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Questions tagged [interpolation]

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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The uniqueness of Hermite's interpolating polynomial in the case of $n=2$ nodes

The problem is : I have tried to prove the problem using proof by contradiction. However, I got stuck. I am not sure which role the derivatives of the function $p$ and $f$ should play, or how we ...
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What is the geometric meaning of null-determinant?

While reading about interpolation I came across the following (linear algebra) equation in Norlund. It involves determinants and I don't understand it in full yet. I do know how Lagrange and Newton ...
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Integrating lagrange polynomial with equispaced points

Suppose we have some second order polynomial interpolant, $P_2$, defined on the equispaced points $x_0, x_1, x_2$, such that $x_{j+1}-x_j=h$. From $P_2$, we have Lagrange polynomials, $L_0, L_1, L_2$. ...
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Interpolate a squared exponential function (Gaussian)

I would like to interpolate a univariate squared exponential $$ f(x) = a\cdot exp\left(-0.5\cdot\frac{(x-\mu)^2}{\sigma}\right)+b $$ using two points $x_1$ and $x_2$, their function values $f(x_1), ...
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1answer
25 views

Maximum interpolation error in lagrange interpolation.

I have the following question: And the following Lagrange interpolation error bound: The way I have started to solve the problem is as follow. For me as a worst case is when all infinitely close to ...
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22 views

Interpolation On A Curvilinear Grid

How would someone interpolate/extrapolate data on a grid such as the one shown? Where can I study about such grids and methods of interpolation?
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Which method is better approximation in a given case?

Let $Q(x)\in\mathbb{R}[x]$ be a polynom. Let 4 sample points out of $Q(x)$: $$ x_0=-1,x_1=0,x_2=2,x_3=3, \\Q(x_0)=0, Q(x_1)=-5, Q(x_2)=15,Q(x_3)=64$$. a) Based on the given points, build an ...
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Trajectory interpolation between two known states

I have a trajectory made of several state vectors $\mathbf{x}_n$ (position and speed). One step forward in time is done with : $$\mathbf{x}_{n+1} = M_n\mathbf{x}_n + q_n$$ where $M_n$ is a matrix and $...
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Error estimate in the approximation of Incomplete Beta Function

In studying a problem related with the negative binomial, I need to approximate some function, which is a polynomial combination of the incomplete Beta function, in this case given by $$f_{a,b}(x):=1-...
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Cubic runout spline (cubic spline)

I was studying cubic spline interpolation and then I stumbled upon "Cubic runout spline". The idea behind it is that you set the boundary conditions for second derivatives to be: $f_1''(x_0) = 2f_1''(...
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Map Chebyshev nodes on an arbitrary shape?

Is it possible to map Chebyshev nodes on an arbitrary shape (e.g. in 2D: on a triangle, in 3D: on a cone or pyramid)? The Chebyshev nodes will be used as interpolation points? Thanks.
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Question about linear splines.

I have a question about linear splines. A computer package I am using has the option to use linear splines, but there is no user manual I can find. I am having problems with the kind of inputs the ...
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Even Functions for RBFs

I have been searching the literature for a while on this and cannot find a clear explanation. This relates to Radial Basis Function Interpolation but I think it applies more generally. I have a ...
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Advanced Methods for Approximating Surfaces based only on partial derivative estimates

I'm looking for information on interpolating a surface function p(x,y) based only on estimates of the partial derivatives at points on a grid. Obviously, any such approximation is subject to a ...
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1answer
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Simplify this expression with divided differences.

The divided differences are defined as follows $$ f[x_i] := f(x_i), \quad f[x_0, \ldots, x_n] := \frac{f[x_1, \ldots, x_n] - f[x_0, \ldots, x_{n - 1}]}{x_n - x_0} \quad \text{for } n \ge 2 $$ For ...
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Find a polynomial with given characteristics

Attempt We can think of this as interpolating $(x_i, f(x_i))$ where $i=0,1,2$. Let $p(x)$ be the the polynomial we want. We can start by $$ p(x) = f(x) + (x-x_0)A $$ taking derivative we obtain $p'...
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Domain of an interpolated polynomial function

I was just wondering what defined the domain of a Lagrange interpolating polynomial. Is it the domain of the function being interpolated? Always the reals? I am asking this as I have constructed a ...
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Interpolation polynomial of first degree.

Let $\alpha \in \mathbb{R}$. If $\alpha x$ is the polynomial which interpolates the function $f(x) = \sin \pi x$ on $[-1, 1]$ at all the zeroes of the polynomial $4x^3-3x$, then $\alpha$ is ? As- $...
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Divided Differences expanded form definition.

