that follows that curve "pretty well". This gives you some advantages:

  • The arithmetic expression is often cheaper to evaluate than it is to find a value of the true curve.
  • The arithmetic expression can be manipulated with symbolic algebra/calculus if the curve is input to some further computation, again saving effort.

The disadvantage is of course that the approximation may not follow the true curve exactly -- especially if it turns out that the true curve is not really as smooth and uniform as the approximation assumes.

It is a trade-off in each case whether the ease of computing with the approximation outweighs the cost of the approximation errors.

And, of course, if you do decide to approximate, a naive polynomial interpolation may not be the best way to do that. There are many other options -- some of which use polynomial interpolation as a component of a more intricate scheme.

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    [+1] thorough treatment. – 2017-02-12
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    Lagrange interpolation *at equally spaced points* is horrible. But if you space the abscissae intelligently, and you're careful how you calculate the value of the interpolant, then using high-degree polynomials works fine. Take a look at some of the approximations produced by chebfun. – 2017-02-13
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If you know the value of some function at certain points only, and you want to know its value at some other point, then your only option is some sort of interpolation.

If you have some knowledge of the specific form of the function you expect, you can use that. You can fit a function of this specific form to your data, and then use that function for interpolation.

If you have no knowledge of the form of the function, then you may as well assume that it's something simple and easy to compute. Lagrange interpolation and spline interpolation satisfy this requirement. The only thing wrong with Lagrange interpolation is that the curves tend to get extraneous "wiggles" if you're not careful.

To answer your question, I'd say that Lagrange interpolation (or, more generally, polynomial interpolation) is quite widely used.

You can make the interpolation as accurate as you like just by increasing the degree of the polynomial. There's a system called "chebfun" that's very good at this.

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    As Henning Makholm as said, Lagrange interpolation is only interesting for theoretical uses. let us say that it **must** always be avoided for practical applications because of its instability. – 2017-02-12
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    According to Lloyd Trefethen (who ought to know), the instability of Lagrange interpolation (for approximation of functions) is a myth. You can read his views on this subject here https://people.maths.ox.ac.uk/trefethen/publication/PDF/2011_139.pdf. The chebfun system uses polynomial interpolants with degrees in the hundreds, with no hint of extraneous wiggles. The key issue is that you must not interpolate at equally-spaced points. – 2017-02-13
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    I appreciate your commentary, and I will surely have a look at the reference you gave. Sure, Lloyd Trefethen, not only is a prominent Numerical Analyst, but moreover he has a wonderful pedagogy. – 2017-02-13