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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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Interpolation error with Legendre/Chebyshev polynomials

I remember seeing somewhere that the Lagrange interpolation over Chebyshev nodes has least possible deviation in the sense of $\|\cdot\|_\infty$-norm, while Legendre nodes are optimal in the sense of $...
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Tri(cubic?)-interpolation on finite sets

I have two sets of data of same size, representing binded triplets, we can represent these two sets like this: $\mathbb{N}_A=\{i \in\mathbb{N}\mid i\in [0;256[\; \mid i \equiv 0\, [8]\}$ $\mathbb{N}...
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How to calculate smoothing spline coefficients

I am attempting to calculate smoothing spline coefficients based on the description in Reinsch's 1967 paper, but I'm having some trouble. The first derivative is not continuous. Here are the ...
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Creating polynomials from interpolation [closed]

) Assume $f(x)= x^3 – 3x^2 + 1$ and $N = \{-1,0,1\}$ a set of nodes. We are looking for the polynomial $P_2(x) = a_0 +a_1x +a_2x^2$ that interpolates $f$ on the set $N$. a. ) Determine $a_0$, $a_1$ ...
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Interpolation Problem?

I'm making a computer program and I ran into a problem of mapping/interpolating number from one range into another. I've formalized it as a mathematics problem. Look over it and I'd appreciate your ...
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Interpolation error for x*atan(x)

I'm trying to interpolate x*atan(x) on [-5, 5] using (n+1) equidistant nodes. The oscillations on the ends seem to be caused by the same reasons Runge phenomenon is. The question is why exactly are ...
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(Tri-?-)interpolation on triplet? (In order to increase image color depth from 15bits to 24bits using a dictionnary)

(I'll refer to a single term from Java, just in order to formalize because my math knowledge is poorly limited, it's the HashMap<X, Y>, it's a really-easy-to-...
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$E$ be a $n+1$ dimensional vector space of functions on a domain $D$ satisfying the following interpolation problem.

$E$ be a $n+1$ dimensional vector space of functions on a domain $D$. Let $x_0,x_1,\ldots,x_n$ are distinct points in $D$. Show that the interpolation problem : Find $f\in E$ such that $f(x_i)=y_i$ ...
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Existence of a holomorphic function on the unit disk

Let $z_1$ and $z_2$ be two distinct points in the unit sphere of $\mathbb{C}$, i.e., $z_1 \neq z_2$ and $|z_1|=|z_2|=1$. I'd like to construct the bounded uniformly continuous holomorphic function $...
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Error in time series forecasting

If I have hourly input data and I want to produce an output forecast of half hourly granularity, I must interpolate the hourly input data - but how can I calculate the average error attributed to my ...
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Integral error $\int_0^3|f(t)-p_n(t)|dt$ when $p_n(t)$ is the interpolating polynomial for $f(t)=e^{-t}cos(4\pi t)$

I have to find the integral error $\int_0^3|f(t)-p_n(t)|dt$ when $p_n(t)$ is the interpolating polynomial for $f(t)=e^{-t}cos(4\pi t)$ when $t\in[0,3]$ I am quite stuck here, however, let me explain ...
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Solve for missing values using linear interpolation

I have been tasked with trying to find a launch angle for a pumpkin launcher for a competition with a club at school! I am given the following data in a table and my goal is to find the launch angle ...
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Are standard function interpolation techniques robust to rotations?

This question may be both too broad and/or ill posed... As a motivation for this question, consider $y = sin(x)$ where the domain is restricted to $R^1$ We can plot this function in two dimensions. ...
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Vector field interpolation

Given a discrete set of vectors in $\mathbb{R}^2$, there seem to be algorithms (https://mathematica.stackexchange.com/questions/77701/interpolate-a-discrete-vector-field) that can 'interpolate' these ...
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Confusion between Wendland RBF functions - missing Wendland functions

I need to compute Wendland functions for a project, and got confused between the formula to construct Wendland functions $\phi(d,k)$ where d is the dimension So in the original paper introducing ...
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Bilinear interpolation example

I just do not understand what we are supposed to do when we want to scale a matrix with using the method of bilinear interpolation. Let's say we hjave a 3x3 matrix as written below. \begin{bmatrix}1&...
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Difference between mathematical interpolation and numerical interpolation?

