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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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Interpolate between 4 points on a 2D plane

I'm trying to 'morph' between 4 values, which I have mapped on 4 corners of a square plane in my user interface (A, B, C, D, see image below). By selecting a point (P) within the boundaries of this ...
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25 views

Excel: inverse interpolation

Through Excel I obtained a calibration curve as shown in this figure The equation of this curve is $$y = 0,6209x^5 - 1,7958x^4 + 2,0116x^3 - 1,14x^2 + 0,3584x + 1,33$$ Now, through some ...
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Natural Cubic Spline beyond boundary guarantees constant slope?

my understanding is that for if I have five points, I can interpolate a natural cubic spline out of any five points with increasing x, because degree of freedom is five. But when I try the python ...
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Lagrange interpolation formula

Give the formula of the $1$st degree Lagrange polynomial $L(x)$ interpolating a function $f$ at the points $0$ and $1$. Give the formula for the error $L - f$. Finally, show that $$\sup_{x \in ...
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What is and how to calculate quartic interpolation?

I was reading the gist on the reward function used in OpenAI Five, but I didn't understand the way they calculate health's reward. This is what they state: Hero health is scored according to a ...
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Best method of spline interpolation for any 2D curve

So I want to make a mobile app where you can draw lines and then interpolate them with a spline. I used this bezier spline method: https://gist.github.com/anonymous/06f4104d93f6cef6f341 But it draws ...
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Combining rational interpolation and trigonometric interpolation?

My teacher mentioned that he never saw an article working out a theory of rational interpolation combined with trigonometric interpolation. (E. g. by using trigonometric polynomials instead of ...
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Monotonic linear interpolation of monotonic data on a scattered grid

Let $x_1,\dots,x_n\in[0,1]^d$ and $y_1,\dots,y_n\in[0,1]$ satisfy: $$\forall i,j\in {1,\dots,n},\hspace{10pt} x_i \le x_j \rightarrow y_i \le y_j.$$ I seek to find a continuous function $f:[0,1]^d\...
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unclamped smoothstep with… not sure if this is the right term, but “dynamic steepness”?

So, I need a smoothstep function for my game, and this seems to be the standard function that most people use: $y = (x^2) *(3-2*x)$ This is computationally friendly, which is important, but it doesn'...
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How to create an equation from linear interpolants

I found this Wikipedia article on Linear Interpolation. The third figure down depicts a data set 0 0 1 0.8415 2 0.9093 3 0.1411 4 -0.7568 5 -0.9589 6 -0.2794 ...
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Constructing a degree-1 Lagrange interpolation polynomial

Construct the Lagrange polynomial $p_1$ of degree $1$ for a continuous function $f$ on $[-1, 1]$ using the points $x_0 = -1$ and $x_1 = 1$. My attempt (note: I am using the notation that is used on ...
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Propagating Uncertainties on Interpolated Data

I have a data set of 2000 $[x, F(x), \delta F(x)]$ triples, where $x$ is exact and $F$ is a measured value with an uncertainty $\delta F$. I can interpolate/fit the function however needed, and this ...
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Integration by interpolation matlab

Hi I would like to write a program for integration in Matlab using the interpolation coefficients from piecewise Lagrange interpolation: $$\int f dx = \sum \frac{c_i}{n+1-i}(x-m)^{n+1-i}$$ This is ...
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2 Dimensional Interpolation with continous 1st and 2nd derivatives

I am looking for an interpolation algorithm for 2 dimensional data that has continuous 1st, 2nd derivatives, and cross-derivatives. I have found bicubic splines, but only the cross derivatives are ...
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Interpolation of a rational function

Assume I am given two polynomials $f(x)$ and $g(x)$ with coefficients from a field $\mathbb{F}_p$, where $p$ is a prime. Now I know that the set of these polynomials is a ring and not a field, meaning ...
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Weighting a cubic hermite spline

I am trying to figure out a function behind the software's curve drawing algorithm. Originally, each node comes with 3 parameters : time, value, and tangent. I have found that it fits cubic Hermite ...
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Cubic Spline Interpolation Derivation Confusion

The derivation of the cubic spline interpolation (from this MIT OCW lecture, page 13) starts off with second derivatives, which are piecewise linear functions. It's written in the form $$ s''(x) = \...
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Convergence rate of cardinal series

