# Tagged Questions

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### There is a unique polynomial interpolating $f$ and its derivatives

I have questions on a similar topic here, here, and here, but this is a different question. It is well-known that a Hermite interpolation polynomial (where we sample the function and its derivatives ...
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### Modified Hermite interpolation

I asked similar questions here and here, but I tried to formulate this one in a sharper way. Is anyone aware of some literature on polynomial interpolation where we sample the function and its ...
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### Estimate the difference between $f$ and $p$ interpolating $f$

Suppose $p$ is the unique polynomial of degree $\leq 2$ that agrees with a function $f$ at points $a_1 < a_2 < a_3$. If the third derivative $f^{(3)}$ exists, and $x\in (a_1,a_3)$, then we can ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $n\in\mathbb{N}$. ...
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### 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]
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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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 $L(x)$: $$L(x) = \sum_{i = 0}^{m} y_i l_i(x) \\ l_i(x) = \prod_{0 \le ... 1answer 255 views ### Wolfram Mathematica - Newton Backward Interpolation? I have the following task: Create a function (in Wolfram Mathematica), called \mathrm{NewtonBackward}[n_,x0_,h_,f_] which interpolates backwards the function f(x) with nodes {x_i = x_0 + ... 1answer 58 views ### Intuitive proof of interpolation polynomial existence Problem: Given a set of n+1 data points (x_i, y_i) where no two x_i are the same, one is looking for a polynomial p of degree at most n with the property p(x_i) = y_i for all i∈ [0, n ... 1answer 105 views ### Interpolation- Barycentric coefficients for nodes that are Chebyshev points of the second kind. So I came across the following theorem: If the interpolation node are Chebyshev points of the second kind given by :$$ x_k=\cos \left( \frac{j\pi}{n}\right) \qquad ( 0 \leq j \leq 0) $$Then the ... 0answers 28 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 ... 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. ... 1answer 424 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 ... 2answers 102 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, ... 0answers 47 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 ... 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 ... 1answer 263 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) = ... 0answers 234 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 ... 2answers 348 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$$for all real x. In my textbook it says as a hint that ... 1answer 127 views ### What is the motivation for this theorem on Polynomial Interpolation Error? My textbook presents this theorem without any sort of introduction. It does cover using the Newton and Lagrange forms of the interpolation polynomial, so I've got that. Anyway, here's the theorem: ... 1answer 245 views ### Least-squares approximation polynomial Consider the function \displaystyle f(x) = \frac{1}{\alpha (x-\beta)^2 + 1} in the interval I = [-1,1]. Set \beta = 0. How do I get the expression for the least-squares polynomial, say \tilde ... 1answer 507 views ### Automatic calculation of the intersection of discrete curves first of all, let me apologize for a poor math-english translation, I'll try my very best. I have the following situation: I have over 16.000 data files which I generated from a biometric ... 0answers 442 views ### Interpolating polynomial with Chebyshev nodes I am interested in constructing an polynomial that interpolates some known arbitrary function f(x) over the domain x \in [0,70]. I want the polynomial to have degree 14 and so need 15 points. ... 1answer 107 views ### Multiplying Polynomials with fewer coefficient multiplications Not sure how this works! Apparently it can be done in 5-6 multiplications Show how to multiply two degree 2 polynomials using fewer multiplications of coefficients than the naive algorithm. 1answer 294 views ### Using Lagrange's Interpolation Formula to show that boolean functions over finite fields are polynomials Let F_2 be the set of all the functions from the finite field GF(2^n) of 2^n elements to GF(2). I am reading a textbook that proves that the elements of F_2 can be represented by ... 1answer 193 views ### Is there a name for these polynomials? Given t \in \mathbb{R}[0,1], consider the following set of polynomials:$$ \left[-{\left(t - 1\right)}^{2} t, {\left(t - 1\right)} {\left(t^{2} - t - 1\right)}, -{\left(t^{2} - t - ...
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 ...
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} ...