# 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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### Where to find proof for the remainder formula of the interpolation in two variables

Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book ...
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### Does Lagrange interpolation at Chebyshev points solve the Runge phenomenon?

I recently came across the concept of the Runge phenomenon while studying numerical methods for special functions in the book "Numerical Methods for Special Functions" by Amparo Gil, ...
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### $f(x)=x^n+a_{n-2}x^{n-2}+...+a_1x+a_0\in \mathbb{R}[X]$. prove$\ \exists$ $i\in[1,...,n]$ so that : $|f(i)| \ge \frac{n!}{\binom n i}$

$f(x)=x^n+a_{n-2}x^{n-2}+...+a_1x+a_0\in \mathbb{R}[X]$. prove$\ \exists$ $i\in[1,...,n]$ so that : $$|f(x)| \ge \frac{n!}{\binom n i}$$ my attempt : i used lagrange interpolation and compared $x^n$ ...
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### How many evaluations do you need to prove that a multivariate polynomial is the zero polynomial?

For a univariate polynomial $f$, you just need to prove that $f(x) = 0$ for $d+1$ distinct $x$ to prove that $f$ is the zero polynomial. But for multivariate polynomials, how does that work? How many ...
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### How can I find a $n$-th degree polynomial for a line with $n-1$ predetermined points?

Essentially, I have a set of points which I want to create a polynomial that passes through a set of points. This is within the context of having set locations on a line to set times. It is preferred ...
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### Polynomial or wave through aligned points [closed]

If I have the distinct points $(x_{1}, 0) ... (x_{n}, 0)$, A) What would be a simple polynomial (non-constant) passing through these points? And could I instead also write a simple continuous curve (a ...
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### Different bounds for truncation error in linear interpolation

I am reading about the Lagrange linear interpolation for approximating a function. In the book, truncation error in linear interpolation is derived as follows : Suppose $f(x)$ is a function which is ...
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### find minimum value of $u$ for $\,u = \frac{ax^2+by^2}{\sqrt{a^2x^2+b^2y^2}}$

Find the minimum value of $u$ where $x^2+y^2=1\;$ and $\;u =\displaystyle{\dfrac{ax^2+by^2}{\sqrt {a^2x^2+b^2y^2}}}$
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### Lagrange interpolation as two points get closer and closer

Suppose, you want to perform a Lagrange interpolation for $x_0$, $x_0+\varepsilon$ and $x_1$. What happens as $\varepsilon \to 0$? Here are the Lagrange-polynomials \begin{gather*} L_0(x)=\frac{(x-(...
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### Newton's Divided Difference Formula without given data point

This is the original question: Let f(x)= sin((pi*x)/6) and P(x) a quadratic polynomial such that f(x) = P(x) at x=0,1, and 2. Find P(x) using Newton's Divided Difference Formula Since Newton's ...
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### Simplifying $\frac{a^4}{(a-b)(a-c)} + \frac{b^4}{(b-a)(b-c)} + \frac{c^4}{(c-a)(c-b)}$ using Lagrange’s polynomial

I have a question of symplifying this expression $$A=\frac{a^4}{(a-b)(a-c)} + \frac{b^4}{(b-a)(b-c)} + \frac{c^4}{(c-a)(c-b)} \tag{1}$$ where $a$, $b$ and $c$ are distinct nonzero real numbers. ...
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### Convex optimization with Ridge regression

If this is the constrained version of ridge regression: min w∈Rd ∥Φw − y∥^2 s.t. ∥w∥2 ≤ s, (1) where Φ ∈ R n×d and y ∈ R n . Answer the following questions. •How can we prove that this is a convex ...
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### Trouble trying to bound an approximation using Lagrange interpolation.

I know this has been asked various times but I do not understand any of the answers given yet. I'm working with the function $f(x)=e^{x}$ in the interval [-4,0] and I need to bound $|f(x)-Q_{n}(x)|$ ...
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### Lagrange Polynomial to estimate the derivatives of a function

We have the following question for our homework. I'm completely lost on what to do. I have no idea how to compute the error of the derivative or how to proceed from there. I've tried googling Lagrange ...
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Statement $\forall L \in \mathbb{R}^+$, and $\forall f:\mathbb{R}\to\mathbb{R}$ for which $f$ isn't differentiable at point $a\in\mathbb{R}$ $f$ is infinitely differentiable everywhere else (on $\... • 2,718 2 votes 1 answer 46 views ### I don`t understand this little incongruence about the error function of Lagrange interpolation polynomial. Given$f \in C^{n+1}([a,b])$and a set of$n+1$points in$[a,b]$. And given$P$the Lagrange interpolation polynomial, the error function is$f - P = \frac{f^{(n+1)}(\eta_x)}{(n+1)!}w_S(x)$where$...
Let $y_1,\dots,y_n$ be integers. Conjecture:$\;$There exists $f\in\mathbb{Z}[x]$ such that $f(i)=y_i$ for $i=1,\dots,n\;$if and only if $(i-j){\,\mid\,}(y_i-y_j)$ for all $i,j$ with \$1\le j < i\le ...