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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. It is necessary because in science and engineering we often need to deal with discrete experimental data.

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15 views

Polynomial interpolation: Construct basis functions such that $\mu_j(L_k)=\delta_{jk}$

We are looking for the solution of the Hermitian-polynomial-interpolation with the data points $x_0=x_1=0, x_2=2, x_3=x_4=1.$ Construct, analogous to the Lagrange-interpolation, basis functions $...
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Variant of Hermitian interpolation

Consider the pairwise distinct data points $x_0,\ldots,x_n$ and values $y_0, y_1^{(1)},\ldots,y_n^{(1)}$. We want to find the polynomial $p \in \Pi_n$ (vector space of polynomials of degree $\le n$) ...
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8 views

Hermite interpolation vs Hermite polynomial

Is there any connection between Hermite interapolation and Hermite polynomials?
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31 views

Interpolation of a 3D curve in space

I'm doing a curious exercise. I have to find a method to interpolate the following ballistic trajectory in 3D space: $$\left\{\begin{matrix} x(t) = \frac{v\cos(\phi)\cos(\beta)}{k}(1-e^{-kt})+d_{0}\...
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3blue1brown level for Devil's Calculator

There's this math game I've been playing called the Devil's Calculator. I saw on twitter that Grant Sanderson made a level and I love 3blue1brown so I wanted to try it (it's free to download). The ...
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15 views

Error estimation of a Taylor polynomial

Suppose $y : \mathbb{R} \rightarrow \mathbb{R}$ is a signal admitting a Taylor series expansion around zero (Maclaurin series) \begin{equation} y(t) = \sum_{n=0}^\infty \frac{y^{(n)}(0)}{n!}t^n \end{...
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29 views

Error estimation of the Taylor polynomial

Suppose that the signal $x: \mathbb{R} \rightarrow \mathbb{R}$ and its first $m$ derivatives are sampled at time $t_k$ and $t_{k+1}$ and there, i.e. the following values are available: \begin{aligned} ...
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58 views

How might I define a parabola in vertex form, such that…

Given the formula for a parabola in vertex form: $y = a(x-h)^2+k$, as $h$ and $k$ are changed, the $a$ value will adjust in order to keep the left hand $x$-intercept anchored to the origin. I'm really ...
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23 views

Error term in polynomial interpolation of non-differentiable function

On Wikipedia it is said that the error when interpolating a function $f(t)$ at $n+1$ distinct points $x_0, x_1, ..., x_n$ using a polynomial $P_n(t)$ of degree $n$, the error term is given by: $$f(t) ...
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Interpolation inequality of Gagliardo-Nirenberg for compactly supported functions

It's obvious that for a standard bounded domain $\Omega$ the interpolation inequality of Gagliardo-Nirenberg for a special case, can be written as the following: If $D^{m}_t v$ and $v$ belong to $L^...
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Linear Interpolation of quadratic function

I'm new in numerical methods. So I'm lost in the following exercise. Given the function $f(x)=3x^2+5x+1$ for $J=[-1,1]$. I've got to determine for the general sample points $x_1,x_2 \in J$ an ...
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Prove $S_{m}^{m}(\Delta)$={$s:s\in C^{m}[a,b]$ and $s$ is a polynomial of order $m$ in each $[x_{i},x_{i+1}]$}=$P_{m}$

Basically, I don't understand clearly. The point is to prove that these two spaces are equal or that the polynomial $s \in S_{m}^{m}(\Delta)$ is unique/the same in each $[x_{i},x_{i+1}]$? where $(\...
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How to recalculate a lerp percentage value, so that it returns the same value, even when the max lerp value is adjusted?

I currently have a lerp function, which is $$ y = p \cdot x_2 + (1 - p) \cdot x_1, $$ where $x_1$ is the min lerp value, $x_2$ is the max lerp value, $p$ is the percentage to lerp between $x_1$ ...
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Trying to understand Catmull-Rom curve

This is in 3D world space. So, I have 4 points P0, P1, P2, P3. I have to create a Catmull curve between points P1 and P2. I need 200 points between P1 and P2 to create a smooth curve. Wanted to know ...
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Intuition for Polynomial Interpolation Error

Math StackExchange, Wikipedia and Proof Wiki all offer roughly similar proofs of the error formula for polynomial interpolation of a $n+1$ differentiable function $f(x)$ with a $n$-degree polynomial $...
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Order Reduction of a field

I have a problem with a static (time invariant, but spatially varying) field expressed only as a set of discrete set of values at some points of an arbitrarily shaped domain in 2 or 3 dimensions. For ...
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43 views

To construct a polynomial using Bézier Curves.

