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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1answer
9 views

Natural cubic spline interpolation - check and suggest better way

I was given the following interpolation nodes: $(0,10),(\frac{1}{2},8),(1,5),(2,2),(3,1)$ and I was asked to find the natural cubic spline interpolation between every 2 points. I want to show you ...
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0answers
27 views

Why do we interpolate - no guarantee of success

this is somewhat of a general question about interpolation, I don't fully understand how can we be confident that our approximation is good, even if we know a lot of points. An example would be: ...
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0answers
19 views

interpolation of the function [on hold]

Suppose u is any function that interpolates f at x_{0},x_{1},\ldots,x_{n-1} and v is a function that interpolates f at x_{1},x_{2},\ldots,x_{n}. Consider the function ...
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0answers
9 views

Are there some scattered point configurations that would yield bad interpolation results using Radial Basis Function (RBF) interpolation?

Is Radial Basis Function interpolation sensible to the scattered point configuration? I seem to be having problems for scattered points $(x_i,y_i)$ that are illustrated below: The values $f(x,y)$ ...
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0answers
28 views

Runge's Phenomenon [on hold]

" Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation ...
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3answers
284 views

The Weierstrass Approximation Theorem Vs The Runge's Phenomenon

I am learning about different interpolation methods in my internship. Today as I was looking this article on Wikipedia to learn about the Runge's Phenomenon exhibited by Polynomial Interpolation. I ...
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0answers
21 views

Taylor's remainder in a compact

My impression is that each function enough regular ($C^\infty$ ) in a compact is equivalent to a polynomial. Is this true? Is there a way to prove it? The expression of the Taylor's remainder just ...
2
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1answer
42 views

Accurate floating-point linear interpolation

I want to perform a simple linear interpolation between $A$ and $B$ (which are binary floating-point values) using floating-point math with IEEE-754 rounding rules, as accurately as possible. Please ...
2
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0answers
20 views

Convex interpolation between two points with given derivatives

Let's say I have two real values $x_1$ and $x_2$, to each of which I associate $y_i$ and $y'_i$ satisfying $$ (y_2-y_1)(y'_2 - y'_1) \geq 0. \tag{1} $$ I would like to find a polynomial ...
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0answers
4 views

Are there other names for multilayer perceptrons or multidimensional interpolants based on Kolmogorov's approximation work?

Are there other names for multilayer perceptrons that are used outside of the neural net community? At its core, multilayer perceptrons form a multidimensional interpolant of the form $$ ...
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1answer
17 views

Different forms of a quadrature

I am solving the following problem: Find the quadrature of the following form: $Q(f) = Af(−1) + Bf(0) + > Cf(1)$, which has the highest degree and interpolates the integral: $\int_{-3}^{3} ...
2
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1answer
516 views

Akima spline interpolation

I want to use Akima interpolation on series of points. I have those points in 3D [x, y, z]. But in all resources, I found, there is only f(x) and x (so [x,y]). In Natrual Cubic Spline I am using this ...
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1answer
27 views

Finding an upper bound on the polynomial interpolation error for $\cos x$ [closed]

Hey guys I need to solve this problem can you help me please: Let $f(x)=\cos(x)$ on $[0,\pi]$ and $P_n(x)$ be an approximating polynomial with degree at most $n$ to $f(x)$. Assume ...
0
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1answer
34 views

Finding an interpolating polynomial [closed]

Hey guys I need to solve this problem but I am not sure how to do: Find an interpolating polynomial $$P(x)$$ with degree at most 2 such that $$P(x_0)=y_1$$ $$P'(x_0)=y_2$$ $$P'(x_1)=y_3$$ And ...
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1answer
15 views

Maximum of error in equidistant interpolation

Interpolating a function, to estimate the error, knowledge of the function $$\omega(x)=\prod_{i=0}^k (x-x_i)$$ with $x_i$ being the sampling points, is required. In the equidistant case, this would ...
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0answers
26 views

Is there a way to do exponential interpolation?

