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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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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51 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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22 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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131 views

FEM solution interpolates the exact solution

Consider the problem $$ -u'' = f \ \ \text{in} \ (0,1), \\ u(0) = (1) = 0. $$ Assume that the Green's functions of the nodal values $G(x_j, \cdot)$ lie in $V_h = \{ v \in C([0,1]) : v \ \text{...
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955 views

The Gherkin (an egg shaped building) - equation for the curve in order to calculate the surface area of revolution

I am trying to calculate the surface area of revolution for The Gherkin, an egg-shaped building in London, UK. Not sure about how to obtain the equation of the curve but I have the data points that ...
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15 views

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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Is this algorithm for 3D spherical interpolation correct?

I am attempting to write a spherical interpolation algorithm for for the application of smooth 3D animation in a game. The scripting language that the game engine uses is Lua. It is often easier for ...
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1answer
842 views

Jacobian determinant for bi-linear Quadrilaterals

Mapping from a square $\left[-\frac{1}{2},\frac{1}{2}\right]\times\left[-\frac{1}{2},\frac{1}{2}\right]$ with local coordinate system $\,(\xi,\eta)\,$ to an arbitrary quadrilateral with global ...
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35 views

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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72 views

Interpolation between iterations of exponentials (and logarithms)

I'm interested in finding a continuous and ideally smooth family of real-valued monotonic functions $H_t(x)$ that interpolate between iterations of exponentials, and also iterations of logarithms. So ...
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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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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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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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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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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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821 views

Generalized parabolic interpolation

Parabolic interpolation is an easy way to estimate the maximum of a function known by three values at equally spaced points, the central value being the largest. Is there an easy way to generalize ...
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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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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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686 views

Upper bound for the error magnitude

For the function $f(x) = \mathrm{e}^x$ on the interval $[0,1]$, by using polynomial interpolation with $x_0 = 0$, $x_1 = 1/2$, and $x_2 = 1$, find the upper bound for the magnitude $$ \max_{0 \leq ...
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10k views

Spatial Interpolation for Irregular Grid

How would I interpolate to a point P if I have four points around it such that: Q1 = (x1,y1), Q2 = (x2,y2), Q3 = (x3,y3), Q4 = (x4,y4) If the coordinates formed a regular 2D grid I would use a ...
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45 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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41 views

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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1answer
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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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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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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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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51 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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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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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 ...
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1k views

Non-linear regression for cumulative distribution function

I have twenty probability distributions based on a simulation. The corresponding cumulative distribution plot for one distribution looks like this: Simulated result I believe that most of the ...
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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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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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What's the best way to calculate all of the points for a curve given only a few points?

I've been reading up on curves, polynomials, splines, knots, etc., and I could definitely use some help. (I'm writing open source code, if that makes a difference.) Given two end points and any ...
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53 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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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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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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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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768 views

Should I use interpolation when finding median, and quartiles?

I am a S1 maths (Edexcel) AS student in the UK. My question: Say we have a stem-and-leaf diagram with 26 values. We want to find the lower quartile. To get the marks for our specification, we need to ...
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1k views

Derivation of linear interpolation?

Anyone know a good derivation of the linear interpolation: $$\frac{y-y_0}{x-x_0}=\frac{y_1-y_0}{x_1-x_0}$$ Wikipedia gives one, which I don't understand.
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Radial Basis Functions Interpolation

$ \let\oldcdot\cdot \renewcommand{\cdot}{\!\oldcdot\!} \newcommand{\e}{\varepsilon} \renewcommand{\p}{\varphi} \renewcommand{\p}{\varphi} \renewcommand{\vp}{\vec{\boldsymbol\p}(x)} \newcommand{\P}{\...
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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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2answers
41 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 ...
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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 ...
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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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1answer
755 views

When fitting a polynomial to data points, how to determine the reasonable degree to use?

I have wondered the following: Suppose that there is a set of data points $(x_i,y_i)$. Then I would like to know if it is more reasonable to assume if there is a polynomial relation of degree $m$ ...