Given a polynomial equation (can be of any degree) how do we find the control points for the Bézier curve that follows the polynomial equation? Note: I know that doing this is counter-intuitive ...
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40 views

Determining locality of piecewise quadratic interpolating

Suppose we wish to interpolate n +1 data points (n > 2) with a piecewise quadratic polynomial. How many continuous derivatives at most can this interpolant be guaranteed to have without becoming ...
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41 views

What is the importance and effect of the smoothness of a spline?

A Catmull-Rom spline is a $C^1$ (but not $C^2$) function, that is, its first derivative is continuous (but its second derivative might not be). However, there are splines that have $C^2$ or, in ...
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How is the recursive evaluation algorithm for Catmull-Rom splines with non-uniform parametrization related to the original formulation?

In the paper A Recursive Evaluation Algorithm for a Class of Catmull-Rom Splines, Barry and Goldman proposed a recursive algorithm to evaluate or calculate the Catmull-Rom spline between two control ...
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46 views

Weighted moving average derived from parabola approximation

Let us have five points at $x = -2,-1,0,1,2$ with ordinates equal to $y_i$, I want to derive the formula for $a_0$, such that the parabola $y = a_0 + a_1x + a_2x^2$ fits the points the best in ...
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Continuous function for day/night with night being c times longer than day

I'm looking for a function to transform domain $[0,1)$ into range $[0,1)$ such that the size of the domain corresponding to the range interval $[.5,1)$ is $c$ times the size of the domain ...
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17 views

Error norm for finite difference approximation

I have approximated the differential equation using finite difference approximation and have the vector $u$. To find the error norm, it says I need the exact solution and the piecewise linear ...
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8 views

Help guesswork an interpolation function

I want to construct a function $f(N, s, b)$ that generates $N$ non-negative numbers with mean equal to $1$. The order of the numbers is irrelevant. The function has two other parameters, the skew $s\...
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Force smoothness of coefficients from a series of least-square-fitting

I have a big XYZ-dataset where for every Z(X=const.,Y) I perform a 2nd order polynomial fitting. This gives me 3 coefficient-vectors p1(X), p2(X) and p3(X), which is of course quite noisy data, such ...
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21 views

How to interpolate the inverse square law?

I am a relative novice in terms of mathematics, but I am trying to understand how to approach a problem I have. I have an area light source that is 0.5 m² that produces 3500 luminous flux. As I ...
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41 views

Newton polynomial interpolation error

I am trying to use Newtons algorithm for polynomial interpolation. The original polynomial is $p(x) = 3x^2+4x+7$ and the Points from which I am trying to interpolate the polynomial are $p(1) = 14$, $p(...
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35 views

how to express the cubic spline tangent vector in the xy coordinate

I'm using piece=wise cubic spline to interpolate $2$-D data. For a given segment, I have the following reparameterization $$ x(t) = a_0 + a_1(t-t_0) + a_2(t-t_0)^2 + a_3(t-t_0)^3,$$$$ y(t) = b_0 + b_1(...
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54 views

Interpolation Polynomial of $\cos(x)$

This is a question from Exercise set 3.1, Numerical Analysis by Faires and Burden: Let $x_0 = 0$, $x_1 = 0.6$, and $x_2 = 0.9$. Construct an interpolation polynomial of degree at most 1 for $f(x) = ...
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Lagrangian interpolating polynomial code outputing an incorrect answer in matlab

I have to write code in matlab for a lagrangian interpolating polynomial that intakes some set of x and y values and a desired x value (named 'c' here) to be estimated and outputs the interpolating ...
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Forming Hessian Approximations from Exchangeable Data

In optimisation and elsewhere, it is of interest to approximate the Hessian of a function $f$, given only evaluations of $\nabla f$ at some collection of points $\{ x^i \}_{i = 1}^N$. That is, given ...
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27 views

Finding a function for $4$ dimension data set

I've got a bunch of data with integer values for let's say variables $X_1, X_2, X_3$ and an also integer value $Y$ for each set of $X_{1-3}$. My goal would be to find something like a formula that ...
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47 views

Minimum rank of a multivariable Vandermonde matrix

Given $m\geq 1$, a multi-degree $\alpha = (a_1, \dots, a_m) \in \mathbb{Z}_{\geq 0}^m$, and an element of $x = (x_1, \dots, x_m) \in \mathbb{C}^m$, let’s create a row of a matrix by writing down all $(...
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Radial Basis Function interpolation: why the multi-quadric basis function increase with distance?

