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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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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1answer
16 views

What is the physical meaning of 2 nodes being same while fitting an interpolating polynomial?

When we are trying to find out constants for Newton's interpolating polynomial, we use divided difference method to find the constants. Then we have Hermite-Genocchi formula to find those constants ...
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Degree of Multilinear interpolation

Supposing you want to interpolate an $n$-variate polynomial on $\{0,1\}^n$, we could take the polynomial to be linear in each coordinate. What is a good interpolation procedure for this that will give ...
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Root finding of a Hermite interpolating polynomial

Consider a Hermite interpolation problem. I have an approach for obtaining the roots of interpolating polynomial. I would like to present an example for this approach. Can you suggest me an applicable ...
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Curve Fitting - When to use Interpolation and/or Best-Fit?

This is my first time posting on Mathematics Stack. Nice to meet you all. I have a question regarding curve fitting, interpolation, and best-fit approximation. My supervisor wants me to write a ...
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32 views

How can I interpolate the following data and get the equation?

I want to interpolate between three vectors which have 8 rows in each. For eg. X = [1 2 3 4 5 6 7 8], Y = [7 8 9 10 11 12 13 14 ], Z = [15 16 17 18 19 20 21 22] I want to find the equation Z = ...
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21 views

Computing integrals in order to find an approximation function

For a project in scientific computing I am trying to find an approximation of an unknown function $f(x)$. Given: data points $(x, f(x))$ A basis with which we can approximate $f(x)$ consists ...
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38 views

How to transform function values to specific interval

I'm doing a project at university about scientific computing and I'm stuck. As in: I seem to lack quite a bit of mathematical background for this project. The program has as input an array of $x$ ...
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3answers
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Where did the idea of hermite interpolation came from?

I am given the Hermite interpolation formula directly in my text book without ANY explanations about how it was first made (obviously it was somehow constructed for the first time with some sort of ...
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2answers
32 views

What interpolation method makes the fewest assumptions about the function?

If I have no information about a function except a regular timeseries of samples, what is theoretically the best interpolation method to use and why? We cannot assume the function is continuous etc. I ...
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1answer
25 views

Error of lagrange interpolation

If the original function I want to approximate using Lagrange interpolation is a polynomial the error function $(x-x_{0})...(x-x_{n})\frac{f^{(n+1)}(\xi)}{(n+1)!}$ is not working because the $n+1$ ...
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28 views

Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
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What is the difference between newton interpolation polynomial and interpolation polynomial with Neville scheme?

I am trying to find the interpolation polynomial by using Neville scheme. It looks like divided difference . What is the difference between newton interpolation polynomial and interpolation polynomial ...
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11 views

“Sigmoid” function with tunable initial slope, upper asymptote and transition period

I'm looking for a function which resembles the transition between the function $f(x)=x$ for small $x$ and the function $f(x)=C$ for large $x$ ($x$ is finite and $\geq 0$). I've found the generalized ...
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1answer
24 views

Interpolating discrete points along a spline

I have 2 lines which i need to connect with a spline (or some other curve, but not the circle; it has to gradually increase its turning angle). The lines cross each other so the curve should make a ...
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1answer
22 views

Express interpolation of a polynomial in other polynomial

Let $x_0$ and $x_1$ be two distinct nodes. Let $P(x)$ be a polynomial of degree 2 or less such that : $P_0' (x_0 ) = f' (x_0 )$ , $P (x_1 ) = f (x_1 )$ , $P_0' (x_1 ) = f' (x_1 )$ , and $P (x) = f' ...
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0answers
29 views

Selecting nodes for spline interpolation

Is there a general method to determine the best sample points for spline interpolation (whether for piecewise linear or piecewise cubic Hermite) given $x$, $f(x)$, and estimating $f^\prime(x)$? Does ...
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2answers
18 views

Interpolate/Increment Vector Rotation

For my 2D physics engine, I'm using the unit vectors of the direction an object is facing to represent its orientation; essentially, [Cos(theta),Sin(theta)] where theta is the object's rotation in ...
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1answer
19 views

Bound for Lagrange interpolant

I'm having difficulty figuring out how the bound in the below example is determined. Consider the function $f(x)=e^{-x}$. For sample points $\{-1,1\}$, the Lagrange basis interpolants are ...
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33 views

$L^2$ vs $L^{\infty}$ norm for interpolation

Under what circumstances should I consider utilizing the $L^2$ norm instead of $L^{\infty}$ when interpolating a function based on sample points? Probably related: Does the answer significantly ...
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14 views

Empirical Mode Decomposition: How to make a suitable spline interpolation when the number of extrama is small?

I am doing an EMD(Empirical Mode Decomposition) project and I am getting a problem with the spline step (find the upper and lower envelopes) because there is always a big deviation in spline ...
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0answers
29 views

From a set of vertices, find smallest polytope enclosing another point

Out of a set of vertices $V=\{\vec v_i\in \mathbb R^D\}$, I am constructing a piecewise linear interpolating function $f:\mathbf{conv}(V)\rightarrow R$ as follows: given a point $\vec d\in ...
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2answers
23 views

Interpolation using rate of change

I have a set of data with missing points, which I estimated using spline interpolation. I've now been given the rates of change at each data point. How will this change/improve my current ...
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4answers
131 views

Find a function with certain requirements

I'm trying to find a function $y=f(x)$ that can be described as follows: $f(x) = g(x) + c/(x-x_a)$. With $f(x)$ I want to design a function with the following properties: $f(0) = 0$; $f(x)$ has a ...
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2answers
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Help to find the best lower bound function for a given set of data, based in the natural logarithm function

