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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Finding polynomial value at a point given values at $n+1$ other points
Let $F_n$ be the $n$-th Fibonacci number ($F_1$ = $F_2$ = $1$ and $F_{n+1}$ = $F_n$ + $F_{n-1}$ for $n \geq 2$). Let $P_n(x)$ be a polynomial of degree $n$ such that $P_n(k)$ = $F_k$ for $k = n+2, n+3,...
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How to apply Lagrange interpolation to mod P polynomials?
I understand how to apply Lagrange interpolation to polynomials whose coefficients are rational numbers, but I always confuse myself when it comes to mod P polynomials. For example, say we have the ...
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Compute function values using Lagrange polynomials for ODEs
Given the Lagrange Basis Functions:
And an ODE of the form:
The lagrange interpolation states that:
Moreover:
I read that the last expression allows to build a algebraic equation system of N ...
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When can Newton's divided differences be $0$?
I want to try and find the possible degrees of the interpolating polynomial of the nodes $(x_j,j), \quad j=0,1,\dots,n$ and $x_j \in \mathbb{N}$. I have a feeling the degree could only be $1$ or $n-1$,...
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Interpolate a countably infinite number of points within a bounded region?
Suppose I have $x\in \mathbb{Q}\cap [0,1]$, where $\mathbb{Q}$ is the set of rational numbers, and let $|x|$ be the length of $x$ when expressed in binary. I wish to find an analytic function of $x$ ...
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Probability of selecting $k$ points such that the resulting polynomial $H$ satisfies $H(0)=0$
I read that $k$ points on some finite field $\mathbb{F}$ completely determine a $k-1$ polynomial $H$ in $\mathbb{F}[x]$, and the polynomial can be computed using Lagrange interpolation.
Suppose that I ...
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Piecewise quadratic interpolation in a symmetric manner
I am trying to derive a reasonable symmetric interpolant for quadratic $C^0$ interpolation. For odd degrees I have no trouble since things are symmetric. For example for a piecewise cubic interpolant ...
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Is $\dot u_h(t_0 + t h) = \sum_{j = 1 \dots s} \dot u_h(t_0 + c_j h) L_j(t)$ a true interpolation formula?
I'm studying collocation mathods to solve ODEs and I got stuck on the proof of the Guillou-Soule theorem which proves that a collocation method is a particular kind of implicit RK method under some ...
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Formula for interpolating polynomial
How can I show that $$P\left(x_{0}+h \theta\right)=\frac{(-1)^{n}}{n !} \prod_{j=0}^{n}(\theta-j) \sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l}n \\ k\end{array}\right) \frac{f\left(x_{k}\right)}{\theta-...
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How to eliminate the denominator in the calculation of the Lagrange interpolation on a finite field? [duplicate]
I am studying Secret Sharing by looking at the lecture notes here:
https://people.eecs.berkeley.edu/~daw/teaching/cs70-s08/notes/n10.pdf
But I could not understand why the following congruence ...
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Interpolation problem (recursive interpolation)
Let $x_0<x_1<...<x_K$ be points on the real line. Let $P$ be the polynomial of degree $K$ such that $P(x_i)=(-1)^i$ for all $i$. Then there exists points $y_0<y_1<...<y_K$ such that ...
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Compare error bound to theoretical error bound
$P_3(x) = 3x^3 +3$ is an interpolating Lagrange polynomial for $\widetilde{P(x)} = x^4-2x^3-x^2+2x$ generated from the data points $$(-1, 0), (0, 3), (1, 6), (2, 27)$$
$\widetilde{P(x)}$ is itself a ...
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Lagrange interpolation at 0's
I tried to find an answer for some time, the answer is simple probably ,say I have a function given by set of points and their values like ${(4,1),(5,0),(6,0)}$,
how do I calculate Lagrange ...
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Show that the sum of Lagrange Polynomials $\sum_{i=0}^{n} L_{i}(t)=1 \quad \forall t \in R$ [duplicate]
I am reviewing a homework problem that is supposed to be really easy but I have trouble wrapping my head around it.
For $j=0, \ldots, n \quad t_{j} \neq t_{i}$ if $ i \neq j $
we define the $n$ ...
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value of $n$ satisfying a polynomial equation
Let $P(x)$ be a polynomial of degree $3n$ so that $P(3i)=2, P(3j-2) = 1, P(3j-1) = 0$ for $0\leq i, j\leq n, j\ge 1$ and $P(3n+1)=730$. Determine the value of $n$ and prove that it is unique.
One ...
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Why does inverse quadratic interpolation converge quicker than inverse linear interpolation
I have used linear inverse interpolation and quadratic inverse interpolation to estimate the root of a function. I found that my linear interpolation procedure required 9 iterations to achieve ...
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orthogonality of Lagrange basis with legend nodes
I came across the following statement and I don't know how to justify it.
If $L_i$ is a Lagrange basis, and $x$ is a zero of Legendre polynomial, then
$$ \int_{-1}^{1} L_i(x)L_j(x) dx = \delta_{ij}w_j$...
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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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how big should n be so that the error be less than a certain number - Lagrange Interpolation
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 $...
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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?
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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 ...
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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:
...
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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 ...
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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 &= \...
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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 $a_1, \dots, a_k \in \mathbb{R}$ ...
