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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Integrating lagrange polynomial with equispaced points
Suppose we have some second order polynomial interpolant, $P_2$, defined on the equispaced points $x_0, x_1, x_2$, such that $x_{j+1}-x_j=h$. From $P_2$, we have Lagrange polynomials, $L_0, L_1, L_2$.
...
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1answer
25 views
Maximum interpolation error in lagrange interpolation.
I have the following question:
And the following Lagrange interpolation error bound:
The way I have started to solve the problem is as follow. For me as a worst case is when all infinitely close to ...
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22 views
Interpolation On A Curvilinear Grid
How would someone interpolate/extrapolate data on a grid such as the one shown? Where can I study about such grids and methods of interpolation?
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14 views
Domain of an interpolated polynomial function
I was just wondering what defined the domain of a Lagrange interpolating polynomial. Is it the domain of the function being interpolated? Always the reals? I am asking this as I have constructed a ...
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29 views
Approximating a function as close as we wish by interpolating polynomials
So yesterday in lecture one of my professors gave a proof that if $f:[a,b]\to\mathbb{R}$ with $f(0)=0,$ then for all $\epsilon>0$ there exists a polynomial $p$ such that $p(0)=0$ and $\|f-p\|_{[a,b]...
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20 views
Finding an interval over which a polynomial interpolant $P_n$ converges to a function $f$ as the degree $n$ increases
I understand that due to Runge's phenomenon, increasing the degree, $n$, of a polynomial interpolant can actually increase the error between the interpolant, $P_n$, and the function, $f$, you are ...
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2answers
45 views
In linear interpolation, what exactly is $\frac{x-x_i}{x_k-x_i}$ in geometric terms?
Thanks to this question: Explanation of Lagrange Interpolating Polynomial, I have an intuition for what $\frac{x-x_i}{x_k-x_i}$ is doing in polynomial interpolation. That is, it is a kind of "on and ...
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2answers
58 views
Linear approximation to xln(x)
Suppose we need to approximate $f(8.4)$ where $f(x) = \mathbb{xln(x)}$ by using a linear polynomial . We have the following points as nodes : $x_0=8.1 , x_1 = 8.3 , x_2 = 8.6 , x_3 = 8.7$ .
I ...
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36 views
Prove $ | u(x) - u_h(x) | \leq h \ \underset{0 \leq y \leq 1}{\max} |u''(y)| $
Prove:
$ | u(x) - u_h(x) | \leq h \ \underset{0 \leq y \leq 1}{\max} |u''(y)| $
for $ 0 \leq x \leq 1 $
Using the fact that
$|| (u-u_h)' || \leq \underset{0 \leq y \leq 1}{\max} |u''(y)| $ and ...
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11 views
Deriving quadratic interpolating function
I want to derive the piecewise interpolating function on the interval $[x_i, x_{i+2}]$, $i\in \{0,\dots, n\}$:
$$p_2(x) = \frac{(x-x_{i+1})(x-x_{i+2})}{(x_i-x_{i+1})(x_i-x_{i+2})}f(x_i) + \frac{(x-x_{...
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1answer
59 views
Lagrange polynomial $x^n$ coefficient
How can we show using Lagrange interpolation polynomial that
$$\sum_{i=0}^n y_i \prod_{j=0, j\ne i}^n \frac{1}{x_i-x_j}$$
is the coefficient of $x^{n}$?
I know that $f[x_0,x_1, .. x_n]$ is the ...
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0answers
45 views
Lagrange Polynomials with Derivatives (lowest order polynomial)
I need to find the lowest order polynomial, $P(x)$, that satisfies the following conditions using Lagrange polynomials:
$P^{'}(x_{0}) = f_{0}^{'}$
$P^{'}(x_{1}) = f_{1}^{'}$
$P^{'}(x_{2}) = f_{2}^{...
