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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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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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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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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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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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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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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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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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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(... 1answer 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) = ... 1answer 23 views 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 ... 0answers 17 views 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 ... 0answers 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 ... 0answers 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 (... 0answers 25 views 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}) = \... 0answers 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+... 0answers 23 views 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 ... 1answer 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 ... 2answers 46 views 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 ... 1answer 37 views 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 ... 0answers 53 views 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 ... 0answers 37 views 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 ... 0answers 20 views 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. ... 1answer 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 ... 2answers 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 ... 1answer 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 ... 0answers 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 + \...
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 ...