Given $f \in C^{\infty}_{0}[a, b]\cap L^{2}(\mathbb{R})$, what can we say about the convergence rate of the cardinal series $$ s(t) = \sum_{j=0}^{n-1} f(a+jh) \mathrm{sinc}\left(\pi\left(\frac{t-a}{h} ...
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General Dimension of The RBF Interpolation

I am reading about the RBF interpolation in order to apply it to interpolate a bunch of data over a three dimensional space. This may sound as a simple question but it is annoying me not knowing it. ...
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Basis Functions For a 3D Source-Free Field

I have a 3D source-free domain, which is governed by the Laplace equation, $\Delta u = 0$. The field represents an electrostatic field and is measured at a point cloud consisting of $n$ non-boundary ...
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Interpolation between log and polynomials using Riesz-Thorin

I consider two $L^1$ weighted spaces, with $m_1(x) = e + x, m_0 = \ln(e+x)$. It is known that the Riesz-Thorin interpolation theorem holds for $L^p$-weighted space. I have an operator $T: L^1(m_1) \to ...
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Algorithm/approach for regridding 3D coordinates

I have a question concerning surfaces of 3-dimensional structures, specifically if there is a way to interpolate or ‘regrid’ points/coordinates on a surface generated from the structure. I ask here on ...
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1answer
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cubic Hermite interpolation

Professor gave us this little bastard of a question and I'm at a complete loss about what to do. Some help or hints would be immensely appreciated, translated to the best of my abilities. Let $x_0=0$,...
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Is filling in grid corners extrapolation or interpolation?

Let's say I measure temperature at locations on a regular 3x5 grid but I am missing data at the 4 grid corners. If I use the available measurements to estimate the grid corner values, am I performing ...
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What is this recursive function approximating? ($x_i = x_{i-1}^2 / x_{i-2}$)

I came across this recursive function in some code, and it is within a function called "interpolate". Essentially the rule is: $x_i = x_{i-1}^2 / x_{i-2}$ which can also be defined as: $x_i = \...
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What determines how many times a polynomial can be differentiated before 0 is reached?

Also, does it relate to the degree of the polynomial in any way? I am struggling to get a high-level understanding of the characteristics of different degrees of polynomials - for example, their shape ...
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The minimum degree of interpolating polynomial that fits table data points.

What is the minimum degree that an interpolating polynomial that fits all five data points exactly can have? The following table data points are given for (x,f(x)): (-0.5,5),(0,15),(0.5,9),(1,3),(1.5,...
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Spline interpolation - why cube with 2nd derivative?

Background For spline interpolation, it looks the degree 3 cubic spline is accepted as the better way and in my understanding it requires 1st and 2nd derivatives at the joints to be the same. ...
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1answer
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Interpolation, identity for the derivative

Let $f\in C^{n+1}([a,b])$, $x_0,\dotso, x_n\in[a,b]$ pairwise different and $p\in \mathbb{P}_n$ the interpolation polynomial (of degree n) with $f(x_j)=p(x_j)$ for all $0\leq j\leq n$ Show that ...
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Hello. How to prove theorem about relationship between the interpolation operator norm and the constant of Lebesgue:

A norm of the linear operator $P_n$ can be defined by $||P_n|| = max_{f \in [a,b]} \frac{||P_n f||}{||f||} $ where on the right-hand side one takes any convenient norm for functions. Taking the $...
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2D Hermite interpolation on rirregular points

I am trying to generalize the principle of Hermite interpolation to 2D, over irregular data points. In 1D, we have a sequence of points on which the value of a function and its first derivative is ...
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1answer
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A question on the existence and uniqueness of a cubic Hermite interpolant

I have been trying to solve a particular problem that establishes both the existence and uniqueness of a cubic hermite interpolant on some generic interval $[a,b]$. Briefly, for a function $f$ we ...
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Polynomial interpolation vs polynomial curve fitting

If we have $n + 1$ points $(x_i,y_i$), then we can use interpolation methods (Lagrange, ...etc) to find a polynomial of degree $n$: $$P_n(x) = a_0 + a_1x + a_2x^2 + \cdots a_nx^n$$ In curve fitting, ...
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Proof involving interpolating polynomials

