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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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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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Conditional Multi Dimensional Data Fitting [closed]

I have data that is dependent on two variable. I need to fit the data using some analytical function. The data, say function of x and y has quite non-linearity about it. But it can be broadly ...
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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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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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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 ...
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Is $f(x) \leq x$ for $0 \leq x \leq \pi$ when sine series $f(x)$ are used to approximate $x$ based on derivatives at $x=0$?

This is a simpler "cousin" question to Would sine trigonometric series $f(x)$ for approximating $g(x) = x$ always be $f(x) \leq x$ for $0 \leq x \leq \pi$? . I am asking this as a separate question, ...
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What is the basic difference between interpolation & inference?

In mathematics we've studied interpolation as predicting the structure of a function from it's given finitely many values from which consequently we use to construct certain function which nearly ...
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How do I expand the Hermite Cubic Spline basis to the nth order?

A useful basis for cubic polynomials are those used for Hermite interpolation: $$h_{00}(t) = 2t^3-3t^2+1$$ $$h_{10}(t) = t^3-2t^2+t$$ $$h_{01}(t) = -2t^3+3t^2$$ $$h_{11}(t) = t^3-t^2$$ It is also ...
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Would sine trigonometric series $f(x)$ for approximating $g(x) = x$ always be $f(x) \leq x$ for $0 \leq x \leq \pi$?

I am trying to use set of $c_k$ with $k \in \mathbb{N}$ such that $f(x) = \sum_{k=1}^{M} c_k \sin kx \approx x$. $c_k$ is determined by setting $$f^{(2k+1)}(0) =0,\,\, k=1,..,n$$ $$f'(0) =1$$ $$f^{(...
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Lagrange Polynomial Interpolation in O(n) Time

Is it possible to evaluate a Lagrange polynomial interpolation in O(n) time after using a preprocessing step with O(n^2) operations? Thank you.
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Error boundary, for shifted Chebyshev interpolation.

Well this question was asked few years ago, but the answer doesn't satisfy me. Here is a link: related question: How to Change the Interval of Interpolation from [-1,1] to [a,b] for Chebyshev Nodes ...
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find interpolation polynomial for $f(x) = x^4+3x^2$ of degree $\le3$ , such that $max_{x\in[-1,1]}|f(x)-p(x)|$ is minimal

find interpolation polynomial for $f(x) = x^4+3x^2$ of degree $\le3$ , such that $max_{x\in[-1,1]}|f(x)-p(x)|$ is minimal. Well I think I don't understand Chebyshev's interpolation points correctly, ...
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Finding Expression for terms in Lagrange Polynomial Interpolation

So for a paper I am writing I am using Lagrange polynomial interpolation: $P(x)=\sum_{i=0}^{N}f(x_i) \cdot \prod_{j=0;j\neq i}^{N}\frac{x-x_i}{x_j-x_i}$ And I need to find an expression that ...
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Looking for an algorithm to create an interpolated curve function with specific requirements

I need this algorithm for an editing program to allow users to adjust curves, and I want those curves to conform to some particular requirements. For simplicity of use I would like to be able to ...
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Linear Interpolation Error estimation

Suppose we have applied trilinear interpolation technique over grid points in the space $A\times B \times C \times D={(x,y,z)|x\in A, y\in B,z \in C}$} . The interpolant, which is piecewise function, ...
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References on Trilinear Interpolation Technique

It's a while that I am looking for a literature on "Trilinear Interpolation", but I didn't find any thorough examination on this topic. I could find a short description in Wikipedia and some papers ...
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From trilinear interpolation to linear interpolation.

I have trouble understanding the following: Let us assume that we have some $\Omega()$ data generating process, which generates {$h,x,y,z$} points. Let us assume that it is possible to construct the ...
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How to generate interpolating polynomial using Newton formula for exponential function?

I am trying to find interpolating polynomial of $f(x) = e^{3x}$ and interpolation nodes $x_0=x_1=x_2 = 0\ and\ x_3 = x_4 = 1 $ with Newton's formula using Wolfram Alpha, but I am stuck with division ...
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Pros and cons of multivariate interpolation techniques for scattered data?