From definition of divided differences we have that $$f[x_0,\cdots,x_n]=\sum_{j=0}^n\frac{f(x_j)}{\Pi_{{k\in\{0,\cdots,n\}-\{j\}}}(x_j-x_k)} $$ It makes completely sense to have $k\neq j$ otherwise ...
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Spline interpolation base on average of function

Let's choose the domain as the unit square and divide it into N^2 subdomain. Now I only know the average of function on each subdomain. I know i can regard the average as the value of the function at ...
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Interpolation between three-dimensional rotations

I have to define a continuous function $g: [0, 1] \rightarrow \mathrm{SO}(3)$ such that $g(0) = I$ and $g(1) = R$ (a given rotation). I know we can do this kind of interpolation using quaternions ...
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Sum of all values along a line

This picture is a representation of a problem I've been trying to tackle for a while now. Basically, I've got a grid with dimensions Latitude, Longitude, and Altitude. Time is also a dimension, but I'...
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Number of points needed to determine a sparse univariate polynomial over a large prime field with known support

Consider a sparse polynomial $f \in \mathbb{F}_p[x]$, of maximum degree $k$, with a known support of $t$ terms. That is, we know a set $I \subseteq \mathbb{N}$ such that $\#I = t$ and $f(x) = \sum_{i \...
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Finding an interval over which a polynomial interpolant $P_n$ converges to a function $f$ as the degree $n$ increases

I understand that due to Runge's phenomenon, increasing the degree, $n$, of a polynomial interpolant can actually increase the error between the interpolant, $P_n$, and the function, $f$, you are ...
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How do you define the sample points used for Chebyshev approximation/interpolation?

It appears there are somewhat conflicting definitions of the points used in Chebyshev interpolation. Wikipedia and Numerical Recipes define the $x_j^{(n)}$ sample points for $(n-1)^\text{th}$-order ...
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1answer
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How to derive the Newton interpolation polynomial

Consider the interpolation problem: find the polynomial through a given set of points $(x_0,y_0),...,(x_n,y_n)$. Suppose we want the polynomial in newtonian form: $$N(x)=\sum _{j=0}^k[y_0,\ldots ,...
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Formula for coefficients of interpolation polynomial

(Question) What is the conventional formula for coefficients of interpolation polynomial? Consider the interpolation problem: find the polynomial through a given set of points $(x_0,y_0),...,(x_n,y_n)...
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Cutting 3D Point cloud and interpolate between points

I have the following data https://www.mediafire.com/file/f8tz1zbpxvyvko7/Waltersdorf_F3.csv/file Which is a 3D point cloud. I can visualize it correctly, but I want to do Cuts like the ones in the ...
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Subsampling in 2d

Consider $g : R^2 → R$. Note that $g(x, y)$ on an interval $[a, b] × [c, d]$ could represent the value of a grayscale photo at position $(x, y) ∈ [a, b] × [c, d]$. Say that you know the value $g$ at $...
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1answer
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Find a two variables function given some values

I have a question. I have to find, for example, a function $f(s,t)$ of $s,t \in [0;1]$ such that : $f(0,t)=1, f(1,t)=0, f(s,1)=0$, but I don't succeed to find it. In general, given a function of ...
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How do we use the picture to get the polynomial?

Calculate the solution $p\in \mathbb{P}^4$ off the following interpolatin exercise: \begin{align*}&p(0)=2 , \ p'(0)=3, \ p''(0)=1 \\ &p(1)=2, \ p'(1)=0\end{align*} $$$$ I have done the ...
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Combination of interpolation types for multivariate interpolation

I have a N-dimensional dataset, for which I need to apply multivariate interpolation. Is there a possible way to use different kind of interpolation methods in different dimensions? I considered to do ...
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Can i find a 3D function given some points?

is it possible to find a 3D function given a set of data points? i tried plane-fitting it did not work, too chaotic for a plane. I am trying to find a 3D equation that cover most of points, how can ...
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polynomial interpolation exam paper

Good evening everybody, I have to calculate the interpolator $g(x)$ of degree at most 2 of values ​​$g(x_o) = f_0 g (x_1) = f_1 g (x_2) = f_2$ on the nodes $x_0 = -1, x_1 = 1, x_2 = 2$ (this passage ...
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1answer
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How to find the particular solution for Augmented Thin Plate Splines in the context of the Dual Reciprocity Boundary Element Method

In the dual reciprocity boundary element method (DRBEM) the non-homogeneous terms are expanded in terms of radial basis functions. This expansion involves approximating the solution to the linear ...
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1answer
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Gradient of an interpolated function

Can anyone please give a explanation: what do you mean by gradient of an interpolated function? Suppose, $f(x, y, z) = 2x^3 + 3y^2 -z$ is a function, and one result of the interpolation for the ...
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In linear interpolation, what exactly is $\frac{x-x_i}{x_k-x_i}$ in geometric terms?

