Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points.

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Computational interpolation?

I need to interpolate this set of equations $$ c_1\left( \frac{c_1}{b_1}-R\ln(x_1)\right)^{-1}=c_2\left( \frac{c_2}{b_2}-R\ln(x_2)\right)^{-1}=c_3\left( ...
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22 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 ...
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1answer
24 views

Is the Lagrange polynomial integer-valued for points with consecutive integer x-values?

What I'm really wondering is, does Lagrange polynomial interpolation have an answer for every question of "what's the next integer in this sequence"? Does it define an infinite integer sequence to ...
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1answer
33 views

Numeric Analysis Interpolation of $f(x) , f'(x) $

There is a problem i'm finding quite difficult to solve, i'd be grateful if anyone could point me to the solution : We want to interpolate the function $f(x)$ and it's derivative $f'(x)$ s.t ...
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1answer
34 views

Find the linear-to-linear function whose graph passes through the given three points

Find the linear-to-linear function whose graph passes through the points $(1, 1)$, $(4, 2)$ and $(30, 3)$. So by using the $$f(x)=\frac{ax +b}{x+d}$$ I got my final answer to be ...
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1answer
77 views

Maximum of $w(x)=\prod\limits_{i=0}^8(x-x_i)$

What is the maximum of $w(x)=\prod\limits_{i=0}^8(x-x_i)$ on the interval $[-1,1]$, with $\bullet$ equidistant nodes $x_i$, $(x_0=-1,x_8=1).$ $\bullet$Chebyshev nodes, $\displaystyle ...
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3answers
134 views

Stuck with integral

Having this: $\int x\sqrt{1-x^2}dx$ Substitution: $t = 1-x^2$ $dt = -2xdx => dx=\frac{-2x}{dt}$ So: $$\int x\sqrt{1-x^2}dx = -\int x t^\frac{1}{2}\frac{2x}{dt} = -\int \frac{2x^2 ...
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1answer
14 views

Inverse distance weighting where neighbouring values are zero

I am trying to interpolate a set of rainfall data in order to find the rainfall at an unknown point. I have been using the inverse distance weighting interpolation method (details given here: ...
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1answer
36 views

Defined matrix in Catmull Spline Curve

I am trying to use Catmull spline curve in my program , I am trying to understand it but why we only use below given Matrix , because the examples I saw I only found the below one In Catmull spline ...
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37 views

Convergence theorem (interpolation)

I am trying to complete a proof of the theorem which we have considered in my numerical analysis course. The tutor made a short sketch, but for me it was not very clear how we prove the statement of ...
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2answers
35 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), ...
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Well-behaved interpolation

I have a set of time-dependent scalar data. I would like to remove certain time intervals and fit the remaining data "well-behaved", which I cannot quite exactly define. The intervals may also be at ...
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1answer
68 views

nonlinear interpolation question

Hi all, here is an screenshot from book called rotation transforms for computer graphics by John Vince. I think here is an error ,if it doesn't please explain how do m+n equals 1 Thank you
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1answer
32 views

Use the zeroes of T3 to construct an interpolating polynomial

Use the zeroes of T3 to construct an interpolating polynomial of degree two for the function x^3 on the interval [-1,1] Okay, so I have been looking at Finding the zeroes using Chebyshev polynomials ...
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27 views

like Gauss-Chebyshev integration formula using Lagrange polynomials

Suppose that $L_k(x)$ is Lagrange Interpolation Polynomial for points $x=1,0,-1$. How to show that: $$\int_{-1}^{1}\frac{f(x)}{\sqrt{1-x^2}}dx=\sum_{k=-1}^1C_kf(k)+E$$ where ...
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1answer
43 views

chebyshev nodes on a 2D grid

I want to use chebyshev nodes for interpolation using lagrange formula. My grid is two dimensional and i do not know how to determine the nodes of chebyshev in a 2D grid point?
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1answer
19 views

Writing a Hermite Interpolation script

Tasked with writing a MATLAB script that computes the Hermite interpolation of a function. Specifically, it asks to find: $$p(x)\in \Pi_{2n+1} \text{, such that } p(x_0)=f(x_0), p'(x_0)=f(x_0)... ...
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10 views

Interpolate between height-maps using feature extraction / interpolation / resynthesis

Imagine a piece of stretchy material, held down to 0 everywhere on the unit disk. So, $x^2+y^2=1 \implies h=0 $ Let us say there is a peak somewhere within that disk, so at (x,y), h=1 And the ...
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2answers
182 views

$f(x)=1/(1+x^2)$. Lagrange polynomials do not always converge. why?

