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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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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Computing the coefficients of the Lagrangian interpolator

The theoretical formula for the Lagrangian interpolation over $n$ given points is well-known, but is not efficient. An improvement is given by the Neville scheme that works by interpolating on subsets ...
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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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Why is my Lagrange polynomial not working?

(Hi! I am a G11 IB Math AA SL student and I am really struggling to understand what I am doing wrong. First time using this site, so please forgive me if I mess up ). My set of data is: {{3.8, 1.69},{...
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Why doesn't Lagrange Interpolation work well in this case? [closed]

So I was asked to use Lagrange interpolation to find the an approximation for the population at a given year. ...
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error estimate Taylor remainder

For f(x)=e^(-2x).9.determine the 3rd taylor polynomial for f about x=0. use the remainder r3 to provide a bound on the error you might make approximating f by t3 on the interval [-1,1]. I found the T3(...
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Given any sequence of numbers of length N, Can it be represented by an explicit formula?

please forgive me if this question is not a good one, I am just a high school student (note: this isn't homework). I was wondering if every sequence of integers could be represented by an explicit ...
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Bound on $(x-x_0)(x-x_1) \dots (x-x_n)$ for equidistant points [duplicate]

I have to estimate the error bound of a lagrange polynomial and to do that i have to show that $|(x-x_1)(x-x_2) \dots (x-x_n)| \leq \frac{1}{4}(n+1)!h^{n+1}$ for equidistant points where $x_{j+1}-x_j =...
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Proving a formula for $\sum_{j=1}^n\frac{x_j^k}{f'(x_j)}$ for $f$ an $n$-th degree polynomial with $n$ distinct real roots $x_j$

If polynomial $f(x)=a_0 x^n+a_1 x^{n-1}+\cdots+a_{n-1} x+a_n$ has $n$ different real roots $x_1,x_2,\cdots,x_n$, prove the following fomular: $$ \sum_{j=1}^n \frac{x_j^k}{f^{\prime}(x_j)}=\left\{\...
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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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I came across this book in Numerical Methods. I need the title. Only found scanned pages of the book. But I don't know its title

I came across this book in Numerical Methods. I need the title. Only found scanned pages of the book. But I don't know its title. Attached are the scanned images. For more images look here: $1$,$2$,$...
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Lagrange polynomial interpolation - question about computation from 4 points

I have these sample data points (0,1), (1,4), (2,11), (3,22), and I have to compute the corresponding lagrange polynomial and express it in the form $p(x)= a_nx^n + a_{n-1}x^{n-1}...+a_1x + a_0$ My ...
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Lagrange Interpolation Basic Proof

I have been given this question, and I am unsure what the intuition behind answering it would be, nor where to start: Let $x_1 < x_2 < \dots < x_n$. Show that, for some given function $f$, ...
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Lagrange polynomial problem - proof/show that..

My problem: Let $x_0, x_1, . . . , x_n ∈ R$ be a set of pairwise distinct sample locations and let $x → L_i(x)$ be the corresponding Lagrange basis polynomials. Show that: $ \sum_{1 = 0}^{n}L_i(x) ≡ ...
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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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What is the theory basis for “Centering improves the numerical properties of both the polynomial and the fitting algorithm.”?

I have a set of $N$ data, $(x_i,y_i)$,which comes from sampling on $sin$ function. Assuming the sampling period is $T$, then $x_i = iT$. If interpolating them without centering, namely $x_i = T, 2T, ...
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Divided differences

I have a problem with the following identity, maybe you can help me to understand it. Given a set of $(n+1)$ data points $(x_i,y_i)$ where no two $x_i$ are the same. And $0 \leq j \leq j+k \leq n$ ...
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Lagrange vs Newton Interpolation Error Term

I have reached the same error term for both Lagrange and Newton's method. Since the interpolation polynomial is unique can I claim that none of the methods is better than the other in terms of ...
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is the series $L_k(t)$ in $T_n$ ($\mathbb{C}$-vector space)

Suppose $$T_n=\{p(t)=\sum_{k=0}^{n}a_k\cos(kt)+\sum_{k=1}^{n}b_k\sin(kt)\mid a_k,b_k\in\mathbb{C}\}$$ $$ L_k(t)=\prod_{\substack{i=0\\i\ne k}}^{2n}\frac{\sin \frac{t-t_i}{2}}{\sin \frac{t_k-t_i}{2}} \...
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Does Lagrange interpolation yield a unique polynomial over finite fields?

