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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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
30 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
31 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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0answers
24 views

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
37 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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0answers
48 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
44 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?
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1answer
23 views

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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0answers
96 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 ...
2
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0answers
108 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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1answer
25 views

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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0answers
10 views

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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0answers
14 views

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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0answers
29 views

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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0answers
34 views

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
41 views

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
602 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
29 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
41 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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1answer
51 views

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 ...
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1answer
77 views

Natural Cubic Spline Unequal Spacing

I'm currently trying to create a natural cubic spline by setting up linear equations to solve for the coefficients of the spline. Where the spline follows as : ...
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0answers
24 views

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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0answers
20 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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0answers
26 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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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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0answers
38 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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0answers
38 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
109 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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1answer
63 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
47 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
113 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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0answers
21 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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0answers
49 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
240 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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1answer
64 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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0answers
34 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
70 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
37 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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0answers
25 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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0answers
30 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
51 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
117 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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0answers
62 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
48 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 ...