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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Non-uniqueness for Fourier Interpolation
I am working through a past exam and I am asked to show that under certain conditions, Fourier interpolation with basis function $e^{inx}, n=0, \ldots N-1$ is unique and to provide an example where it ...
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Is this second proof of Lagrange Polynomial known or published before?
This proof came to my mind and I think it out by myself, so I'm not certain if this proof is published by other sources before. The Lagrange Polynomial used for interpolation can be proved by ...
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Oscillations of Lagrange interpolation polynomials
Let $I = [a,b]$ be a real closed interval. Let $n$ be a positive integer and let $x_i = a+i\frac{(b-a)}{n}$ for $i=0,...,n$. Let $p_j(x)$ be the Lagrange interpolating polynomial of the $n+1$ points ...
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Understanding Remark I.12.14(c) from Amann Escher Analysis I on certain arithmetic sequences
The remark mentioned in the title reads as below. My bracketed comments are for extra context. I have also labelled each sentence so that I can refer to it in my questions.
(1) A function $f \in E^\...
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Approximation Error on Arc Length of Quadratic Bezier curve
Given a quadratic Bezier curve defined by:
$$ B(t) = (1-t)^2P_0 + 2t(1-t)P_1 + t^2P_2 $$
The arc length $ s(t) $ from $0$ to $ t $ is:
$$ s(t) = \int_0^t |B'(τ)| dτ. $$
It's known that the arc length ...
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Determinant calculation and Lagrange interpolation polynomial
I've been trying to solve the following IMO-related question.
Let $x_1,x_2,...,x_n$ be $n$ different reals, prove that $$\sum\limits_{i=1}^n\prod\limits_{\substack{j=1\\ j\neq i}}^n\frac{1-x_ix_j}{x_i-...
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Lagrange interpolation and orthogonal polynomials
Suppose that $\{p_i(x)\}_{i=0}^{n}$ are pairwise orthogonal polynomials on the interval $[a,b]$, It means,
$$
\int_{a}^{b} p_i(x)p_j(x)dx = 0\ , \;{i\neq j}
$$
wherein $p_i(x)$ for all $i$ is a ...
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Lagrange interpolation on a Galois Field in time $O(n \log (n))$
I have the values of a polynomial $p(x)$ defined on the Galois field mod $p$ (with $p$ prime) at the points zero to around two million. I need an algorithm to find the coefficients of the original ...
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Maximum degree of polynomial interpolating a two-degree polynomial with an additional condition
Let $f$ be a twice differentiable function over real line. Let $a\in \mathbb{R}$ and $h\in \mathbb{R}^{+}$. Let $P_f$ denote the interpolating polynomial of degree $2$ of $f$, i.e.,
$$f(a)=P_f(a), \...
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how to prove the following equation $\sum_{i=1}^{i=n}\prod_{j=1,j\neq{i}}^{j=n}{\frac{1}{x_{i}-x_{j}}}=0$
$\sum_{i=1}^{i=n}\prod_{j=1,j\neq{i}}^{j=n}{\frac{1}{x_{i}-x_{j}}}=0$
The product on the denominator is what I derive from the derivative of an n degree polynomial with different solutions $x_{1}$ to $...
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Where to find proof for the remainder formula of the interpolation in two variables
Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book ...
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Showing recursive Formula for Polynomial Interpolation
Let $f : [a, b] \rightarrow \mathbb{R}$. Define
$$
f[a_0, ..., a_n] := \frac{f[a_1, ..., a_n]-f[a_0, ..., a_{n-1}]}{a_n - a_0}
$$
with $f[a]=f(a)$. Consider now $(m + 1)$ distinct nodes $x_j \in \...
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Lagrange Interpolation as generalized polynom
I need to write an algorithm that constructs a function of the form
$f(x) = \sum_i^n q_i x^i$
that exactly goes through the points $p_i = (a_i, b_i)$.
In general building such a function is not the ...
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Does there exist $f \in \mathcal{C}^\infty\left([a, b]\right)$ such that $M_n \rightarrow \infty$ but $\left\|f - p_n\right\| \rightarrow 0$?
Let $f \in \mathcal{C}^\infty\left([a, b]\right)$ and $n \in \mathbb{N}$.
Let $p_n$ be the Lagrange interpolating polynomial for a partition of equispaced points $a = x_0 < x_1 < \cdots < x_n ...
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Does Lagrange interpolation at Chebyshev points solve the Runge phenomenon?
I recently came across the concept of the Runge phenomenon while studying numerical methods for special functions in the book "Numerical Methods for Special Functions" by Amparo Gil, ...
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