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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34 views

$\left( 1 - \frac{1}{n} \right)\left( 1 - \frac{2}{n} \right) \cdot … \cdot \left( 1 - \frac{k-1}{n} \right) = \frac{n!}{n^k r! (n-k-r)!}$

I'm trying to understand a proof in "Interpolation and Approximation by Polynomials" by Phillips. Let me quote (page 253): "For $k\geq 1$ we begin with $$B_{n+k}^{(k)}(f;x)=\frac{(n+k)!}{n!} ...
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1answer
62 views

Show that S is a cubic spline (natural or clamped)

Please see question. I believe the answer should be: $S_0(2)=\frac12(x^3-3x+2)=2$ $S'_0(2)=\frac12(3x^2-3)=\frac{9}{2}$ $S''_0(2)=\frac12(6x)=6$ $S_1(2)=\frac12(x^3-12x^2+45x-46)=2$ ...
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55 views

prove this using lagrange and newton divided difference error!

suppose f(x) is polynomial with degree of three.prove $f[{x}_{0},{x}_{1},{x}_{2}] = \frac{1}{2}{f}^{(2)}(\frac{{x}_{0}+{x}_{1}+{x}_{2}}{3})$ and ${x}_{0},{x}_{1},{x}_{2}$ are distinct point. I ...
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1answer
68 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
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11 views

Does the condition $S \in C^2[x0, x2]$ lead to a meaningful solution, when conctructing quadratic spline

we have three data $x_0 , x_1 , x_2$ we want to find the quadratic interpolation . $$S_0(x) = a_0 + b_0(x − x_0) + c_0(x − x_0)^2 on [x_0, x_1]$$ $$S_1(x) = a_1 + b_1(x − x_1) + c_1(x − x_1)^2 on ...
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12 views

cubic interpolation and data on a straigh line

Suppose the data we want to interpolate lie on a straight line. What can be said about the natural cubic spline ? i think i should show that the cubic spline in fact becomes a line . so if the spline ...
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1answer
20 views

Given an interpolating polynomial, how do I find another polynomial that interpolates at 1 less point?

The polynomial $$p(x) = x^5-2x^4-5x^3+15x^2+4x-12$$ interpolates x = -2,-1,1,2,3,0 with p(x) = f1, f2, f3, f4, f5, -12 respectively. In general, how do I find another polynomial of a lower degree ...
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33 views

how to learn the fast method of finding the cubic splines

in finding cubic splines if we have n points we get system of equations of magnitude 4(n-1) in the naive approach . in more sophisticated approach one only solves a system of equations with (n-1) ...
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solving $c_j$ system of equations for cubic splines?

the problem is like this : There are $N$ points $(x_0,y_0),(x_1,y_1),\dots,(x_{N-1},y_{N-1}) \in \mathbb{R}^2$ where $x_0 < x_1 < \cdots < x_{N-1}$. Cubic spline interpolation should give ...
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1answer
130 views

What is the difference between nodes and knots in interpolation?

I have been reading literature about polynomial interpolation (Lagrange) where the principles are described around nodes. The literature I have read about spline interpolation, however, talks only ...
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1answer
189 views

Calculating the x, y coordinate a set distance between two points

I'm trying to calculate the x and y coordinates that are a set distance between the coordinates of two pixels in an image. For example, if I travel from my original location (x1=4, y1=3) to a new ...
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3answers
52 views

Parameterizing cliffs

I am looking for a function $f(x; \alpha, X_1, X_2, Y_1, Y_2)$ that has the following property: For $\alpha=0$ it behaves linearly between $(X_1, Y_1)$ and $(X_2, Y_2)$, and as $\alpha$ gets closer to ...
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0answers
20 views

$(x,y)$ points extrapolation

My process is generating $x,y$ points. The points are result of tracking movement of an object so its a bit inertial. The tracked object can't change direction rapidly. I need to extrapolate data ...
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29 views

Re-initialize a 3D spline surface using different control points

I have a 3d spline surface is that is modeled with 65 points where the x,y,z position of each point is known. I want to keep the surface shape, but have different x,y positions for my control ...
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1answer
82 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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1answer
23 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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24 views

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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0answers
13 views

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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0answers
66 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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0answers
31 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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1answer
65 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
124 views

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
56 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
60 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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2answers
63 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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0answers
25 views

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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0answers
27 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
57 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
25 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
38 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
62 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
30 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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0answers
43 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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0answers
21 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 ...
2
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0answers
46 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
32 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
137 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
48 views

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
29 views

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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0answers
21 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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0answers
32 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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0answers
12 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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0answers
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
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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
38 views

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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0answers
68 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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50 views

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
63 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
31 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?