0
votes
0answers
25 views

Interpolation point selection for Rational Polynomial Interpolation

people, 1st time on math.stackexchange so aloha to all!! The Question: I have a certain data set and I am using Thiele's rational Polynomial Interpolation to interpolate some data but the curve will ...
0
votes
1answer
28 views

Product of Chebyshev polynomials of the second kind?

So Wikipedia has this formula for a product of two Chebyshev polynomials of the second kind evaluated at a fixed $x$ with different indices: $$ U_n(x)U_m(x)=\sum_{k=o}^{n}U_{m-n+2k}(x) $$ Which would ...
0
votes
0answers
13 views

Changing the order of the elements of the divided difference Polynomial Interpolation

Apparently this is rather trivial but I don't understand why what I've highlighted in green is correct.
1
vote
3answers
100 views

Lagrange Interpolation Theorem?

The polynomials $p(x) = 5x^3 - 27x^2 + 45x - 21$ and $q(x) = x^4 - 5x^3 + 8x^2 - 5x + 3$ both interpolate the points $(1,2) , (2,1) , (3,6), (4,47)$. Even though these polynomials are of different ...
0
votes
0answers
37 views

How can I cleverly use the error term of polynomial interpolation?

Let $f(x):=x^2$. We're interested in the closed form of the error $|I(f)-T_n(f)|$ where ...
0
votes
0answers
29 views

Derivation of composite Gaussian quadrature error formula

I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following: Derive the error term for the composite Gaussian quadrature rule with ...
0
votes
1answer
39 views

Why does the interpolation error go to zero if we increase the number of sampling points?

This question is motivated by polynomial interpolation. We know that for $f\in C^{n+1}[a,b]$ and $a=x_0<\dots<x_n=b$ holds $$\| f - p_n \|_\infty \leq \frac{1}{(n+1)!} \| f^{(n+1)} \|_\infty ...
0
votes
0answers
29 views

Approximate function algorithm using a polynomial and Boor splines

I have a defined function and a set of points with equal distance between them. The problem is that I have to approximate the graphic of that function using a polynomial function of 3rd degree and a ...
0
votes
2answers
36 views

Polynomial interpolation using derivatives at some points

Given $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), (x_5, y_5)$, we can interpolate a polynomial of degree 4 using Lagrange method. But, when we are given $(x_1, y_1), (x_2, y_2), (x_3, y_3), ...
0
votes
1answer
54 views

Equivalent condition for interpolation polynomial

Let $(x_1,y_1),...,(x_n,y_n)\in \mathbb{R}^2 $, where $x_i\neq x_j$ if $i\neq j$. Let $p$ be a polynomial such that $$\det\begin{pmatrix} p(x)& 1 & x & x^2 &\dots & x^n \\ ...
2
votes
2answers
90 views

Uniform convergence of Lagrange polynomials

There is a well-known theorem that states that on a closed interval $[a,b]$ any continuous function is the limit of a uniformly convergent sequence of polynomials. Proofs for this theorem usually ...
0
votes
0answers
35 views

How to differentiate Lagrange Basis Polynomial?

How to differentiate Lagrange Basis Polynomial ? I don't know, if the term is correct, but the question is: If $x_0,...,x_n\in\mathbb R$ are pairwise distinct ...
0
votes
0answers
29 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve c(u) of degree 4 at the points -1, 0, 1 as: c(-1) = 4; c'(-1) = 4; c(0) = 6; c(1) = -4; c'(1) = -6; find the generalized ...
4
votes
1answer
145 views

Integer valued polynomial through some known points

I have 2 questions, but I'll put both of them here since they are closely related: An integer valued polynomials $P(x)$ is a polynomial whose value $P(n)\in\mathbb{N}$ for every $n\in\mathbb{N}$. ...
0
votes
1answer
27 views

Legendre polynomials verification

I'm confuse on how to answer this question: Verify that the first 4 Legendre polynomials are indeed mutually orthogonal on the interval [-1,1]
10
votes
1answer
75 views

For a fixed degree, is there always a Lagrange polynomial below the original function?

Let $x_1<x_2< \ldots <x_n$ be $n$ real numbers, and let $y_1,y_2,\ldots,y_n$ be real values to be interpolated. Let $r\leq n$. For any $I\subseteq \lbrace 1,2,\ldots,n\rbrace$ of cardinality ...
1
vote
1answer
46 views

polynomial interpolation

I have a function, for example $f(x)=\frac{-x^2}{2}+|x|$, which is divided on $[-1,0)$ and $[0,1]$. How do we interpolate this function with a polynomial $p$ in the maximum degree 4 with ...
1
vote
2answers
80 views

Polynomial Interpolation

My professor gave the following question as a practice for study guide. Any assistance in terms of helping me to solve this would be much appreciated. Suppose that $f$ is continuous and has ...
1
vote
2answers
65 views