I am trying to find a lower bound function for a set of data I have, and I am struggling with it. In the following graph the blue color is the set of data and the red color is my lower bound function. ...
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1answer
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Mathematics and Algorithms for Interpolation

I am doing some programming, where I am interpolating point a to point b, against a timer that is constantly incrementing by ...
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18 views

Combining samples for interpolation

I'm writing a program where values at a position in a 3D field should be estimated, based on a number of existing samples. In this case it is the density of a point cloud at different positions in ...
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18 views

Increase number of points after interpolation

I have a set of points which would give a curved line.I want to increase the number of points on the curve. eg- In the diagram, the original points are shown in highlighted black. If I interpolate ...
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9 views

Calculating 4 components (W,G,R,B) via interpolation

I am trying to find a formula for calculating the 4 components via some type of interpolation. Being very new to the subject, never studying it, I would appreciate if anyone could point me in the ...
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13 views

Healing a curtailed function

Suppose we have a function f : $\mathbb{R} \rightarrow \mathbb{R}$ and a ceiling M. We could define a second function $ g(x) = \left\{ \begin{array}{lr} f(x) & : f(x) < M \\ ...
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1answer
30 views

How can I approximate a function that is not derivable with derivable ones?

Suppose that I have a function whose graph has many angles (i.e. my function is not derivable). How can I approximate this function with derivable ones? Thank you!
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Find $a,b,c$ of polynomial function (Hermite interpolation)

Given abscissae $x_1 < x_2 < \dots < x_N$ and corresponding data values $\{y_i\}_{i=1}^N$ and derivative values $\{y_i'\}_{i=1}^N$, consider the following Hermite interpolation method: For ...
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1answer
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What to do when the chebyshev point is equal to data point in lagrange interpolation?

I am going to use Lagrange interpolation using Chebyshev nodes using the following formula $$\sum_x \prod_{k=0,k\not={j}}^n \frac {x-y_k}{y_j-y_k} f(x) $$ in which $x$ in my data points, $y_k $s ...
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58 views

proving linear interpolation of Level Set

I tried to explain figure below in mathematics form. As you can see I have got triangle (v1, v2, v3). The signed shortest distance form red interface (level set value) is calculated for each vertex ...
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Polynomial Evaluation

My question is to some extent related to cryptography, but I'd like the mathematicians answer my question, please (as their answers are usualy more clearer than cryptographers). Consider I have a ...
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1answer
32 views

Upper bound for the error magnitude

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

Find $f(x)$ given $f(0), f(1)$ and $f[x1,x2,x3]$

I need to find f(x) given $f(0) = 0$, $f(1) = 2$, and the divided difference $f[x_1,x_2,x_3] = 1$ for any three points $x_1, x_2, x_3$ How do I go about solving this?
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1answer
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Using linear interpolation between two points to find the three remaining points

I am taking a graphics programming course, and I am looking at how Linear interpolation can be used to move points from one location to another location within a certain time. My mathematics ...
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80 views

Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by \begin{align} Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t}, \end{align} that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast ...
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81 views

Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
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Partition of function into pieces for interpolation needs

I've got some experimental data obtained from my mate's research. There are two sets of (x,y) points for each curve. He asked me to interpolate function values between this points, so for each curve I ...
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using approximation to find interpolantion function?

my question is can we actually find the interpolating polynomial if we solve the approximation problem for degree m = n-1 ( where we have n data ).(i know we usually solve the approximation problem ...
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Recovering a continuous density function from its discretized version.

The probability density function is defined on [0,10], and it's discretized by taking integration over short intervals [0,0.01],[0.01,0.02], etc. Is there any kind of interpolation method ( I'm not ...
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Do nth degree polynomials derived using Least Squares Interpolation always have n+1 intersections with the function?

I have recently studied Interpolation Techniques in my College Numerical Methods class and I have this question: If we have a function $f(x)$ and we are asked to use Least Squares Interpolation(LSI) ...
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Creating FEM mesh for image region — what is the most suitable shape function?

I wish to create a FEM mesh to solve an inverse elasticity problem, for an irregular domain. This domain is given by a medical image, so it is discretised and each square on the grid has one scalar ...
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1answer
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deriving an integral quadrature rule on a triangle

I'm trying to look for references on this but I've not found any. I'm probably using the wrong keywords ... Let's suppose that our domain of integration $\Omega$ is the triangle in $R^2$ with a ...
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1answer
83 views

Lagrange Interpolating Polynomials - Error Bound

Let $f(x) = e^{2x} - x$, $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$. Construct interpolation polynomials of degree at most one and at most two to approximate $f(1.4)$, and find an error bound for the ...
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1answer
25 views

Best fit in logarithmic chart

I have several variances ($\sigma^2$) which value depends on the velocity ($v$). As you can see in the graph, if increase the velocity, the variance does the same. I am studying this dependency, but ...
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1answer
35 views

Newton form of the interpolation polynomial?

Let $ f(x)=\sin(x) $ and let $ p_1 $ be the first degree polynomial that interpolates $ f $ at $0$ and $\pi/2$. Then $ p_1(x)=(2/pi)x. $ How did they get this result; how is $ p_1(x) $ found?
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Prove that the difference between a continuous function and its interpolation polynomial..?

Let $ g \in C^2 [x,y] $ and $ P $ be its interpolation linear polynomial at $ a_0 $ and $ a_1 $ in $ [x,y] $. Prove that $ \lVert g-P\rVert _\infty < 1/8(a_1-a_0)^2B $ where $ \lvert ...