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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Sum of $p$th powers using polynomial interpolation

It is well known that the sum of the first $n$ $p$th-powers is polynomial in $n$ and is given by: $$ \sum_{k=1}^n k^p = \frac{1}{p+1} \sum_{j=0}^p (-1)^j {p+1 \choose j} B_j n^{p+1-j} $$ where $B_i$ ...
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8 views

Finding the missing points from available points

I'm currently working on a project to improve the resolution of an image.The basic idea of resolution enhancement is finding out the points in between the currently available points(In image ...
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7 views

Interpolation of the metric tensor

I am currently facing the following problem. I have a Riemannian manifold, where the metric is only known at certain points. Are there some standard strategy to interpolate the metric in other points ...
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19 views

What equation would be used to be draw a curve like this?

I have created a little visual based programming system and am not working on the visuals, if two nodes are connected and B.x<A.x then I want a curve to be drew in the fashion above.
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17 views

Convergence of quadrature formulas and interpolating polynomials

There is a theorem of Polya (1933), which says: 1) If a interpolatory quadrature formula converges for all continuous functions on [a, b] and quadrature weights are all positive, then the formula ...
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32 views

Quaternions: Finding the two quaternion vectors knowing the inital and final positions of interpolation?

Basically I have this really simple problem where I have the two vectors <0,1,0> (initial) and <0,-1,0> (final) for ...
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27 views

Regarding the Lebesgue constant for interpolation

I have a question regarding Lebesgue constant $\Lambda_{n}\left(\boldsymbol{\chi}\right)$, with which the worst case error between an interpolant $p\left(\boldsymbol{x}\right)$ and the function which ...
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Interpolating polynomials question?

For a function $f$ , the Newton’s interpolatory divided-difference formula gives the interpolating polynomial: $P_{3}(x)=1+4x+4x(x−0.25)+\frac{16}{3}x(x−0.25)(x−0.5)$ , on the nodes $x_{0}=0$ , ...
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22 views

Find nonlinear growth formula out of a set of values

I got two sets of values: $(x=240,y=20)$ and $(x=960,y=480)$. How can I find a formula to get any values in between? Anyone knows? Also if I distribute the value of $x$ among subvalues (imagine $x$ ...
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1answer
32 views

Newton's forward-difference formula question?

Use Newton's forward-difference formula to construct interpolating polynomials of degree two, and three for the following data. Approximate the specified value using each of the polynomials. I ...
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13 views

Interpolation with a constrained range between given control points

I am trying to create an algorithm that creates smooth color gradient functions, given control points in the red, green, and blue components. Mathematically, each curve would have a domain [0, 1] ...
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1answer
23 views

Lagrange polynomials question

Construct the Lagrange interpolating polynomials for the following functions, and find a bound for the absolute error on the interval $[x_0, x_n]$. $f(x) = e^{2x}\cos3x, ~~~~~~~x_0=0, ~~~~~~~x_1=0.3, ...
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1answer
75 views

Possible generalizations of Hadamard's three line lemma

Let $f$ be an analytic function on a sector $$ S=\left\{re^{i\theta}:0<r<\infty,\; 0<\theta<\gamma<\frac{\pi}{2}\right\} $$ with opening angle $\gamma$ at the origin. Suppose $f$ is ...
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12 views

Linear interpolation vs polynomial interpolation

Why linear interpolation is better than polynomial interpolation when we want to approximate $f(0.25)=e^{0.25}$? I can't formulate a concrete explanation. I thought that maybe it has a link with the ...
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32 views

Estimate the error of the Lagrange Interpolating polynomial

consider the function, f(x) = 1/(1+x^2) Estimate the error of the Lagrange Interpolating polynomial of 5 equally spaced points in the interval [-5,5]. Is there a proper way to find this logically ? ...
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1answer
41 views

Evaluate derivative of Lagrange polynomials at construction points

Assume, that we have points $x_i$ with $i=1,...,N+1$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align} Now according to my ...
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22 views

Reverse spline interpolation

Say I have a number of sets $(x, y)$ for $x \in \{0, 1, \dots, 255\}$. I want to find the least number of points to reproduce the set with a certain accuracy using linear interpolation. What is the ...
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27 views

Finding the period of the function

My question is as follows:- Find the trigonometric interpolant $ \bar{f}(x)$ for $f(x)= \frac{\pi}{x+3\pi}$ and $n=1$. Thant is, find coefficients $c_{-1}, c_0 ,c_1$ such that ...
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1answer
30 views

General name for interpolation and extrapolation

I would like to know if there is a technical term to cover both interpolation and extrapolation. The reason why I am asking is that I am writing a computer program to do interpolation and ...
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1answer
29 views

Getting error of interpolating polynomial by subtraction.

