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.
0
votes
1answer
32 views
Lagrange interpolation formula
Give the formula of the $1$st degree Lagrange polynomial $L(x)$
interpolating a function $f$ at the points $0$ and $1$. Give the formula for the error $L - f$. Finally, show that
$$\sup_{x \in ...
2
votes
1answer
30 views
Burden Numerical Analysis Lagrange Interpolation Question
I have been trying to solve a problem on Lagrange Interpolation from the book Numerical Analysis 10th Edition by Richard Burden. I have been stuck on the first question it for hours and cannot figure ...
1
vote
0answers
12 views
Constructing a degree-1 Lagrange interpolation polynomial
Construct the Lagrange polynomial $p_1$ of degree $1$ for a continuous
function $f$ on $[-1, 1]$ using the points $x_0 = -1$ and $x_1 = 1$.
My attempt (note: I am using the notation that is used on ...
1
vote
2answers
88 views
Lagrange interpolation formula question
Find the number of values of $x$ satisfying the relation
$$
\alpha_1^3 \left( \frac{\prod_{i=2}^n (x - \alpha_i)}{\prod_{i=2}^n (\alpha_1 - \alpha_i)} \right) + \sum_{j=2}^{n-1} \left( \left( \frac{\...
0
votes
0answers
15 views
Using Lagrange interpolation formula, deduce partial fraction
Question no 3 , kindly explain the relation btw lagrange and partial fraction](https://i.stack.imgur.com/EYVIk.jpg)
I am unable to figure out the relation btw lagrange interpolation formula and ...
0
votes
1answer
71 views
Lagrange Polynomial Interpolation, centered coefficients
My textbook states that the Lagrange Interpolant on the interval $[a,b]$, with the data points $(x_0,y_0),...,(x_n,y_n)$, written as:
$\prod_nf(x)=\sum_{i=0}^ny_i\phi_i(x)$, with $\phi_i(x)$ being the ...
0
votes
0answers
22 views
Analyzing the error of the Lagrange Interpolation of $x^2-\sin(10x)$
Given the function
$f(x)=x^2-\sin(10x)$, find degree, $N$, of the lagrange interpolation satisfying that the error $\vert f(x)-p(x)\vert < \epsilon = 10^{10}$ in the interval $I= [0,3]$. Here is my ...
0
votes
3answers
46 views
Solving $\int dx/(x^{2}+1)$ with polynomial interpolation on the complex plane and with trigonometric substitution
Using the Lagrange interpolation theory for $x_{0}=-i$ and $x_{1}=i$ we have
$\displaystyle
\int \frac{dx}{(x^{2}+1)}= \frac{-1}{2i}\int \frac{dx}{x+i} + \frac{1}{2i}\int\frac{dx}{x-i}=\frac{1}{2i}(\...
0
votes
1answer
25 views
Lagrange polynomials basis
The problem is prove that $\beta$={$f_0(x),f_1(x)...f_n(x)$} is basis for $P_n(R)$
($P_n(R)$={$a_0+a_1x+..+a_nx^n|$ $a_i\in$R} )
$$f_i(x)=\frac{(x-c_0)...(x-c_{i-1})(x-c_{i+1})...(x-c_n)}{(c_i-c_0).....
0
votes
0answers
30 views
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$.
...
0
votes
1answer
30 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 ...
0
votes
0answers
24 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?
1
vote
0answers
16 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 ...
3
votes
0answers
34 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]...
2
votes
0answers
23 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 ...
1
vote
2answers
52 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 ...
1
vote
2answers
64 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 ...
0
votes
0answers
44 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 ...
0
votes
0answers
15 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_{...
1
vote
1answer
69 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 ...
1
vote
0answers
67 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}^{...
0
votes
0answers
47 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 ...
0
votes
1answer
29 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 $I_n(f):=\sum_{i=1}^n w_i f(x_i)$.
...
0
votes
0answers
66 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 ...
1
vote
0answers
19 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.
1
vote
0answers
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 ...
1
vote
2answers
74 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 ...
0
votes
0answers
35 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 ...
1
vote
0answers
33 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 ...
0
votes
1answer
27 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 ...
2
votes
2answers
44 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 ...
1
vote
0answers
62 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{\...
0
votes
1answer
68 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)?...
1
vote
2answers
74 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,...
0
votes
0answers
35 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)=\...
1
vote
0answers
20 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 ...
0
votes
0answers
43 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 $...
0
votes
1answer
52 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)\...
0
votes
1answer
23 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 ...
0
votes
0answers
26 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 ...
0
votes
1answer
39 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 ...
0
votes
0answers
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 ...
2
votes
2answers
54 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 $$\ln\Big(L_{j}(x)\Big) = \ln\Big(\prod_{i\neq j} \frac{x-x_{i}}{x_{j}-x_{i}} \Big) = ...
0
votes
0answers
47 views
Sum of Lagrange polynomials: $\sum_{i=0}^{n}L_i(0)x_i^{n+1} = (-1)^{n}x_0\cdot\cdots\cdot x_n $.
Given $\{x_0,...,x_n\}$ I am asked to show that $\sum_{i=0}^{n}L_i(0)x_i^{n+1} = (-1)^{n}x_0\cdot\cdots\cdot x_n $.
I already showed that $\sum_{i=0}^{n}L_i(x)x_i^{j} = x^j$ for $j=1,...,n$ and that ...
0
votes
1answer
135 views
Can the Lagrange Interpolating polynomial be used in a machine learning algorithm?
My understanding of the Lagrange interpolating polynomial is that given $\space n \space$ points, we can fit a polynomial of degree $\space n+1 \space$ as a means of approximating the values between ...
0
votes
0answers
44 views
Why can’t we cancel $l(x)$ from barycentric interpolation?
Recently, I read up on the barycentric interpolation which was apparently much better than Lagrange interpolation due to the ease of adding new data points.
There is
$$w_j = \frac{1}{\prod_{k={0..n},{...
0
votes
2answers
1k 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 ...
0
votes
0answers
88 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 ...
0
votes
1answer
103 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 ...
0
votes
0answers
28 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 ...