I understand that part of this question has been addressed before (here), but I am confused about the uniqueness of a Lagrange interpolant. Given any field $F$, I am trying to prove the existence and ...
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Let $L_0(x), L_1(x),L_2(x), L_3(x)$ be the Lagrange's basis polynomials

Let $L_0(x), L_1(x),L_2(x), L_3(x)$ be the Lagrange's basis polynomials with $4$ interpolating nodes. Suppose \begin{align} L_0(x)&=l_{00}+l_{01}x+\frac{5}{36}x^2-\frac{1}{18}x^3 \\ L_1(x)&=...
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Solve polynomial interpolation $f(x)=\frac{bc(x - b)(x-c)}{(a - b)(a-c)} + \frac{ac(x - a)(x - c)}{(b - a)(b - c)} + \frac{ab(x-a)(x-b)}{(c-a)(c-b)}$

I think the polynomial $$f(x)=\frac{bc(x - b)(x-c)}{(a - b)(a-c)} + \frac{ac(x - a)(x - c)}{(b - a)(b - c)} + \frac{ab(x-a)(x-b)}{(c-a)(c-b)}$$ may be easy to find, but I am a bit unexperienced. So, I ...
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Simplifying $\frac1{a(a-b)(a-c)}+\frac1{b(b-c)(b-a)}+\frac1{c(c-a)(c-b)}$ without opening brackets using interpolation theorems

I have such expression: $$\frac{1}{a(a - b)(a-c)} + \frac{1}{b(b-c)(b - a)} + \frac{1}{c(c - a)(c-b)}$$ I need to use interpolation in order to simplify it and I basically have no idea how.
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$f(x) = (x-1) (x-2) (x-3) (x-4) $ and given 4 points to restore polynomial $g$, find remainder after dividing $g$ over $f$

We have information about values of $g$ in four points: $g(1) = 4$, $g(2)=1$, $g(3)=2$, $g(4)=13$ I used Lagrange’s interpolation formula and got polynomial $g(x) = x^3 - 4x^2 + 2x + 5$ And I have no ...
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About Newton-Cotes numerical integration for degree >8.

I'm studying Newton-Cotes integration for nine points of interpolation, let's say $$(a_i)_{0 \leq i \leq 8}=\{0,\frac{1}{8}, \frac{2}{8}, \frac{3}{8}, \frac{4}{8}, \frac{5}{8}, \frac{6}{8}, \frac{7}{8}...
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Lagrange Interpolation Polynomials and functions of Matrices

Say we are given a matrix $A$ over the field $\mathbb{C}$ with minimal polynomial $\mu_A(\lambda)=\lambda^2(\lambda-1)^2$ so we know $A \in \mathbb{C}^{m \times m}$ where $m \in \mathbb{N}_{\geq 4}$. ...
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Bound Lagrange interpolating function constructed with nodal values on Gauss points to a minimum and a maximum value

I need to keep a Lagrange interpolating function constructed on $P+1$ Gauss quadrature nodes to a minimum and a maximum values by tweaking nodal values of the exact function at Gauss nodes. The ...
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Divided Differences Proof - Show the following formula is True

I am struggling with this problem on my numerical analysis homework. We're talking about polynomial interpolation and we just learned about divided differences like two days ago. "Show that the ...
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Problem require Lagrange interpolation

Given $P(x) = ax^3 + bx^2 + cx + d$ where $ \vert P(x) \vert \le 1\; \forall \vert x \vert \le 1$ Prove: $$ \vert a \vert + \vert b \vert + \vert c \vert + \vert d \vert \le 7$$ Hopes you guys can ...
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Find points to uniquely determine the product of two polynomials defined with points, in linear time

There are two polynomials $A$ and $B$, such that $\deg(A) = m$ and $\deg(B) = n$, and the coefficients are not given. However, there are $m+1$ known points on $A$ $((x_0,y_0),(x_1,y_1),\ldots,(x_m,y_m)...
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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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Lagrange polynomial on $E_n$, projection on $E_{n-1}$

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 $\...
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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 ...
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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 ...
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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 ...
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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 ?
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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 $...
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When is the Lagrange interpolation polynomial exact?

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 ...
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Lagrange polynomial extrapolation errors

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 \...

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