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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.

Let $(x_j)_{j=0}^{m-1},(y_j)_{j=0}^{m-1}$ be real numbers such that no $2$ $x_j$s are the same. The Lagrange interpolating polynomial is given by $$ l(x) = \sum_{j=0}^{m-1} y_j \prod_{j \neq k \in [0..m-1]} \frac{x-x_k}{x_j - x_k} $$

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Lagrange polynomial in 3-D. Variable of interest is a vector.

I have familiarity with lagrange interpolation polynomials in 1-D for a scalar variable of interest. I am currently interested in a 3-D interpolation of a variable that is a vector $\vec{u} = [u, v, w]$. So essentially, I think what I need is 3…
David
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How can we derive the Interpolation formula for higher order?

Let, $$y = mx + c \tag 1$$ is the equation of a straight line. Let, it pass through the point $$(x_0, y_0).$$ So, from (1) we find, $$c = y_0 + mx_0 \tag 2$$ On the other hand, from the formula of slope we find $$m=\frac {y_1-y_0}{x_1-x_0} \tag…
user6704
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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 $$ p_n(x)=\sum_{i=0}^nL_i(x_i)f(x_i), $$ where $$ L_k(x)=\prod_{i=0,i\neq…
Sha Vuklia
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How to show this Lagrange Polynomial formula?

For a Lagrange Polynomial, how to show $$l_0(x) = 1 + \frac{x-x_0}{x_0 - x_1} + \frac{(x-x_0)(x-x_1)}{(x_0 - x_1)(x_0 - x_2)} + ...+ \frac{(x-x_0)(x-x_1)...(x-x_{n-1})}{(x_0 - x_1)(x_0 - x_2)...(x_0 - x_n)}$$ especially the 1 at the very front,…
saladham
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Prove Lagrange form of interpolation for $x^j$

To interpolate a polynomial of degree $n$ using the Lagrange form, $$p(x)=\sum_{i=0}^ny_iL_i(x)$$ with $$L_i(x)=\frac{\prod_{i\not=j}(x-x_j)}{\prod_{i\not=j}(x_i-x_j)}$$ How can I show that for $y_i=x^j_i$ for $j=0, 1, ..., n$…
bowen.jane
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Lagrange polynomials: representation

For some complex numbers $\lambda_1, \lambda_2, ..., \lambda_k$, define $$p_i(\lambda) = \prod\limits_{j=1, j\neq i}^k \frac{\lambda - \lambda_j}{\lambda_i - \lambda_j}$$ We now observe that, for any polynomial $P(\lambda)$ of degree less than…
man_in_green_shirt
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Intuition for Linear Lagrange Interpolating Polynomial

I'm trying to understand how the formula for Lagrange Interpolating Polynomials comes about by looking at the basic case of Linear Lagrange Interpolating Polynomials. I found this derivation but it gives me no understanding as to how the final line…
Robin Andrews
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Problem with Lagrange's Interpolation formula

Problem: Given $u_1 = 1, u_2 + u_3 = 6$ and $u_4 + u_5 + u_6 + u_7 + u_8 = 30.$ What is the approximate value of $u_4$ when computed by using Lagrange's interpolation formula? I tried to solve it as described here:…
Soumya Boral
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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(…
Abhishek Kumar
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Numerical differentiation by interpolation

In numerical differentiation by using Lagrange interpolating polynomial (Numerical Methods For Scientific And Engineering Computation By M.K. Jain , ch. # 5) the error may be obtained by using the relation : $\frac{1}{(n+1)!}\frac{d^j}…
Abbeha
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Lagrange's interpolation, property proof

$x_0, x_1, ... x_n$ are distinct points and $A(x)=\prod_{j=0}^n{(x-x_j)}$, $A_k(x)=\prod_{j=0, j\neq k}^n{(x-x_j)}$, $L_k(x)=\frac{A_k(x)}{A_k(x_k)}$. Prove each of the following: a) $\sum_{k=0}^{n}{L_k(x)}$=1 and…
user250273
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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 $\;u =\displaystyle{\dfrac{ax^2+by^2}{\sqrt {a^2x^2+b^2y^2}}}$
Kiran Kiran
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Lagrange interpolation at 0's

I tried to find an answer for some time, the answer is simple probably ,say I have a function given by set of points and their values like ${(4,1),(5,0),(6,0)}$, how do I calculate Lagrange interpolation polynomial for them? We know that $$w(x) =…
user378298
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Given any sequence of numbers of length N, Can it be represented by an explicit formula?

please forgive me if this question is not a good one, I am just a high school student (note: this isn't homework). I was wondering if every sequence of integers could be represented by an explicit equation. I have read another similar question whose…
cole-wilson
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Problem require Lagrange interpolation

Given $P(x) = ax^3 + bx^2 + cx + d$ where $ \vert P(x) \vert \le 1\; \forall \vert x \vert \le 1$ Prove: $$ \vert a \vert + \vert b \vert + \vert c \vert + \vert d \vert \le 7$$ Hopes you guys can help me with this problem. Thank you so much!!!!!!
HAHAHAHHA
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