Let $f(x) = \frac{1}{1+x^2}$. Error of Interpolation with Lagrange polynomials for $n+1$ points is given by $$ e(x)=f(x)-P_n(x)=\frac{f^{(n+1)}(\eta_x)}{(n+1)!}\prod_{i=0}^n (x-x_i) $$ Carl Runge ...
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1answer
69 views

Hardy-Littlewood-Sobolev fractional integration inequality fails at endpoints

Here's a version of the theorem: If $1 < p, r < \infty$ and $ 0 < \alpha < n $ be such that $ \frac{1}{p} + \frac{ \alpha }{ n} = \frac{1}{r} + 1 $. Then for any $ f \in L^p ( \mathbb R ^n ...
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41 views

Lagrange interpolation polynomial and Vandermonde matrix?

I am trying to solve a problem about Lagrange interpolation polynomial, but i have the feeling i am stuck in something very trivial...So, may i ask you for a little help? Given is the polynomial ...
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1answer
36 views

Two sets of number ranges where one influences the other, how to find the intersection point?

Suppose I have the following number ranges: a = [x = 1, y = 5] b = [x = 9, y = 3] Now say a donkey is travelling between the number range 'a' and a horse is moving between number ranges 'b'. The ...
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Fitting a surface to scalar functions given on the edges of a triangulation

Given a triangle mesh $\mathcal{T}$ with vertices $V=\{\mathbf{v}_i\}_{i=1}^n$ in $\mathbb{R}^3$ and triangles $T_{ijk}=[\mathbf{v}_i, \mathbf{v}_j, \mathbf{v}_k]$. For each vertex $\mathbf{v}_i$, I ...
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29 views

Constructing quadratic equation from three points

I'm trying to wrap my head around the Muller method for approximating the roots of non-linear equations. The method is quite clear, my main concern is the part where you need to construct the ...
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2answers
51 views

How to find the second derivative?

I use this article from Wikipedia to build it in my program. How to find the second derivative in $(x_i, y_i)$ point of this cubic interpolation, if I know other $(x_j, y_j)$ points?
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1answer
32 views

Interpolation. Linear algebra [closed]

Find values of a, b, and c such that the graph of the polynomial passes through the points (-1, 0) and horizontal tangent (2, -9). I know who to find if we have only points but how find if we have ...
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1answer
47 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 \\ ...
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2answers
87 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 ...
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29 views

Interpolation of Polynomial using Lagrange

$f(x) = x^3 + 2x^2 + x + 1$. Find a polynomial of degree $4$ that interpolates the values of $f$ at $x = -2, -1, 0, 1, 2$. I was trying to use the Langrange algorithm, but I think i'm doing it ...
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23 views

Interpolation of Polynomial

Let $f(x) = x^3 + 2x^2 + x + 1$. Find the polynomial of degree $2$ that interpolates the values of $f$ at $x = -1,0,1$. I was able to do the an initial part of this problem (not written), but I ...
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1answer
22 views

Polynomial Interpolation - Bound on Error

Let the function $f(x) = \ln(x)$ be approximated by an interpoation polynomial of degree of 9 with 10 nodes uniformly distributed in the interval $[1,2]$. What bound can be placed on the error? I've ...
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1answer
36 views

re-arranging polynomial (newton interpolation)