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$\int_a^{a + n \cdot h} (x-a)(x-(a+h))(x-(a+2h))\cdots(x-(a+n\cdot h))\, \text{d}x = ?$
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 ...
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Is there a name for a set of linear functionals $f_i$ 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 ...
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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 ...
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Expressing linear interpolation as sum
So I have the discontinuous function $f = [3 \; 4 \; 7 \; 4 \; 3 \; 5 \; 6]$ for $x \in [1,7]$. Finding the linear interpolation can easily be done by just finding the linear polynomials that cross ...
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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 $...
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What is wrong with my derivation of Simpson's Rule via Lagrange interpolation? + Is my alternative a better approximation?
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 ...
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Multiple Lagrange Interpolation [closed]
I have a data table looking similar to the one below.
There I have for given x- and y-values respective z-values ($z(x,y)$).
How can I interpolate the matrix that I find a function that describes $f(...
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How to estimate the next step in a time series for a given function?
In my application, I ask users to perform a sequence of loopback latency measurements from an output (ex: cellphone speaker) to an input (ex: built-in microphone).
Below, there are some usual measured ...
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Proof of Lagrange's interpolation formula
Given $n+1$ points $(x_i,f_i)$, $i=0,1,2,\ldots,n,$ with $x_i\neq x_j$ for $i\neq j$, define $L_k(x)$ to be
$$\frac{(x-x_0)(x-x_1)\cdots(x-x_{k-1})(x-x_{k+1})\cdots(x-x_n)}{(x_k-x_0)(x_k-x_1)\cdots(...
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Are multivariate polynomials determined by their values on a lattice?
In one variable, a polynomial (of any degree) is determined by its values on a finite set of points. More specifically if $p$ is a polynomial of degree $k$, and $x_0 , \dots x_{k}$ are points for ...
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Basis function finite element method
In the finite element method we consider a grid with nodes $x_i$, $i=1,...,n$ and define basis functions $\psi_i$ such that
$$\psi_i(x_i)=1$$
$$\psi_i(x_j)=0 \text{ for } j\neq i$$
and $\psi_i$ is ...
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What could I use to show that triangular numbers should grow quadratically?
I'm learning to work with polynomials, specifically right now I'm working with progressions like triangular/tetrahedral/pyramid numbers etc.
I know there are different ways to show the nth triangular ...
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How to transform computationally efficient Lagrange Interpolation method, from integer arithmetic, to finite-field Arithmetic?
Im implementing SSSS in a programming language and I have a polynomial. I have to find f(0) from a minimum number of points.
From my understanding, I can use ...
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Polynomial interpolation of a polynomial with degree at most $n$ with $n$ data points is same as polynomial [duplicate]
I'm trying to solve following question from D Kincaid & W Cheney, Numerical Analysis section 6.2 and need a proof check
Prove that if $p$ is a polynomial of degree at most $n$ then
$$ p(x) = \...
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Interpolation with function values and derivatives (at different points)
My question is similar to the one found here. There is an answer which does seem to provide what I am looking for. However either I am missing something or such answer is not complete, and since that ...
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Exponential Interpolation Inspired from Lagrange Interpolation?
I am currently working on expanding formulas from the closed Newton-Cotes Formulas. I wanted to try interpolating functions using exponentials, and I am trying to do the same thing for different ...
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Intuition for Linear Lagrange Interpolating Polynomial
I'm trying to understand how the formula for Lagrange Interpolating Polynomials comes about by looking at the basic case of Linear Lagrange Interpolating Polynomials. I found this derivation but it ...
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Does Lagrange interpolation work for complex numbers?
I am attempting a question that regarding Polynomial and remainder theorem with quadratic divisors ( $x^2 + 1 $) and ($x^2 + 2$) specifically. The resulting remainder is supposed to be a degree 3 ...
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Finding Lagrange Polynomials from LU-Decomposition
The linear system $V p = f$ can be solved to find coefficients of the polynomial $P(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 $ that interpolates the points $(-1,f_1), (2,f_2), (1,f_3), (0,f_4)$, where $V$ ...
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Proving Uniform Convergence of a Polynomial with bounded degree with no specific domain
I have the question:
Let $(p_n)$ be a sequence of polynomials with $\text{deg}(p_n)\leq N \in\mathbb{N}$ $\forall n$, and suppose $p_n\to f$ pointwise. Show that $(p_n)$ converges uniformly to $f$ ...
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Strange Analysis Polynomial Question (Lagrange Interpolation?)
We have $0\leq x_0\lt x_1\lt\,...\,\lt x_N\leq1$, where all $N+1$ distinct points are in $[0,1]$ and I must show:
If $p$ is a polynomial with degree $\leq N$, and $$Q_k(t)=\prod_{\substack{0\leq i\leq ...
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What's the difference between Lagrange interpolation and piecewise (Lagrange) interpolation?
For my numerical maths course I have to compare two types of interpolation:
Lagrange interpolation and piecewise (Lagrange) interpolation.
I've created codes in matlab for both and now I'm trying to ...
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Asking abount Finite element interpolation
At first, sorry if my question is a stupid thing. It is about mapping a function into semi-discrete space. Let's see, for the basis function $\{\phi_i(x)\}_{i = 1}^n $, the projection of function $u(x,...
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