# 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.

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### Approximating a Value by Using Lagrange Interpolation

I am getting a mathematics course and there is a homework question to solve. If $e^{1.3}$ is approximated by Lagrangian interpolation from the values for $e^0 = 1$, $e^1 = 2.7183$ , and $e^2=7.3891$,...
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### How to get the general term for a quartic sequence really need help [closed]

Is it possible to find the general term for a quartic sequence and if so how do you do it? The sequence I am using is 1,9,36,100,225,441, 784, 1296, 2025, 3025 I am only interested in finding the ...
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### the maximum value of a fundamental complex function

Consider $n$ distinct points $z_1,z_2,\cdots,z_n$ in the unit closed disc $\mathbb{D} = \{z \in \mathbb{C}: |z| \leq 1\}$, i.e., $|z_i| \leq 1$ for any $i = 1,2,\cdots,n$. Define the complex function \...
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I am having trouble understanding how to solve this question: if $f(x) = e^x$, then how big must $n$ be, so that $\vert p_n(x)-e^x \vert \leq 10^{-6}$ for all $x \in [-1,1]$ The interpolation points $... 0 votes 1 answer 28 views ### What is the difference between linear lagrange interpolation and defining secant lines? From what I am noticing, linear Lagrange interpolation just looks like another way to find a secant line given two points. What is the difference? 3 votes 2 answers 121 views ### Prove that there exists 3 numbers$a,b,c$so that$P(a)=b, P(b)=c, P(c)=a$The problem Given that$P(x)=x^3-3x$. Prove that there exists$a \neq b \neq c$such that$P(a)=b, P(b)=c, P(c)=a$My ideas So the conditions here makes it quite clear that I intend to use the ... 3 votes 1 answer 74 views ### FUN with f̶̶l̶̶a̶̶g̶̶s̶ Newton Cotes Quadrature formula and Bernoulli polynomials of the second kind I was told to phrase my question in a more exciting way when I asked it last time. The following is a preliminary consideration. If you don't need it, just scroll down to START HERE. Here we go then: ... 0 votes 1 answer 98 views ### why it is not possible to use quadratic spline The following four points are known to lie on a closed curve in the (x, y)-plane: (−1,0), (−1/2,3), (1/2,−3), (1,0) and the goal of this question is to fit a piecewise-polynomial approximation of the ... 1 vote 1 answer 33 views ### Integral of$n-th Bernoulli polynomial of the second kind We have \begin{align} \int x(x−1)(x−2)...(x−n)\,dx=(n+1)!\cdot\psi_{n+2}(x), \end{align} where,\psi_n(x)$is the$n-th Bernoulli polynomial of the second kind. We have \begin{align} I &= \... 7 votes 2 answers 183 views ### Existence of real analytic diffeomorphisms with prescribed values on a finite set My question is the following: given a finite set F = \{ x_1, \dots, x_k \} \subset \mathbb{R} $such that$x_i < x_{i+1}$for all$i = 1, \dots, k-1$and numbers$a_1, \dots, a_k \in \mathbb{R}$... 2 votes 2 answers 99 views ###$\int_a^{a + n \cdot h} (x-a)(x-(a+h))(x-(a+2h))\cdots(x-(a+n\cdot h))\, \text{d}x = ?I would like to dissolve this expression: \begin{align} f(h) = \int_a^{a + n \cdot h} (x-a)(x-(a+h))(x-(a+2h))\cdots(x-(a+nh)) \ \text{d} x \end{align} I have a guess by working out the first two ... 2 votes 1 answer 26 views ### Is there a name for a set of linear functionalsf_i$that is "sufficiently rich" to uniquely identify a polynomial from the values$f_i p$I have found this statement in some old lecture notes on interpolation in my lab. Let$\mathcal{P}_{n}(I)$be the vector space of polynomials over some open interval$I\subset\mathbb{R}$. Suppose some ... 1 vote 1 answer 152 views ### Find$P(-\frac{1}{2})$given a infinite points of a polynomial The problem Given$P(x)a polynomial with real coefficients and satisfies \begin{align} P(n)= 1^{2010} +2^{2010}+ \cdots + n^{2010}, \forall n \in \mathbb{N*} \end{align}, then calculate the value of ... 0 votes 0 answers 23 views ### Expressing linear interpolation as sum So I have the discontinuous functionf = [3 \; 4 \; 7 \; 4 \; 3 \; 5 \; 6]$for$x \in [1,7]$. Finding the linear interpolation can easily be done by just finding the linear polynomials that cross ... 0 votes 0 answers 64 views ### How to prove that coefficients of Hermite interpolation polynomial are differentiable? Definitions: Let's say we are given interval$I=[a,b]\subset\mathbb{R}$, function$f\in C^{m}(I)$and$n\in\mathbb{N},n\leq m$. Now for a vector$p=(p_0,p_1,...,p_n)$we define the Hermite polynomial$...
I understand that correct derivations exist on this site. I am, however, interested in why my workings are incorrect. Let $f(x)\approx L(x)=\sum_{n=0}^2\ell_n(x)\cdot f(c_i)$, where $c_i$ are three ...