I have a data set of points. This data is not of full points. Instead, I have to generate a python code using tck = interpolate.bisplrep(xx, yy, zz, s=0) to ...
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$\underset{v \in S_h}{\Vert u - v_h \Vert_{1,\Omega}} \leq ch\Vert u \Vert_{2,\Omega} $ implies $H^2(\Omega) \hookrightarrow H^1(\Omega)$

We have the following that is derived from the Bramble-Hilbert Lemma $\underset{v \in S_h}{\Vert u - v_h \Vert_{1,\Omega}} \leq ch\Vert u \Vert_{2,\Omega} $ implies $H^2(\Omega) \hookrightarrow H^1(\...
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Convex curve interpolation

I have a set of points $(x_0,y_0)$ ... $(x_N,y_N)$ with the $x_i$ increasing and the $y_i$ such that $\frac{y_{i+1} - y_{i}}{x_{i+1} - x_{i}} > \frac{y_{i} - y_{i-1}}{x_{i} - x_{i-1}}$ Is there a ...
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Interpolating a number of equidistant points.

I have the coordinates: $(0,3125),(1,3125), (2,2500), (3,1500), (4,600), (5,120), (6,0), (7,0), (8,0), (9,0), ...$ And I want a way to construct a smooth curve through the points that increases in ...
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Constructing families of $C^p$ curves which fit a finite number of points

Let $b > a$ be real numbers and consider the space $C^p(a,b)$ of $p$-times differentiable functions $f:(a,b)\rightarrow\mathbb{R}$. Let $x_1,\dots,x_k\in(a,b)$, where $i\neq j$ implies $x_i\neq x_j$...
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Interpolate values in 2D space + time (3D interpolation) of multiple objects in concert

With a given dataset, I have multiple objects that move, in concert, inside a 2D eucledian space. The dataset of one object is given like this: ...
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Newton's forward and backward interpolation method

Can we solve an interpolation question where the desired point x is near the end of the table by using forward interpolation rather than backward interpolation? If yes, then what's the process? I am ...
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Summation with vectors - Incorrect result?

While researching some different equations to use for smooth interpolation, I came across a generalized form of the smoothstep function that I've never seen before: $$S_N(x) = x^{N+1}\sum_{n=0}^N \...
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Truncation error in first and 2nd derivative of a function using forward difference method

I have been given a function $y=x^2$ and 4 set of values: (2, 4), (2.5, 6.25), (3, 9), (3.5, 12.25) Now, the problem is then to calculate the truncation error and the actual errors for the 1st and ...
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Hybrid implicit FFT resampling, does it make sense?

In signal processing there exist so many different methods of interpolation that one could probably write a book about it. Or ten. Or a hundred. When learning about the Fast Fourier Transform, one of ...
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Whittaker-Shannon-Kotel’nikov theorem in higher dimensions

The fundamental result in sampling theory states that if a signal $f(t)$ contains no frequencies higher than $\omega$ cycles per second, then $f(t)$ is completely determined by its values $f(\frac{k}{...
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Minimal error chebyshev interpolation

Let's say the n-degree Chebyshev polynomials : $$ T_{n} (x)=\cos(n\arccos(x))$$ Make a polynomial such that: $$\mid y- P (x) \mid$$ be minimal, using the first three Chebyshev polynomials for the ...
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Estimate for matrices

Can anybody help me with this? Is the following statement true? Can anybody prove it (or prove it wrong)? For $m<n$ let $\mathcal{M}:=\{A=(v_1,\ldots,v_m)\in \mathbb{R}^{n\times m}:\; <v_i,v_j&...
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Question on Levinson's proposed discrete form of Wiener filter: the stationarity assumption

The whole foundation of Levinson's discrete version of Wiener filter is based on the assumption of stationarity of a time series, and aims to predict a value based on the past observed values. Now, if ...
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What is the inverse of this transformation from a rectangle to a quadrilateral?

Given the following rectangles: one can map any point $(x, y)$ from the rectangle to a point $P$ on the quadrilateral using the following steps: Define linear interpolation as $l(p0, p1, t) = p0 + t(...
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Interpolation polynomial for $f=|x|$.