I have a numerical simulation $f$ that takes 6 input parameters $\mathbf x = x_1, x_2, \ldots x_6$. I have randomly selected $25,000$ random combinations of these inputs and calculated $f(\mathbf x)$. ...
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Bounding at Newton Divided Difference Formula

Let $a=x_0,x_1,...,x_n=b$ are $n+1$ points which are equally spaced in $[a,b]$. The distance between consecutive terms is $h= \frac{b-a}{n}$ and $x \in [a,b]$ Show that $ \biggr|\prod_{i=}^n (x-x_i) \...
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1answer
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Hermite interpolation code in MATLAB

This is my MATLAB code for divided differences and Hermite interpolation, but it doesn't work properly. Could you take a look at it? Thank you. I'm sorry for the layout, but it's the best I could do. ...
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Choice of the family of the Basis Functions

As I have learned for now, there are several families of Polynomial-type Basis functions (Lagrange, Serendipity, Hermite, ...). My question is beside the order of the elements which affects the order ...
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Lattice vs. Random sampling for function interpolation

Suppose $f: [0,1]\times[0,1]\rightarrow \mathbf R$ is smooth. I am to interpolate the function from the function values of $f$ at $n^2,\, n\in \mathbf N$ samples points. I have the freedom to pick ...
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Estimate the remainder term of an unusual interpolation: $f(-1)=f(1)=f'(0)=f''(0)=0$

Suppose that $f(x)\in C^{4}[-1,1]$ and $$f(-1)=f(1)=f'(0)=f''(0)=0$$ Show that for every $x\in[-1,1]$, there exist a $\xi_x\in[-1,1]$, such that $$f(x)=\frac{x^4-1}{4!}f^{(4)}(\xi_x)$$ where $f^{(4)}...
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Recommendations for Numerical Analysis book covering specific requirements?

I have a numerical analysis course, Course content is as follows can anyone recommend me a good book or several books which covers these areas. If the book gives an intuitive idea it would be better. ...
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Given interpolating function on an interval, find the upper bound of the error for that function

Let p(x) be a linear function interpolating sin(x) at $x=0, x=\frac{\pi}{2}$. Prove that $|p(x)-\sin x| \leq \frac{1}{2}(\frac{\pi}{4})^2$ on $[0, \frac{\pi}{2}]$. I've already done a bit of work and ...
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Construction and Uniqueness of a Periodic Interpolating Function with Special Contraints

Consider the following points: $(t_{1},y_{1})$ $(t_{2},y_{2})$ … $(t_{n},y_{n})$ where $t_{1}<t_{2}<...t_{n}<t_{1}+P$ for some $P \in \mathbb{R}$ Define $G(t)$ as a function $\mathbb{R}\...
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Feasible way to find interpolating complex polynomial based on absolute value

Consider a complex degree-$(n-1)$ polynomial $p(z) = \sum\limits_{i=0}^{n-1} a_i z^i$. Given a number $0 < m < 2n$ of positions in the complex plane with absolute value requirements, i.e. $|p(...
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Would the use of cubic splines increase the number of data points to interpolate from result in smaller error and avoid Runge's phenomena?

I am implementing the cubic spline method to interpolate the function: $$f(x)=\sin(x);\ -π≤0≤π$$ I am using this paper by Antony Jameson to construct my code, with a few modifications to correct the ...
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2d interpolation using derivatives

I have four points on a rectangular grid $(x_1,y_1)$, $(x_1,y_2)$, $(x_2,y_1)$ and $(x_2,y_2)$. I also have the value of a third variable $z$ at each of these points, as well as the partial ...
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Beginner question about interpolating polynomial error estimation

Given the function $f: \mathbb{R}_{\gt 10} \to \mathbb{R}$ with $f(x) = \ln(x + 10)$, and given the data points $(0, f(0)) , (1, f(1))$ & $(2, f(2))$ I need to estimate the interpolation error ...
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For $f(x) = \tan(\pi \cdot x)$, find the Interpolation $Q(x) = b_0 + b_1 x + b_2 \frac{1}{x-\frac{1}{2}}$