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37 views
Interpolation on Chebyshev point with octave
I have to solve this numeric problem on octave:
(A) Check the correctness of the Lagrange (or Newton) interpolation method on some functions, of which the analytical formula is known, considering ...
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1answer
28 views
On highest degree of precision of numerical integration scheme that comes from interpolating polynomial
Let $x_1,...,x_n$ be distinct points in $[a,b]$ and $l_i(x):=\prod_{k\ne i}\dfrac {x-x_k}{x_i-x_k} $. Let $w_i=\int_a^b l_i(x)dx$. For every $f \in C[a,b]$, let $I_n(f):=\sum_{i=1}^n w_i f(x_i)$.
...
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59 views
What is the basic difference between interpolation & inference?
In mathematics we've studied interpolation as predicting the structure of a function from it's given finitely many values from which consequently we use to construct certain function which nearly ...
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16 views
Lagrange Polynomial Interpolation in O(n) Time
Is it possible to evaluate a Lagrange polynomial interpolation in O(n) time after using a preprocessing step with O(n^2) operations? Thank you.
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45 views
Existence of a complex polynomial with given zeros and critical points?
Suppose we are given two families of complex numbers (need not to be distinct) $z_1, z_2, \cdots, z_n$ and $w_1, w_2, \cdots, w_{n-1}$ such that
the second family lies in the convex hull of first ...
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2answers
52 views
Finding Expression for terms in Lagrange Polynomial Interpolation
So for a paper I am writing I am using Lagrange polynomial interpolation:
$$P(x)=\sum_{i=0}^{N}f(x_i) \cdot \prod_{j=0;j\neq i}^{N}\frac{x-x_j}{x_i-x_j}$$
And I need to find an expression that ...
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34 views
Polynomial Interpolation using large set of data.
I have a large set of {$x_i$}$_{i=1}^n$ and corresponding values of the function {$f(x_i)$}$_{i=1}^n$. My aim is to estimate the function $f(x)$. Therefore, I think that appropriate technique is ...
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31 views
Choice of the family of the Basis Functions
As I have learned for now, there are several families of Polynomial-type Basis functions (Lagrange, Serendipity, Hermite, ...).
My question is beside the order of the elements which affects the order ...
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1answer
25 views
Interpolation- Lagrange polynomial
Let $x_0,x_1,...,x_n$ will be different real numbers.
Show, that: $f[x_0,x_1,...,x_n]=\sum_{i=0}^m\frac{f(x_i)}{\Phi '(x)}$ where $\Phi (x)=(x-x_0)(x-x_1)...(x-x_m)$
So, I have some problems.How to ...
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2answers
38 views
Approximation using Lagrange Interpolation
I am aware of the formula of this method. However, is it true that the method produces more accurate polynomial when the $x$ points are closer to each other? if so or not, why?
Moreover, If I am ...
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23 views
How to calculate interpolation error with repeated nodes
Given $f(x)=\cos(\pi x)$
a) Calculate the interpolation polynomial such that $p(-1)=f(-1)$, $p'(-1)=f'(-1)$, $p(0)=f(0)$, $p(1)=f(1)$ and $p'(1)=f'(1)$.
b) Show that $|f(x)-p(x)|\leq\frac{\...
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1answer
60 views
Lagrange Interpolation how does it work?
I am making a program to find a polynomial given a set of data. I understand the summation of the formula. Given the image below can someone explain what the square looking symbol is next to the Li(x)?...
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2answers
55 views
Existence of a interpolating polynomial of degree less than or equal to 3, given only 3 support points and 1 derivative value
I'm trying to show the existence and uniqueness of an interpolating polynomial $p$ of degree less than or equal to 3 that interpolates a differentiable function $f$ ( i.e. $f(x_i) = p(x_i)$ for $i = 0,...
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31 views
Lagrange Interpolation Coefficients/ Frobenius Covariant
I am having problems trying to show that
$$ \sum_{k=1}^{n} L_k(A)=I$$
where A is a $nxn$ matrix,$I$ is the $nxn$ identity, and $L_k$ is the Lagrange interpolation coefficient defined as: $L_k(A)=\...