Let $f$ be sufficiently differentiable on $[a,b]$ and write $a=x_0$, $x_1 = \frac{a+b}{2} = x_0+h, x_2 = b = x_1+h$. Prove that $a)$ there exists a cubic polynomial $q$ such that $f(x_i) = q(x_i)$, $...
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Interpolation exercise of coordinates

The next problem of interpolation I do not know how to approach it. The problem says: A way to interpolate in two dimensions the values ​​of $f (x, y)$ in the vertices $(x_1, y_1)$, $(x_1, y_2)$, $(...
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Solving $\int dx/(x^{2}+1)$ with polynomial interpolation on the complex plane and with trigonometric substitution

Using the Lagrange interpolation theory for $x_{0}=-i$ and $x_{1}=i$ we have $\displaystyle \int \frac{dx}{(x^{2}+1)}= \frac{-1}{2i}\int \frac{dx}{x+i} + \frac{1}{2i}\int\frac{dx}{x-i}=\frac{1}{2i}(\...
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How to find dimensions by using experimental data?

I don't have the equations of the curves, instead i have a set of data points which describe the two red curves, i know the actual postion of the point $Q$ and $M$, I also know the angle between the ...
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Are my results new? [closed]

I'm eighteen and sometimes I like doing math on my own when I'm inspired. I would like to know if some of my "discoveries" are new (I don't think so :) ). These are some of the results I found in the ...
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The uniqueness of Hermite's interpolating polynomial in the case of $n=2$ nodes

The problem is : I have tried to prove the problem using proof by contradiction. However, I got stuck. I am not sure which role the derivatives of the function $p$ and $f$ should play, or how we ...
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What is the geometric meaning of this null-determinant?

While reading about interpolation I came across the following equation in Norlund. It involves determinants and I don't understand it in full yet. I do know how Lagrange and Newton follow by using the ...
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Integrating lagrange polynomial with equispaced points

Suppose we have some second order polynomial interpolant, $P_2$, defined on the equispaced points $x_0, x_1, x_2$, such that $x_{j+1}-x_j=h$. From $P_2$, we have Lagrange polynomials, $L_0, L_1, L_2$. ...
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Interpolate a squared exponential function (Gaussian)

I would like to interpolate a univariate squared exponential $$ f(x) = a\cdot exp\left(-0.5\cdot\frac{(x-\mu)^2}{\sigma}\right)+b $$ using two points $x_1$ and $x_2$, their function values $f(x_1), ...
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Maximum interpolation error in lagrange interpolation.

I have the following question: And the following Lagrange interpolation error bound: The way I have started to solve the problem is as follow. For me as a worst case is when all infinitely close to ...
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Interpolation On A Curvilinear Grid

How would someone interpolate/extrapolate data on a grid such as the one shown? Where can I study about such grids and methods of interpolation?
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Which method is better approximation in a given case?

Let $Q(x)\in\mathbb{R}[x]$ be a polynom. Let 4 sample points out of $Q(x)$: $$ x_0=-1,x_1=0,x_2=2,x_3=3, \\Q(x_0)=0, Q(x_1)=-5, Q(x_2)=15,Q(x_3)=64$$. a) Based on the given points, build an ...
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Trajectory interpolation between two known states

I have a trajectory made of several state vectors $\mathbf{x}_n$ (position and speed). One step forward in time is done with : $$\mathbf{x}_{n+1} = M_n\mathbf{x}_n + q_n$$ where $M_n$ is a matrix and $...
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Error estimate in the approximation of Incomplete Beta Function

In studying a problem related with the negative binomial, I need to approximate some function, which is a polynomial combination of the incomplete Beta function, in this case given by $$f_{a,b}(x):=1-...
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Cubic runout spline (cubic spline)

I was studying cubic spline interpolation and then I stumbled upon "Cubic runout spline". The idea behind it is that you set the boundary conditions for second derivatives to be: $f_1''(x_0) = 2f_1''(...
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Map Chebyshev nodes on an arbitrary shape?

Is it possible to map Chebyshev nodes on an arbitrary shape (e.g. in 2D: on a triangle, in 3D: on a cone or pyramid)? The Chebyshev nodes will be used as interpolation points? Thanks.
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Question about linear splines.

I have a question about linear splines. A computer package I am using has the option to use linear splines, but there is no user manual I can find. I am having problems with the kind of inputs the ...