Is there a way to assign a set of values to an exponential function just like polynomial interpolation?
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0answers
13 views

Find the polynomial interpolating function $f(x)=\cos\left(\frac{\pi}{2}x\right)$ [closed]

Find the polynomial interpolating function $f(x)=\cos\left(\frac{\pi}{2}x\right)$ at points: $\{-1,0,1,2\}$ Write this polynomial as Lagrange, Newton and power polynomial.
1
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1answer
24 views

Finding polynomial optimal in terms of least squares approximation

Find polynomial $w$ of degree at most $2$ optimal in terms least squares approximation for a function $f(x)=x^3$ in the norm $\|g\|=\sqrt{(g,g)}$, given that: $$ (f,g) = \int\limits^1_0 f(x)g(x)dx. ...
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1answer
15 views

Collective term for interpolation and extrapolation

Is there a collective term for both interpolation and extrapolation? If there is such a term, what is it?
1
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1answer
252 views

Wolfram Mathematica - Newton Backward Interpolation?

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
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1answer
20 views

How can missing data be organised or classified (Interpolation vs Approximation)?

I'm looking for a way to distinguish between the various types of missing data techniques? Can someone help to clarify or organize these categories in sub-sections or indicate similarities or ...
4
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1answer
155 views

Integer valued polynomial through some known points

I have 2 questions, but I'll put both of them here since they are closely related: An integer valued polynomials $P(x)$ is a polynomial whose value $P(n)\in\mathbb{N}$ for every $n\in\mathbb{N}$. ...
1
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1answer
449 views

How to evaluate Newton's Divided Difference Polynomial in MatLab with an unknown degree?

I already have the code that finds the coefficients for the polynomial, but how do you find a value for the polynomial if given an x coordinate in MatLab code?
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2answers
111 views

Intepolate from linear to step function, and one application for shading colors

I'm running after a particular function $f_\sigma : [−1,+1] \rightarrow [-1,+1]$ that could take three different forms depending on the value of its parameter $\sigma$. Could anyone help me ...
2
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1answer
837 views

Interpolation of a logarithmic function

I have a logarithmic function $$m \ln(x) + b$$ And three points $$(x_0, y_0), (x_1, y_1), (x_2, y_2)$$ The task is to find $m$ and $b$. Do I understand right that the third point is redundant? ...
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2answers
2k views

How to determine function based on input and output

I'm not very good at mathematics, so please bear with me. How can you determine / define a function based on sets of values of its input and output parameters. You have: $f(x_{1_1}, x_{1_2}, \ldots ...
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0answers
31 views

How to find a graph's equation from its points

I have a set of data that constitutes the graph on the picture. What I want to know is how would I find the equation equivalent to that kind of graph? The X are on the interval $[1,10]$.
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1answer
272 views

Reconstruct Control points in a Bézier Curve?

I have a curve that I know is a (non-periodic) Cubic Bézier Curve (because I constructed it as such). I stored each ordered pair in the curve, but not the control points. Is it mathematically ...
4
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2answers
142 views

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal ...
4
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1answer
120 views

Interpolation inequality on Holder space

Let $0< \beta < \gamma <1$. Show that the interpolation inequality holds. $$||U||_{C^{0,\gamma}(U)} \le ||U||^{\frac{1-\gamma}{1-\beta}}_{C^{0,\beta}(U)} ...
0
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1answer
224 views

How to find B-Spline represenation of an Akima spline?

Given points $t_i$ and values $y_i$, I'd like to use Akima interpolation to interpolate to a different set of locations $x_j$. This means I need to calculate the cubic polynomials $A_{3,t}(x)$. Given ...
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1answer
233 views

Spline with varing tension, selection of tension factor

I need to perform a special interpolation, using that kind of basis : $$\varphi_{i,j}(x) = a_i + b_ix + c_i(\cosh(\tau\ x) - 1) + d_i(\sinh(\tau\ x) - \tau\ x)$$ where the $a_i$, $b_i$, $c_i$ and ...
1
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1answer
42 views

How to fit a formula to three data points?