$f(x)= \frac{1}{1+x^2}$ and when I computed the interpolating polynomial of 5 equally spaced points in [-5,5] I got $ p(x)= 0.0053x^4 -0.1711x^2 +1$ Now I need to estimate the error in the ...
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21 views

numerical errors near at the borders

I use some kind of partitioning on my data and then I do some interpolation and some other mathematical operations using chebyshev points. I have noticed that in the borders of each partition, It ...
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28 views

Help on Lagrange Error Calculation [duplicate]

Here is an example in Burden's Numerical Analysis book. My problem is in bold In example 2 we found the second Lagrange polynomial for $f(x)=1/x$ on $[2,4]$ using the nodes $x_0=2$, $x_1=2.75$, ...
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1answer
29 views

computing interpolating polynomials of 5 equally spaced points in a given interval

The question I faced is as follows:- Consider Runge's function. $f(x)= \frac{1}{1+x^2}$ Compute and graph the interpolating polynomials (atop a graph of Runge's function itself) of with 5 equally ...
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1answer
41 views

3D Interpolation for Irregular Grid

I have a cloud of data points in three dimensions (x,y,z) that carry a value of some sort (lets say temperature T). I was wondering what the best options were for interpolating the temperature to a ...
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61 views

Trouble doing simple polynomial interpolation [duplicate]

I need to do a polynomial interpolation of a set of $N$ experimental points; the functional form I have to use to interpolate is this: $$ f(x) = a + bx^2 + cx^4$$ as you can see the coefficient that ...
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1answer
42 views

Trouble doing polynomial interpolation

I need to do a polynomial interpolation of a set $N$ of experimental points; the functional form I have to use to interpolate is this: $$ f(x) = a + bx^2 + cx^4,$$ as you can see the coefficient that ...
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1answer
86 views
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Is there a generalization of the Lagrange polynomial to 3D?

What is a way to construct a smooth polynomial surface ($\mathbb{R}^2 \rightarrow \mathbb{R}$) with Lagrange-polynomial properties in every partial derivative? I want to try this for image ...
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126 views

estimating a particular analytic function on a bounded sector.

Let $f(z)$ be an analytic function on $C^+=\{\Re z>0\}$, and we have the following (weaker) estimates $$ |f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ...
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1answer
63 views

Looking for help for building a Spline's algorithm 10th order

I'm trying to code the following algorithm in C++ and need help to understand the build of Splines from a mathematical point of view (found on page 129 on this paper). $$ f(t) = \boldsymbol{t} \cdot ...
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1answer
44 views

A big contradiction in interpolating point and number of it's

For calculating divided (fraction) difference table for interpolating $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, $n(n+1)/2$ difference fraction was used. I ...
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1answer
58 views

Multivariate polynomials at bounded evens

Univariate polynomials Given $n$, is there a degree $cn^{c'}$ polynomial $p(x)\in\Bbb R[x]$ and a degree $dn^{d'}$ polynomial $q(x)\in\Bbb R[x]$ with fixed $c,c',d,d'>0$ such that $$m\in\Bbb ...
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1answer
51 views

Why does the Bezier Curve work?

Recently I've been looking at Bezier curves and trying to understand how they work. I know that a general Bezier curve is given by the equation $$ \vec{\mathbf{B}}(t) = \sum_{k=0}^n{b_{k,\ ...
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29 views

Formula for $s_n = \sum_{i = 1}^n i^3$ Newton's Forward Difference Interpolation

Use Newton's Forward Difference formula to find an expression for $$ S_n = \sum_{i = 1}^{n} i^3$$ This is from an Introductory Numerical Analysis paper. I cannot figure out the connection ...
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1answer
37 views

Formal interpolation derivation of polynomial

I have a polynomial $$F(x_1,x_2,x_3,x_4)=k(x_1+x_2)(x_3+x_4)$$ The polynomial can be described by $$F(x_1,x_2,x_3,x_4)=0\iff (x_1+x_2)=0\mbox{ or }(x_3+x_4)=0$$ Is there a way to formally derive ...
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2answers
96 views

Transform polygons into one another?