Why the interpolation polynomials for $f=|x|,x\in[-1,1]$ will oscillate near the endpoint of $[-1,1]$ as $n$ increases? I know that one explanation of the Runge's phenomenon is that the interpolation ...
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Optimization for recovery of partial observation

I'm working on an optimization problem where I'm trying to recover a real vector from a subset of that vector's elements that are observed, subject to a regularizing function and some constraints. $$ ...
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Find a, b, c from second degree polynomial interpolation formula

Linear interpolation formula $$ g(x)=f_j+\frac{x-x_j}{x_{j+1}-x_j}(f_{j+1}-f_i) $$ where answer is $ y=ax+b $ can be changed to $$ a=\frac{f_{j+1}-f_j}{x_{j+1}-x_j} \\b=f_j-x_j\frac{f_{j+1}-f_j}{x_{j+...
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How to compute $\prod_{i=1}^ny'{_i}^\left(\prod_{\genfrac{}{}{0}{1}{j\not=i}{j=1}}^n\frac{x_j}{x_j-x_i}\right)$with modular arithmetic for Lagrange

What I would like to do is an exponentiation with a public constant $(c=9)$ to the power of a secret number $(s=4)$ without revealing it and everything with modular arithmetic. $$c^s= 5 \pmod{11} \...
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Questions about polynomial interpolation problem

I have just started to read something about interpolation problem, and in a book it is written: Interpolation problem. Given $n+1$ points $(x_i,y_i)$ we look for a polynomial of degree $n$ passing ...
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Very small constants from poly-bernoulli

If we define $$a_{n}(m)=\sum\limits_{k=0}^{n}k!{n\brace k}(k+1)^m(-1)^{n-k}$$ $$\prod\limits_{k=2}^{n+1}1-kx=\sum\limits_{k=0}^{n}t(n,k)x^k$$ for $n>0$, $m\geqslant0$, so $$\sum\limits_{k=0}^{n}t(n,...
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Interpolation inequalities

Let $\Omega$ be a regular domain of $\mathbb{R}^d$, $d=2,3$. Let $\mathcal{T}_h$ be a triangulation of $\Omega$ of size $h>0$. Assume we can prove \begin{equation*} \begin{aligned} \|v\|_{L^2(\...
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Interpolation method that does never overshoot

for implementing a system that will control hardware, I need an interpolation between points on a graph that does never overshoot. By overshooting I mean that between two points there may be no y-...
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Almost simple Hermite interpolation

I'm trying to use Example 4 in Section 2.5 of Philip J. Davis's book Interpolation and Approximation (Dover 1975). The aim is to fix an error in an answer I posted last night. This gives the problem a ...
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Converting to cartesian equation from parametric equation for a Cubic curve

Considering the Cartesian form of the cubic equation: $$y(x)= ax^3 + bx^2 + cx + d $$ And considering the parametric equation of this above equation: $$x(t)=et^3 + ft^2 + gt + h \\ y(t)=pt^3 + qt^2 +...
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What is the standard method of proof for an INTERpolation?

Proof by induction is good for proving an extrapolation to the nth term, but what if instead you want to interpolate between two values? Like using $log(x^{1})=log(x)$ and $log(x^{-1})=-log(x)$ to ...
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Why can’t we cancel $l(x)$ from barycentric interpolation?

Recently, I read up on the barycentric interpolation which was apparently much better than Lagrange interpolation due to the ease of adding new data points. There is $$w_j = \frac{1}{\prod_{k={0..n},{...
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What is difference between newton interpolation and lagrange interpolation?

I know that they both represent same polynomial and their formulas, but what is difference between them. If they are not different, then why do we study them separately? Don't need detailed proof ...
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Iterated backward difference quotient from splines

I am working on a certain problem: say we have a function $f$ defined on an interval $[a,b]$ and we want the spline $Sf$ of order $k$ (and knots needed for the B-splines are in $[a,b]$) to agree to $f$...
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What are some current best practices for function approximation using neural networks?

There are lots of guides out there for current best practices for using neural networks for classification tasks. However, these guides don't always apply to function approximation. What are some of ...
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newton's interpolation error for non-differentiable function

I was given this function: $$ f(x) = \begin{cases} x^3 & {\text{if}}\ x>0 \\ 0 & {\text{if}}\ x\leq0\ \end{cases} $$ and I was asked to give an upper bound on it's interpolation error ...
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Morrey embedding for Potential Spaces

I'm trying to prove that for $f\in H^{s,p}:=\{f/f=G_s*g,\,\,g\in L^p\}$ where $G_s$ is the bessel potential: $G_s(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}(1+|w|^2)^{-s/2}e^{iw\cdot x}dw$ the following ...
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Inverse of trilinear interpolation: three degree-1 polynomials in three variables

I'd like to find the interpolation weights of a trilinear interpolation of a 3D-vector field. I'm given the vectors $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x\in\mathbf{R}^3$ and I want to find the scalar ...
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1answer
23 views

numerical integration - quadrature of second degree

Short Task: How can I find a interpolational quadrature formula with 2 points which interpolates all polynomials of second degree exactly.