Polynomial Interpolation and Error

I have numerical analysis final coming up in a few weeks and I'm trying to tackle a practice exam. Assuming $p(x)$ interpolates the function $f(x)$, find the polynomial $p(x)$ that satisfies the ...
1
vote
1answer
48 views

Other way to write Lagrange's form (with derivative)

Prove that we can write polynomial $L_{n}\in\Pi_{n}$ which is interpolating function $f(x)$ in $n+1$ nodes $x_{0},\,\ldots,\, x_{n}$ in following form: ...
0
votes
1answer
99 views

Degrees of interpolating polynomials

Given a collection of $m+1$ points $\{(x_0,y_0), (x_1,y_1), ..., (x_m,y_m)\}$, we can form the interpolating Lagrange polynomial $L(x)$: $$ L(x) = \sum_{i = 0}^{m} y_i l_i(x) \\ l_i(x) = \prod_{0 \le ...
1
vote
1answer
246 views

Wolfram Mathematica - Newton Backward Interpolation?

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
0
votes
1answer
54 views

Intuitive proof of interpolation polynomial existence

Problem: Given a set of $n+1$ data points ($x_i, y_i$) where no two $x_i$ are the same, one is looking for a polynomial $p$ of degree at most $n$ with the property $p(x_i) = y_i$ for all $i∈ [0, n ...
1
vote
1answer
97 views

Interpolation- Barycentric coefficients for nodes that are Chebyshev points of the second kind.

So I came across the following theorem: If the interpolation node are Chebyshev points of the second kind given by : $$ x_k=\cos \left( \frac{j\pi}{n}\right) \qquad ( 0 \leq j \leq 0) $$ Then the ...
0
votes
0answers
26 views

Polynomials of the form $g(x)=f[x,x_1,x_2,…,x_m]$.

Can anybody point me to some materials about polynomials of the form $g(x)=f[x,x_1,x_2,...,x_m]$, meaning that for given $x$ they will give back a leading coefficient of function $f$ interpolated at ...
1
vote
1answer
49 views

$p$-polynomial of $n$'th degree, $q(x)=p[x,x_1,x_2,…,x_k]$, prove that q has the same leading coefficient.

So I have a polynomial $p$ of $n$'th degree and q given by $q(x)=p[x,x_1,x_2,...,x_k]$, meaning that for $x$ it gives back the leading coefficient in interpolation of $p$ on points $x,x_1,...,x_k$. ...
1
vote
1answer
402 views

Derive the error term of Basic Corrected trapezoidal Rule for a Cubic Hermite Polynomial

The basic trapezoidal rule for approximating $I_f = \int_{a}^{b}f(x)dx$ is based on linear interpolation of $f$ at $x_0=a$ and $x_1 = b$. The Simpson rule is likewise based on quadratic polynomial ...
1
vote
2answers
99 views

Simple Polynomial Interpolation Problem

Simple polynomial interpolation in two dimensions is not always possible. For example, suppose that the following data are to be represented by a polynomial of first degree in $x$ and $y$, ...
1
vote
0answers
44 views

Given a finite set of points construct a polynomial that meets the points.

Say I have a set of points in $\mathbb{Z}^3 \times \mathbb{Z}_2$ each of which represent part of a mapping $(z_1, z_2, z_3) \mapsto z_4 \in \mathbb{Z}_2$. How do I find the the simplest polynomial ...
2
votes
0answers
20 views

Analogue of Helly’s theorem for non-exact interpolation

Let $\overrightarrow{x}=(x_1,x_2, \ldots ,x_n),\overrightarrow{a}=(a_1,a_2, \ldots ,a_n)$ and $\overrightarrow{b}=(b_1,b_2, \ldots ,b_n)$ be vectors in ${\mathbb R}^n$, with $a_k \leq b_k$ for every ...
0
votes
1answer
254 views

Coefficients of Newton interpolation polynomial

Given distinct $y_0,...,y_m$ in $\mathbb R$, let $N_m(x)$ be the Newton interpolation polynomial of degree $m$. That is, $N_m(x) = \sum_{n=0}^{m}a_nw_n(x)$ where $w_0 = 1$, $w_n(x) = ...
5
votes
0answers
228 views

Runge's phenomen: interpolation error using Chebyshev nodes oscillates

We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
0
votes
2answers
336 views

For n=3 Lagrange interpolation why is it equal to 1?

I'm studying Lagrange's formula for polynomial interpolation and I cannot seem understand why for $n=3$ $$L_0(x)+L_1(x)+L_2(x)+L_3(x) = 1$$ for all real x. In my textbook it says as a hint that ...
1
vote
1answer
123 views

What is the motivation for this theorem on Polynomial Interpolation Error?