I am aware that there must be no standard way to achieve this, but I don't know what has been done so far. I feel like I'm missing keywords to investigate further. I have any two 2D polygons $a$ and ...
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33 views

Is absolute continuity enough for uniform convergence of Chebyshev interpolation

Wikipedia says For every absolutely continuous function on [−1, 1] the sequence of interpolating polynomials constructed on Chebyshev nodes converges to f(x) uniformly. $^{[\text{citation ...
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3answers
157 views

How to create a computationally cheap function passing through given points?

I am trying to develop a function which goes through the follow points. The function will be calculated on a microprocessor which has 20 mHz. List of given points: ...
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82 views

Lagrange interpolating polynomial

How can one find $$ L[x_0,x_1,..,x_n;\frac{1}{x+a}]?$$ The original problem asks for $$ L[x_0,x_1,..,x_n;\frac{x^{n+1}}{x\pm 1}]$$ I know there is a formula for $[x_0,x_1,..,x_n,\frac{f(x)}{a-x}]$, ...
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1answer
47 views

Find cubic Bézier control points given four points

What I need is to generate an SVG file while having a series of (x,y) ready. P0(x0,y0) P1(x1,y1) P2(x2,y2) P3(x3,y3) P4(x4,y4) P5(x5,y5) ... I need to make a ...
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16 views

Flipping X, Y Known Values with Result Values; Table Data and Linear Interpolation

I am not knowledgeable in the terminology I need to be searching for to accomplish what I need in Excel. I have the following table of values which gives me the resulting RPM if I know the Pressure ...
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1answer
85 views

Polynomials with specified ranges in intervals

Say I have two finite intervals $[a,b],[c,d]\subsetneq\Bbb R$ where $a<b<c-1<c<d$ and $b-a=d-c=s<1$. I want to find a polynomial $f \in \Bbb R[x]$ such that $$\forall x\in[a,b],\mbox{ ...
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1answer
16 views

Meaning of indices for cubic hermite splines

While digging through some code about Perlin noise, I noticed, that a Cubic Hermite Interpolation polynome is used at some point. At this point, I wanted to know, which of the Hermite basis ...
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1answer
39 views

Surface fitting to a mesh grid of data points

I wonder if there is a technique for fitting a surface to a given mesh grid of data points? I've seen interpolating a polynomial to $2$D data, but not $3$D. E.g. say I was given the matrix $$ ...
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2answers
47 views

How to smooth a list of angles.

I'm not a math guy so maybe there is a super simple thing that my eyes cannot see. And sorry if my math terminology is not good at all. Please address me the right math terminology to use because ...
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3answers
48 views

Polynomial interpolation

I need to find the polynomial of degree 3 with respect to these conditions: $ p(0) = 1 $ $ p(1) = -1 $ $ p'(0) = 1 $ $ p''(0) = 0 $ How do I deal with the condition on the second derivative?
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1answer
69 views

Interpolation of polynomials

let $f(x)=2^x$ and $x_0=1$, $x_1=2$, $x_2=3$. Use divided differences to compute the interpolation polynomial $P(x)$ satisfying $P(x_i)=f(x_i)$, i=0,1,2 and $P'(x_1)=f'(x_1)$ and estimate error ...
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30 views

Lagrangian interpolation

I have a high-school student doing a modeling project using interpolation. (Hopefully someone understands what I mean without having to write out explicit examples, as I only have screenshots.) When ...
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1answer
23 views

Functions for interpolation

Do we always need to be given points to do interpolation? Or can we be given only a function? For lagrangian interpolation we require points, and does it apply for others also?
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1answer
16 views

Orders of data in Divided Differences and Lagrangian Interpolation

As we know that the order of data points i.e. x values do not matter in Divided Differences and The Lagrangian Interpolation. Why is that? What happens if we arrange them in order? better ...
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21 views

Neville's Method for Approximation

What do the numbers mean in Neville's method? Neville's method performs interpolation of numbers of the previous values, but what do the numbers mean?