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19 views
$N$ random points make a random polynomial?
Let's say I have a finite field $A$, and I choose $N$ (distinct) points uniformly at random from $A \times A$. Then using Lagrange interpolation I can find a polynomial $f(x)$ of degree $N-1$ that ...
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42 views
Interpolation error with Legendre/Chebyshev polynomials
I remember seeing somewhere that the Lagrange interpolation over Chebyshev nodes has least possible deviation in the sense of $\|\cdot\|_\infty$-norm, while Legendre nodes are optimal in the sense of $...
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1answer
39 views
Will Lagrange interpolation formula give unique polynomial for modulo composite integer?
MWE:
Let us consider a polynomial :
\begin{equation}
f(x)=a_0+a_1x \tag{1}
\end{equation}
where integer coefficients $a_0,a_1\in [0,2^r-1], r\in \mathbb{N}$.
Let $y_1=f(1)\pmod{2^r}$ and $y_2=f(2)\...
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1answer
21 views
Interpolation error for x*atan(x)
I'm trying to interpolate x*atan(x) on [-5, 5] using (n+1) equidistant nodes. The oscillations on the ends seem to be caused by the same reasons Runge phenomenon is. The question is why exactly are ...
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24 views
Integral error $\int_0^3|f(t)-p_n(t)|dt$ when $p_n(t)$ is the interpolating polynomial for $f(t)=e^{-t}cos(4\pi t)$
I have to find the integral error $\int_0^3|f(t)-p_n(t)|dt$ when $p_n(t)$ is the interpolating polynomial for $f(t)=e^{-t}cos(4\pi t)$ when $t\in[0,3]$
I am quite stuck here, however, let me explain ...
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1answer
29 views
Lagrange interpolation accuracy
I computed cot(0.0015) as accurately as possible using the following table by a) interpolating for cot(x), b) by interpolating for cos(x) and sin(x) and using them to calculate cot(x).
So why using ...
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19 views
Are lagrange polynomials equal to B-spline polynomials?
I am having a discussion with a friend about whether or not the set of B-splines contains the set of lagrange interpolants or not.
I think it is not the case because the basis for B-splines are ...
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2answers
50 views
Finite Taylor series of Lagrange Basis functions
Lagrange basis functions are defined as follows $$L_{j}(x) = \prod_{i\neq j} \frac{x-x_{i}}{x_{j}-x_{i}} $$
then $$\ln\Big(L_{j}(x)\Big) = \ln\Big(\prod_{i\neq j} \frac{x-x_{i}}{x_{j}-x_{i}} \Big) = ...
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28 views
Sum of Lagrange polynomials: $\sum_{i=0}^{n}L_i(0)x_i^{n+1} = (-1)^{n}x_0\cdot\cdots\cdot x_n $.
Given $\{x_0,...,x_n\}$ I am asked to show that $\sum_{i=0}^{n}L_i(0)x_i^{n+1} = (-1)^{n}x_0\cdot\cdots\cdot x_n $.
I already showed that $\sum_{i=0}^{n}L_i(x)x_i^{j} = x^j$ for $j=1,...,n$ and that ...
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1answer
99 views
Can the Lagrange Interpolating polynomial be used in a machine learning algorithm?
My understanding of the Lagrange interpolating polynomial is that given $\space n \space$ points, we can fit a polynomial of degree $\space n+1 \space$ as a means of approximating the values between ...
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41 views
Why can’t we cancel $l(x)$ from barycentric interpolation?
Recently, I read up on the barycentric interpolation which was apparently much better than Lagrange interpolation due to the ease of adding new data points.
There is
$$w_j = \frac{1}{\prod_{k={0..n},{...
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2answers
815 views
What is difference between newton interpolation and lagrange interpolation?
I know that they both represent same polynomial and their formulas, but what is difference between them. If they are not different, then why do we study them separately?