Thanks to this question: Explanation of Lagrange Interpolating Polynomial, I have an intuition for what $\frac{x-x_i}{x_k-x_i}$ is doing in polynomial interpolation. That is, it is a kind of "on and ...
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Prove $ | u(x) - u_h(x) | \leq h \ \underset{0 \leq y \leq 1}{\max} |u''(y)| $

Prove: $ | u(x) - u_h(x) | \leq h \ \underset{0 \leq y \leq 1}{\max} |u''(y)| $ for $ 0 \leq x \leq 1 $ Using the fact that $|| (u-u_h)' || \leq \underset{0 \leq y \leq 1}{\max} |u''(y)| $ and ...
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1answer
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Newton-Gregory interpolation with divided differences calculations for new extra interpolation point

So lets suppose that we have the following $x_0=-1, y_0=2$ $x_1= 0, y_1=1$ $x_2=1, y_2=2$ $x_3=3, y_3=10$ and we know that all the above $x_i,y_i$ belong to $p_2(x)=x^2+1$ , and we want to add ...
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1answer
59 views

Lagrange polynomial $x^n$ coefficient

How can we show using Lagrange interpolation polynomial that $$\sum_{i=0}^n y_i \prod_{j=0, j\ne i}^n \frac{1}{x_i-x_j}$$ is the coefficient of $x^{n}$? I know that $f[x_0,x_1, .. x_n]$ is the ...
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How does one approximate a second derivative with ATPS interpolation

When using the Dual Reciprocity Boundary Element Method ( or any radial basis function method ) to solve a nonlinear differential equation it is necessary to approximate some derivatives of a ...
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1answer
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Interpolating data.

Consider the following formula $$_{n}q_x=1-exp[-n\times _{n}m_x-.008 \times n^3 \times _{n}m_x^2]\ldots(1).$$ Page 867 of this book shows values of $_{5}q_x$ associated with $_{5}m_x$ by the above ...
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Lagrange Polynomials with Derivatives (lowest order polynomial)

I need to find the lowest order polynomial, $P(x)$, that satisfies the following conditions using Lagrange polynomials: $P^{'}(x_{0}) = f_{0}^{'}$ $P^{'}(x_{1}) = f_{1}^{'}$ $P^{'}(x_{2}) = f_{2}^{...
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1answer
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Bilinear Interpolation - Alternative Calculation

Problem description: Given four points $P_i$ with coordinates $(x_i, y_i, z_i)$ find the $z$-value at point $C$ with known $(x_c, y_c)$ that lies within the quadrilateral formed by the $P_i$s. I am ...
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Mitigating the Runge Phenomenon with Constrained Norm Minimization

I am interested in polynomial interpolation of a set of points in $\{(x_1, y_1), \ldots, (x_n, y_n)\} \subset \mathbb{R}^2$. On the wikipedia page for Runge's phenomenon, the Constrained ...
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Estimating Spline curve by OLS. Is a good idea to fix the knots at Chebyshev sites?

I am writing my master's degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I've found many references to the so-called Chebyshev sites. This sites or ...
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1answer
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On highest degree of precision of numerical integration scheme that comes from interpolating polynomial

Let $x_1,...,x_n$ be distinct points in $[a,b]$ and $l_i(x):=\prod_{k\ne i}\dfrac {x-x_k}{x_i-x_k} $. Let $w_i=\int_a^b l_i(x)dx$. For every $f \in C[a,b]$, let $I_n(f):=\sum_{i=1}^n w_i f(x_i)$. ...
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How to approximate identity function using Fourier sine series

I want to approximate identity function $g(x) = x$ for $x \in [0,x_c]$ with $x_c<\pi/2$ by finite (sum) Fourier sine series $f(x)$. $f(x)<x_c$ is required for $x \in [0,\pi]$, $f(x)$ is assumed ...
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27 views

Shepard interpolation algorithm

Classic way to get a value based on other points in Shepard (Inverse distance weighting) method: $$F(x,y) = \Sigma_{k=1}^{N}w_{k}(x,y)f_{k} / \Sigma_{k=1}^{N}w_{k}(x,y) $$ where $w$ can be compute as ...
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1answer
33 views

Interpolating/estimating f on subset of integers

Here is the question I have been struggling to solve lately. Imagine we have two integers $x, y \in \mathbb Z, x \le y$ and $Y = \{ a | a \in \mathbb Z\ and\ x \le a \le y \}$; $X\subset Y$. In the ...