Progress so far: In a previous task, I determined a polynomial interpolation using a system of linear equations. The data points to be used were $(0, f(0)), (\frac{1}{6}, f(\frac{1}{6})), (\frac{1}{...
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Interpolation with a new point of $f'$

Background: (Lagrange Interpolation) Let $f\in C^{n+1}([a,b])$ and $x_0,...,x_n\in[a,b]$. If they are different there is a unique $p_n\in\mathcal{P}_n$ such that $p_n(x_i)=f(x_i)$. Also, we have ...
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How to trace equation from values of f(x) using Interpolation formulae

I have the following data, representing the values of a function at given points: $f(1)=0$ $f(2)=5$ $f(3)=12$ $f(4)=21$ $f(5)=32$ I want to know the exact quadratic equation which was used ...
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Approximation using Lagrange Interpolation

I am aware of the formula of this method. However, is it true that the method produces more accurate polynomial when the $x$ points are closer to each other? if so or not, why? Moreover, If I am ...
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Aggregate and interpolate overlapping time-series data

I'm trying to aggregate counter data from two different types of measurements. The first type of measure gives an exact value of the counter on a given day. ...
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Rescaling and linear interpolation

Is rescaling as explained in this question and linear interpolation two names for the same thing or are there any differences between them?
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How can I use RBF interpolation on a highly stretched rectangular domain?

I performed a 2D parametric analysis where one variable is much larger than the other. Basically I sampled a function in many points: let's say 5 points for $x_1$ and 5 points for $x_2$, where the ...
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How to interpolate/extrapolate a complex function?

Let us assume that we have $f:\mathbb{R} \to \mathbb{R}$. Also let us assume that $x_1\in \mathbb{R}$ and $x_2\in \mathbb{R}$ are given too. With this we can calculate $y_1 = f(x_1)$ and $y_2 = f(x_2)$...
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Smooth a graph as if placing a rope across the data…?

I'm not sure how to correctly phrase this question, in fact if I knew exactly what I needed to ask I could probably work it out myself, so please bear with me. What I need to do is smooth out a line,...
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Error bound of interpolating polynomial for equally spaced points

I am using the following error formula for polynomial interpolation $\frac{f^{(n+1)}(\xi)}{(n+1)!}$$\prod_{j=0}^n x-x_j$ where n is the degree of the interpolating polynomial and j+1 is the number ...
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$[t_j,t_{j+1},…,t_{j+k}]f$ Divided Difference on B-splines.

While reading Moments and Fourier Transform of B-splines (Neuman,E.) I found a different notation for the B-splines. The author define $M_{j,k}(x)$ with knots $t_j<t_{j+1}<...<t_{j+k}$ as: $$...
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Linear Interpolation for scattered 3-D data

I have a dataset of scattered 3-D points (non-regular) that carry some variable and am trying to interpolate that variable to a new point. I have currently implemented a couple of methods, but don't ...
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Calculation of a interpolating polynomial

Let $x_i=i+1$ for all $i=0,1,...,20$ and let $p(x)$ be a polynomial of degree at most 20 satisfying the following property: $$p(x_i)=(x_i)^{21}$$ for all $i=0,1,...,20$ I need to compute $p(0)$ I ...
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Construct a polynomial $p$ of degree $\leq n-1$ such that $p(a_i) = b_i$

Theorem. Given two countably infinite sequences of complex numbers $\{a_{n}\}_{n}$ and $\{b_{n}\}_{n}$ with $\lim_{n \to \infty}|a_{n}| = \infty$, it is always possible to find a entire function $F$ ...
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Multiplicative Perpetuity of Dividend

"A share pays dividends annually and the next dividend payment is due in 3 months time and is expected to be 5 cents per share. It is expected that future dividends will grow at a rate of 4% p.a. ...
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Monotonic and smooth interpolation between three points

The problem I have is the following: Given three $x,y$ points, $(-1, -y_{0})$, $(0,0)$ and $(1,y_{2})$, where $y_{0} \geq 0$ and $y_2 \geq 0$, I want to interpolate smoothly between them with a ...
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Inverse Trig Identities

I've been attempting to interpolate between two trig functions for a piece of software I'm writing, but feel I've finally reached the end of my mathematical knowledge. $$\frac{\pi x-\sin^{-1}\left(\...