How does: $$ f(x_2)= f(x_0)+\frac{f(x_1)-f(x_0)}{x_1-x_0}(x_2-x_0)+a_2(x_2-x_0)(x_2-x_1)\tag1 $$ rearrange to: $$ a_2 = \frac{\frac{f(x_2)-f(x_1)}{x_2-x_1}-\frac{f(x_1)-f(x_0)}{x_1-x_0}}{x_2-x_0} ...
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64 views

How to obtain Lagrange interpolation formula from Vandermonde's determinant

Assume that we have An interval $[a,b]$ A function $f(x)$ that is continuous on $[a,b]$ $n+1$ distinct points $a = x_0<x_1<x_2<\cdots<x_n = b$ And $f(x_0),f(x_1),\ldots,f(x_n)$ Now we ...
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Newton Interpolation jump in notes

I am told, The general expression for a second order polynomial that passes through 3 points $(x_1,y_1),(x_2,y_2)$, and $(x_3,y_3)$ can be written as: $$ p_2(x)=b_0+b_1x+b_2x^2 \tag1$$ which I am ...
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1answer
66 views

2D cubic B-splines

I have been looking at B-splines to interpolate points. Having 1-D B-splines makes perfect sense to me, but haven't been able to find something that explains 2-D B-splines well for me nor provide me ...
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2answers
35 views

Lagrange interpolation, syntax help

I am told, the basic interpolation problem can be formulated as: Given a set of nodes, $ \{x_i, i=0, ..., n\} $ and corresponding data values$\{y_i, i=0, ..., n\}$, find the polynomial $p(x)$ of ...
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33 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 ...
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1answer
21 views

how to show a data set satisfying a equation?

I have a set of data points like $(1,6)$, $(4,9)$, etc., and I am given a specific linear equation with two variable like $y = a/b + b x$. How can I show that the data points fit the curve?
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28 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 ...
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How to use derivatives at points with interpolation

I am given given $n$ points with $x$ and $y$ values. I am also given the derivatives at each of these points. How can I use the derivatives to calculate or to improve my interpolation? I've been ...
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Simple multi-variate interpolation algorithm improvements

I currently have some code to perform multi-variate interpolation across arbitrary data sets with an unlimited number of variables and a sparse grid of data. However, the current algorithm ...
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10 views

Interpolating on the borders of differently-resolved images

I'm creating a three-dimensional model of the earth based on SRTM height data. The data set is pretty huge, so only a small fraction of the data is available at any given time. The height data is ...
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How to process this data simultaneously?

I have a few maps obtained at different energies (500x600 pixels). Energy distribution is not very homogeneous . 1st dim is X, 2nd dim is Y, 3rd dim is energy (different levels for blue, green, yellow ...
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1answer
117 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 $n\in\mathbb{N}$. ...
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1answer
86 views

Hermite Polynomials Triple Product

Similar to the question Legendre Polynomials Triple Product, I would like to ask whether there are any explicit formulas for the inner product of the Hermite polynomial triple product \begin{align} ...
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1answer
34 views

Solving sub-triangles for Barycentric Interpolation (Triangle Geometry)

I'm trying to solve this triangle, so I can implement a barycentric interpolation, but I'm having trouble solving everything. I have all the base values for each of triangular sections and with a ...
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1answer
39 views

Reverse range of numbers, scaling

I have a float that goes from 1 to 0 .Im trying to make it so that the order is reversed and scaled so it goes from 0 to -80 Just wondering if there is a straight forward way to do this?
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Smallest set of Liner equations, which exactly fit a set of points

I have a set of 2-d points,(it can be of any arbitrary dimension n). I want to find the minimum set of straight lines(linear equations) which exactly passes through the given 2-d points (unlike ...
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1answer
25 views

interpolation inequality- how we use it to get into this inequality

I was working to get some inequality, and the author use the following inequality and call it "interpolation inequality" $$\|u\|_{L^2} \leq c\|u\|_{H^{-1}}^{\frac{1}{2}}\|\nabla ...
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1answer
67 views

Bézier curves and optimization

I have a very peculiar problem. Assuming that you know how B-Splines or Bézier Curves work, you may also know that if we assume the result of the function, let's say tri-dimmensional, as a position in ...