My textbook presents this theorem without any sort of introduction. It does cover using the Newton and Lagrange forms of the interpolation polynomial, so I've got that. Anyway, here's the theorem: ...
0
votes
1answer
240 views

Least-squares approximation polynomial

Consider the function $\displaystyle f(x) = \frac{1}{\alpha (x-\beta)^2 + 1}$ in the interval $I = [-1,1]$. Set $\beta = 0$. How do I get the expression for the least-squares polynomial, say $\tilde ...
2
votes
1answer
453 views

Automatic calculation of the intersection of discrete curves

first of all, let me apologize for a poor math-english translation, I'll try my very best. I have the following situation: I have over 16.000 data files which I generated from a biometric ...
2
votes
0answers
438 views

Interpolating polynomial with Chebyshev nodes

I am interested in constructing an polynomial that interpolates some known arbitrary function $f(x)$ over the domain $x \in [0,70]$. I want the polynomial to have degree 14 and so need 15 points. ...
0
votes
1answer
107 views

Multiplying Polynomials with fewer coefficient multiplications

Not sure how this works! Apparently it can be done in 5-6 multiplications Show how to multiply two degree 2 polynomials using fewer multiplications of coefficients than the naive algorithm.
1
vote
1answer
287 views

Using Lagrange's Interpolation Formula to show that boolean functions over finite fields are polynomials

Let $F_2$ be the set of all the functions from the finite field $GF(2^n)$ of $2^n$ elements to $GF(2)$. I am reading a textbook that proves that the elements of $F_2$ can be represented by ...
6
votes
1answer
193 views

Is there a name for these polynomials?

Given $t \in \mathbb{R}[0,1]$, consider the following set of polynomials: $$ \left[-{\left(t - 1\right)}^{2} t, {\left(t - 1\right)} {\left(t^{2} - t - 1\right)}, -{\left(t^{2} - t - ...
2
votes
3answers
8k views

Given four points on a cubic function curve, how can I find the curve's function?

Say I have a curve $$y = ax^3 + bx^2 + cx + d.$$ I don't know $a$, $b$, $c$ or $d$, but I do know the $(x,y)$ values of four points on this curve. How can the values of $a$, $b$, $c$ and $d$ be ...
1
vote
0answers
119 views

Extending Hermite polynomial interpolation

Working with the definition of Hermite polynomials $x_0,\ldots,x_n$ are distinct in $[a, b]$, $f''(x)$ is continuous on [a, b], then $$H_{2n+1}(x)=\sum_{j=0}^{n} [f(x_j)H_{n,j}(x)] +\sum_{j=0}^{n} ...
1
vote
1answer
411 views

Lagrange Coefficients in Maple

I'm trying to compute Lagrange coefficients in Maple. Having found the $n$ roots of a Lagrange polynomial, I want to calculate the $j$-th coefficient: $$L_j(x) = \prod_{{i=0}\atop{j \neq ...
1
vote
1answer
372 views

Piece-wise linear interpolating polynomials

Somebody please help me to obtain piece-wise interpolating polynomials for the function $f(x)$ defined by the below data: $x=1$, $f(x)=3$; $x=2, f(x)=3$; $x=4, f(x)=21$; $x=8, f(x)=73$ I know the ...
2
votes
2answers
963 views

Determine the coefficients of an unknown black-box polynomial

Let $p$ be a polynomial of known degree $n$: $$p(x) = a_0 + a_1 x + \ldots + a_n x^n$$ Suppose we have a magic black box that can evaluate the polynomial for us. How could one then determine the ...
1
vote
3answers
116 views

Polynomial interpolation $n+1$ distinct points

How would you show that $p(x)= \sum\limits_{i=0}^n b_i(x-c)^i$ is equivalent to $p(x)=\sum\limits_{i=0}^n a_ix^i$ by expressing the $a_i$ in terms of $b_i$ and $c$? Also we know that the polynomial ...
3
votes
1answer
196 views

Polynomial interpolation of the residues of a rational function

Let $g(z) = a\prod_{i=1}^N (z-\lambda_i) \in \mathbb{Q}[z]$ be square-free. At each root $\lambda_i \in \mathbb{C}$, let $r_i$ denote the residue $\mathrm{Res}_{\lambda_i} 1/g(z)$. Let $I_g(z)$ ...
1
vote
1answer
382 views

A problem on Lagrange interpolation polynomials

Based on a previous question, I had the following conjecture and was wondering if anyone knew how to prove it or find a counterexample. Consider the polynomial $$ ...
4
votes
1answer
754 views

Remainder term of Lagrange Interpolation Polynomial

Suppose $x_0,x_1,\ldots,x_n$ are $n+1$ distinct numbers in the interval $[a,b]$ and $f\in C^{n+1}[a,b]$. Then for each $x$ in $[a,b]$, there is a number $\xi$ in $(a,b)$ such that $$f(x) = P(x) + ...
3
votes
0answers
914 views

Computation of coefficients of Lagrange polynomials

For our homework we should write a program, that creates Lagrange base polynomials $L_k(x)$ based on a few sampling points $x_i$. Now i am eager to develop a formula to be able to compute the ...