I need a very basic formula that will be used to determine a CSS line-height based on a provided font pixel size. So in essence, I need the formula to covert ...
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2answers
32 views

Find a formula that describes the given day-price relation

Given data: Days Prices 0 0 1 1000 2 1000 3 1000 4 1600 5 2200 6 2800 7 3400 8 4000 9 4600 10 5200 Price is function of day. I had my formula but always put negative ...
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0answers
23 views

Relationship between Lagrange interpolation and Taylor expansion

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h_{-1}$ and $x_{1} = x_0 + h_1$ with $h_1, h_{-1}$ > 0. Given a smooth function f, and an approximation to $f'(x_0)$ given by the ...
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0answers
14 views

Interpolating graph path

I have a directed graph with very few nodes, and I would like to add many more redundant nodes between them. So if I have A->B->C, I'd like to get A->A0->A1->A2->B->B0->B1-B2->C The intermediate ...
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24 views

Blended surface

Partially blended surfaces are extensively used in the literature for shape preserving interpolation. Most of these shape preserving partially blended surface interpolation is based on the result that ...
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2answers
223 views

General method to “naturally interpolate” to a complex map?

Given a region of the complex plane and a map $z \to f(z)$, is there a general way to "naturally interpolate" the point $z$ to $f(z)$ in such a way that the movement follows a "natural" smooth path ...
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0answers
25 views

Interpolation point selection for Rational Polynomial Interpolation

people, 1st time on math.stackexchange so aloha to all!! The Question: I have a certain data set and I am using Thiele's rational Polynomial Interpolation to interpolate some data but the curve will ...
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1answer
63 views

Calculating a cubic spline goes wrong

I am trying to solve a old exam and really stuck at the cubic splines. We have the function $f(x) = \cos^2(\frac{x}{2})$ and the points $x_0 = \frac{\pi}{2}$, $x_1=0$ and $x_2 = \frac{\pi}{2}$. ...
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0answers
16 views

estimate an upper bound for the error of an interpolation polynom

The task is to estimate the error of an interpolation polynom $p(x)$ to an function $f(x)$. The sampling points are $x_0 = -1,\ x_1= 0,\ x_2=1,\ x_3=3$ So i already calculated the polynom which does ...
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1answer
35 views

Formula to link two exponential values together - doesn't quite work

Basically, I've done a script, and I'm stuck on a formula for it. After I run the code on a cube, based on two different inputs (detail level and vertex average iterations), the resulting size will be ...
0
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1answer
28 views

Product of Chebyshev polynomials of the second kind?

So Wikipedia has this formula for a product of two Chebyshev polynomials of the second kind evaluated at a fixed $x$ with different indices: $$ U_n(x)U_m(x)=\sum_{k=o}^{n}U_{m-n+2k}(x) $$ Which would ...
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2answers
3k views

Newton's Interpolation Formula: Difference between the forward and the backward formula

I was taught that the forward formula should be used when calculating the value of a point near $x_0$ and the backward one when calculating near $x_n$. However, the interpolation polynomial is unique, ...
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30 views

Find the hermite interpolating polynomial

$$\begin{array}{ccc}x&f(x)&f'(x)&f''(x)\\0&1&\frac12&0\\1&2&1&-\end{array}$$ Find the interpolating polynom using divided difference table with the given ...
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0answers
14 views

Changing the order of the elements of the divided difference Polynomial Interpolation

Apparently this is rather trivial but I don't understand why what I've highlighted in green is correct.
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3answers
31 views

what is difference between numerical integration and interpolation?

I am studying finite element method.While studying i am confuse with numerical integration and interpolation.Is this two methods are same or different?. If they are different then is there any ...
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0answers
14 views

Fast way to compute barycentric lagrange interpolation

Is there any fast way to compute the barycentric Lagrange interpolation using matlab?
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3answers
102 views

Lagrange Interpolation Theorem?

The polynomials $p(x) = 5x^3 - 27x^2 + 45x - 21$ and $q(x) = x^4 - 5x^3 + 8x^2 - 5x + 3$ both interpolate the points $(1,2) , (2,1) , (3,6), (4,47)$. Even though these polynomials are of different ...
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1answer
33 views

Is there a closed form solution for slope lines of bilinear function?

Given a bilinear function $f(x,y) = a + bx + cy + dxy$, is there a closed form solution for a slope line passing through point $(x_0, y_0, f(x_0, y_0))$? It can exclude degenerate cases, e.g. $b = c = ...