Mathematics Stack Exchange
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Questions tagged [lagrange-interpolation]

A method of generating a polynomial that crosses through a set of data. The degree of this polynomial is equal to the size of the data.

Let $(x_j)_{j=0}^{m-1},(y_j)_{j=0}^{m-1}$ be real numbers such that no $2$ $x_j$s are the same. The Lagrange interpolating polynomial is given by $$ l(x) = \sum_{j=0}^{m-1} y_j \prod_{j \neq k \in [0..m-1]} \frac{x-x_k}{x_j - x_k} $$

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question about the definition of linear Lagrange polynomial

How the following relation can be proved: If $$l\left(x\right)=\left(x-x_{0}\right)\left(x-x_{1}\right)...\left(x-x_{k}\right)$$ if a polynomial,then $$l_{j}\left(x\right)=\frac{l\left(x\right)}{l^{'}\left(x_{j}\right)\left(x-x_{j}\right)}$$ …
Absurd
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Polynomial Interpolation using large set of data.

I have a large set of {$x_i$}$_{i=1}^n$ and corresponding values of the function {$f(x_i)$}$_{i=1}^n$. My aim is to estimate the function $f(x)$. Therefore, I think that appropriate technique is Lagrange Polynomial Interpolation. The problem is: as…
David
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Lagrange Interpolation how does it work?

I am making a program to find a polynomial given a set of data. I understand the summation of the formula. Given the image below can someone explain what the square looking symbol is next to the Li(x)? Can you explain the Li(x) function to me as…
User
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Is it possible to avoid the Runge phenomenon by other norm instead of $L_{\infty}$ norm?

Most of error analysis of polynomial interpolation employs $L_{\infty}$ norm. However, $||\prod\limits_{i=1}^{n}(\cdot-x_i)||_{L^{\infty}(a,b)}\leq\frac{n!h^{n+1}}{4} $ when equidistant points are used, i.e., $x_i=a+ih$ with $h=(b-a)/n$. The bound…
Lin Xuelei
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Lagrangian interpolation: degree of the interpolation polynomial

Given $n+1$ points in the plane, Lagrangian interpolation gives the unique polynomial of degree less or equal to $n$. From the examples I've seen so far, interpolation polynomials all have degree exactly $n$ but I don't know if there might be a set…
Conjecture
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How to convert a linear vector to a polynomial using Lagrangian interpolation?

I'm trying to understand this article by Vitalik Butarin on conversion to QAPs as a pre-requisite to understanding zk-SNARKS. However, I seem to have hit a dead-end when he says to convert a vector into its corresponding polynomial. I am able to…
goelakash
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Second lagrange interpolating polynomial f(x) = 1/x

https://i.stack.imgur.com/tD7bn.jpg I do not understand how these polynomials are equal to each other at all. For example when I calculate the first one, I find 1*(x-2.75)*(x-4)
Thunfische
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Lagrange interpolation polynomial, approximation

I'm sorry for my bad english, I translated it from a French exercice. Let $f$ be a function of class $C^3$ defined on $[a,b]$, and $c=(a+b)/2$. We want to approximate the quantity $d(f) = f''(c)$ by an expression of the following form : $\delta(f) =…
Mélanie De la Cheminée
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