# 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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### Properties surrounding Polynomial Interpolation

A polynomial $P(x)$ of degree $n-1$ interpolates $n$ distinct points $(x_i, f_i)$, i.e $P(x_i) = f_i$, $i = 1,\dots, n$. Assuming that $P(x)$ is known, we wish to construct, without using the Lagrange ...
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### Lagrange polynomial interpolation maximum degree

I want to prove that no polynomial of degree $1$ that passes through $(0, \cos(0))$, $(0.6, \cos(0.6))$ and $(0.9, \cos(0.9))$. By the following theorem: Theorem 1. If $x_{0}, x_{1}, \ldots, x_{n}$ ...
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### Multivariate approximation - interpolating polynomial

How many points uniquely determine the interpolating polynomial of degree at most $d$ if there are $k$ variables?
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### MatLab: Lagrange interpolation function graph deviates when nodes are greater than 28

I don't know if this is the right place to ask this question. But I have the following code that interpolates a function $y$ base on the node $x$, which based on the ...
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### Construct an analytic function given a countably infinite set of coordinate pairs.

Suppose we have a function $f(x), x\in\mathbb Z$. Is there any way to construct an analytic function $g(x), x\in\mathbb R$, such that: $$f(x) = g(x), x\in \mathbb Z$$
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### Construction of Hermite Polynomials

I am studying Hermite Interpolation and the most common practice I came across was the use of Lagrange Polynomials. I tried to construct a Hermite Polynomial for 2 points $x_0$ and $x_1$ but instead I ...
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### Gaussian expectation of Lagrange polynomial

Let $\{x_i\}_{i=1}^N$ be the zeros of the probabilists' Hermite polynomial and $L_j(x)$ for $j = 1, \ldots, N$ the Lagrange polynomials with nodes $\{x_i\}_{i=1}^N$. Moreover, let $\rho \in [0, 1]$ ...
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### Generalized Lagrange interpolation

Find a polynomial of degree smaller than five such that: $P(1)=2$, $P(2)=-1$, $P'(2)=-1$,$P'(1)=3$,$P''(2)=1$ I know I need to use a combination of Lagrange Interpolation and Taylor series but the ...
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### Find the error of the interpolation (either Lagrange or Newton) of the following set of points

Find the polynomial that interpolates a function given by $f(x) = x(\ln(x)-1)$ on the points 2 and 3. Use values for f with 4 correct decimal places (?) and give an estimation for the error of the ...
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### Lagrange polynomials

I learned that Lagrange polynomials have a special structure . So, if we are given 3 points we can use $$P(x) = \sum_{k=0}^2 L_{k,2}(x)f(x_k).$$ My question is if I have a new point added, is there a ...
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### Lagrange Polynomials Linear Algebra [closed]

I'd appreciate some help with this problem! My professor in my graduate linear algebra class is giving us some tough homework. $$P(x) = p(x) = a_0 + a_1x + .... + a_nx^{n}$$ such that $a_i$ is real ...
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### How to find the error of Lagrange Interpolation Method without knowing the original function

If we have 3 points lets say (0,7), (2,11), (3,11). By using Lagrange Interpolation Method, we obtain $$P_n(x)= 5x^2-8x+7$$ Now the problem is, I want to find the error of my calculation without ...
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### Lagrange interpolation clarifications

Use excel to construct a quadratic polynomial $P_2(x)$interpolating three data points $(x_0,f(x_0))$,$(x_1,f(x_1))$, $(x_2,f(x_2))$. Use it to check the accuracy of $P_2(x)$ as an interpolating ...
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Let : $n \geq 1$ $E_n= \mathbb{R}^n[X]$ $\forall ~ 0 \leq j \leq n, \quad L_{n,k}(j)= \delta_{k,j}$ Scalar product $<P_1,P_2> =\sum_{k=0}^n P_1(k)P_2(k)$ Let $P \in E_{n-1}^{\perp}$ and $\... 3answers 98 views ### What does “discrete” really mean, in plain English? Can someone explain what a "discrete" function really means, in a philosophical sense, in plain English? As a guess, does discrete mean there are only points with known values, and nothing ... 0answers 60 views ### Alternative derivation for Lagrange interpolation error I am reading the book, Computational Differential Equations by Erikson, Estep, Hansbo and Johnson. My problem setting is as follows - Given the nodal points$\zeta_0, \zeta_1 \in [a, b]$, and the ... 0answers 51 views ### Modified toom-cook algorithm For these transformation matrices generated using modified toom-cook algorithm , why are they different compared to the following transformation matrices in the picture below ? We can solve the ... 1answer 20 views ### toom-cook algorithm matrix G For this toom-cook algorithm at https://arxiv.org/pdf/1803.10986v1.pdf#page=6 , how do I get the value 4/2 in the matrix G ? 0answers 72 views ### Lagrange interpolation polynomial with a matrix Assume I have a Lagrange polynomial $$p(\lambda) = \sum_{k=1}^n L_k(\lambda) = 1 \\ L_k(\lambda) = \prod_{j=1 \\ j \neq k}^n \frac{\lambda - \lambda_j}{\lambda_k - \lambda_j}$$ where I derived that$...
Find the Lagrange interpolation polynomial for data points $x_k=k$ and $f(k)=k^2$, where $k=0,1,2,3$. Also, find the Lagrange interpolation polynomial for the same data points but with $g(k)= k^4$. I ...
Lagrange polynomial interpolation error term is: $$E(x)=\frac{f^{(n+1)}(ζ)}{(n+1)!}π_{n+1}(x)$$ where $π(x)=(x−x_0)…(x−x_n)$ and $ζ∈(a,b)$. However, the theorem doesnot include the situation when \$x \...