I'm trying to understand the underlying logic in the Radial Basis Function interpolation. I understood that we estimate the value of the underlying function in any unknown point as $ y(\vec{x}) = \...
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54 views

Prove the relation between divided differences and derivatives of a function

Divided differences are defined like this: \begin{align} [y_n] & = y_n \quad (1) \\ [y_{n}, y_{n+1}] & = \dfrac{[y_{n+1}] - [y_n]}{x_{n+1} - x_n} \quad (2) \\ [y_{n}, y_{n+1}, \dots, y_{n+...
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Interpolation of function that includes a step change (discontinuous)

I have the following function that includes a step change: function image For those who cannot access the image, the function is linear between two times $t_1$ and $t_2$ with gradient $m_a$, there ...
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70 views

A possible typo in Problem I.12.10 about Newton polynomial from textbook Analysis I by Amann/Escher

I'm proving a remark in problem I.12.10 from textbook Analysis I by Amann/Escher. The remark is as follows: $$(x_{n}-x_{0}) a_{n}+a_{n-1}-b_{n-1}=0$$ Here are relevant definitions from my ...
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Cosine interpolation reverts to linear interpolation in higher dimensions?

Paul Bourke's article on interpolation explains different types of interpolation including linear, cubic, Hermite spline and cosine. He goes on to state (emphasis mine): In most cases the ...
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Method to rescale signals to mean length

I have a set of signals of varying lengths. I have provided an example of the same below - Time Series Their lengths vary between 186 to 202, with a mean length of 197. I am looking to rescale them ...
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For $p \in \mathbb{K}_{k}[X]$, the function $n \mapsto p(x_{0}+h n)$ is an arithmetic sequence of order $k$

I'm reading Remark 12.14.c from textbook Analysis I by Amann/Escher. $\mathbb{K}_{k}[X]$ is the ring of polynomials whose degrees are less than or equal to $k$. Here is Formula 12.15: where ...
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Vapnik-Chervonenkis dimension for (polynomial) interpolation/regression

I may have some misconceptions here, but by my understanding, the Vapnik-Chervonenkis dimension of a class of functions describes the number of points it can be guaranteed to classify correctly. I'm ...
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How to move an object's world space matrix to another matrix by linear interpolation?

I have got the world transform of an object and I want to animate it. I thought about creating a handful of matrices at different positions with different rotations, just like key frames in blender. ...
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29 views

Radial Basis Functions in $\mathbb R^3$ - How to interpolate?

I am fairly new to radial basis functions. I get the concept, but I need a little explanation or coaching when it comes to the actual application. Let's assume that I have $n$ points in $\mathbb R^3$ ...
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54 views

spherical interpolation in triangle

Is there a formula or algorithm with which one can interpolate the points of a triangle that lies on the unit sphere in a spherical manner? Let me elaborate: If you want to interpolate two points on ...
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29 views

There is a way to determine error of an interpolation outside of the given range

I'm trying to think about this problem: Given a set of points $x_i$, such that $x_i \in [a,b]$ and the value on this points of an function $f(x_i)%$ is given too,we know that the derivative is ...
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26 views

Trigonometric interpolation from DFT on sampled data

I want to interpolate N points on the $[0,2\pi)$ interval using trigonometric functions. Looking at my old notes from university courses, I can build the trigonometric polynomial $$ P(t) = a_0 + \...
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57 views

How to transform a shape in a uniform way?

I have two lists of ordered points in $\Bbb{R}^2$, for example in the following figure the lists are $S_1=\{1,2,3,4,5\}$ and $S_2=\{a,b,c,d,e\}$. A list of ordered points models my idea of shape so ...
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28 views

Interpolation polynomial of data with errors

Good day, everybody! I have an assumption that I've tried to proof for a fortnight, but still have no results. Let's say we have a function $f(x)$ and a set of evenly distributed data points $\{x_0,...
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22 views

Is there a common name for interpolation of reciprocal linear function?

For linear function of the form $y(x) = a * x + b$ there exists a linear interpolation term calculated as: $$lerp(A,B,alpha) = A + (B - A) * alpha$$ Is there a similar term for reciprocal linear ...
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49 views

Natural linear spline function representation

Let $s: \mathbb{R} \to \mathbb{R}$ be a natural spline function of degree one (that is it is piecewise a polynomial of degree at most 1) and let $x_0 \leq x_1 \leq ...\leq x_n$. Show that $s$ can be ...