Don't need detailed proof ...
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0answers
51 views
newton's interpolation error for non-differentiable function
I was given this function:
$$
f(x) = \begin{cases}
x^3 & {\text{if}}\ x>0 \\
0 & {\text{if}}\ x\leq0\
\end{cases}
$$
and I was asked to give an upper bound on it's interpolation error ...
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1answer
67 views
Deriving Adams-Moulton 2 step method with variable step size
I am trying to derive the Adams Mouton 2 step method for
$$y'(t)= f(t,y(t))$$
by interpolating $f$ at the nodes $(x,f_j)$, $(x+h,f_{j+1})$ and $(x+h+\vartheta h,f_{j+2})$. I get the following lagrange ...
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27 views
Abel Ruffini theorem for Lagrangian interpolation?
I've recently tried to recreate a Lagrangian interpolation for a recursive polynomial function of the form $f_n(x)=g(x,f_{n-1}(x))$, where $n$ is the degree of the equation. It interpolated exactly ...
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58 views
Choose support for polynomial interpolation
I have the following table, where $t$ is defined in minutes and $v(t)$ in $m / s$:
\begin{array}{|c|c|c|c|}
\hline
t& 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\ \hline
v(t) & ...
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1answer
46 views
How do you find polynomial function of the image below using Lagrange interpolation?
Sorry for my bad english, the problem have so many missing parts that I dont even know how to start solving it (https://i.stack.imgur.com/hAdoY.jpg)
2
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1answer
144 views
Lagrange basis function is for the set of all polynomials of a given degree?
Wikipedia says that
The set $\{(t−1)(t−2)/2, −t(t−2), t(t−1)/2\}$ is another basis for
quadratic polynomials, called the Lagrange basis.
I do not understand the concept of Lagrange basis ...
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71 views
Is it possible to avoid the Runge phenomenon by other norm instead of $L_{\infty}$ norm?
Most of error analysis of polynomial interpolation employs $L_{\infty}$ norm. However, $||\prod\limits_{i=1}^{n}(\cdot-x_i)||_{L^{\infty}(a,b)}\leq\frac{n!h^{n+1}}{4} $ when equidistant points are ...
3
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0answers
91 views
polynomial interpolation error; is it continuous?
Let $f$ be a $n+1$-times continuously differentiable function on the interval $[a,b]$, and let $p_n$ be the $n$th degree Lagrange interpolation polynomial, defined by
$$
p_n(x)=\sum_{i=0}^nL_i(x_i)f(...
1
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1answer
80 views
Lagrange interpolation with multiplicities
I was wondering if it were possible to do Lagrange interpolation with multiplicities. Lagrange interpolation gives us a polynomial that obtains certain values on a given set of points. More precisely, ...
0
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1answer
93 views
Polynomial interpolation of Bessel function
I used polynomial interpolation of degree 5 to approximate one 0 of the Bessel function $J_0$. I used the following data:
With the data above, I computed a polynomial using the images of the Bessel ...
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0answers
168 views
Lagrangian interpolation at Chebyshev points - estimate on coefficients in monomic basis
First, let us fix some Notation:
Let $n\in\mathbb{N}$ and $x_i=\cos(\tfrac{(i+1/2)\pi}{(n+1)})$, $i=0,\dots,n$, be the Chebyshev points. Let
\begin{align}L_i(x)={\displaystyle\prod_{\substack{0\leq j\...
2
votes
1answer
225 views
Show $ \sum_{i=0}^n \lambda_i(0)x_i^{j} = 0 $ Lagrange-interpolation-coefficients.
Consider the Lagrange-interpolation-coefficients $\lambda_i$ ( $0 \leq i \leq n$) with nodes $x_0, x_1,..., x_n$.
Show: $$ \sum_{i=0}^n \lambda_i(x) = 1 $$ and
$$ \sum_{i=0}^n \lambda_i(0)x_i^{j} = ...
