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
28 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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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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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
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 ...
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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 ...
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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.
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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?
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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 ...
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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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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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0answers
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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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 ...
8
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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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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 ...
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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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 ...
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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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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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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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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 ...
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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 = ...
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1answer
49 views

How obtain a (accurate) function from this graph with these points?

I need obtain the function from 0 to 20 from this graph: I have the even numbers in the {x, f(x)} format: {0, 0}, {2, 1.8}, {4, 2}, {6, 4}, {8,4}, {10,6}, {12,4}, {14,3.6},{16,3.4}, {18,2.8}, ...
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0answers
37 views

How can I cleverly use the error term of polynomial interpolation?

Let $f(x):=x^2$. We're interested in the closed form of the error $|I(f)-T_n(f)|$ where ...
2
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1answer
51 views

Is this interpolation, does it have a name?

I was waching Signle Variable Calculus MIT lectures (I stop on 9 about linear approximation) I was also learn interpolation at my university and I thought that I'll create my own equation for ...
0
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1answer
26 views

How to calculate the length of a cubic hermite spline between two points

I am using the following equation to create a cubic hermite spline: $$p_n(t) = a_nt^3+b_nt^2+c_nt+d_n$$ $$1\geq t\geq 0$$ $p_n(t)$ is the unit interval interpolation equation for dimension n. $t$ is ...
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2answers
35 views

using lagrange multipliers to fit a curve through a point

So this is part math/ part statistics. I have a set of data I'm fitting a 2nd order curve through using least squares method (matrix form). However, I've been given the requirement to pass the curve ...
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0answers
26 views

How to interpolate and obtain this graph?

I was interpolating with software mathematica v9, but how can I obtain a curve similar to this?, in a easy way
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0answers
21 views

How to calculate a bezier curve given derivative of endpoints, location of endpoints, and points on the curve?

I know how to calculate a hermite spline, which has known derivatives and locations for each point, and I know how to calculate bezier curves which go through certain points, but I need to be able to ...
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0answers
31 views

How can I generate a spline with a maximum second derivative without specifying first derivative for mid points?

I've done interpolation before with bezier splines and cubic splines, but I need to find a way to limit the second derivative throughout the curve so that there is a limit to how sharp the corner can ...
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0answers
38 views

NURBS surface fitting for a closed region on mesh

I'm developing a tool that allows users to select a closed boundary (a polygon) on the triangle mesh and then from this boundary, generate a NURBS surface fitting the original mesh surface. My idea ...
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2answers
80 views

Approximate formula for the series: $\sum_{k=1}^{+\infty}\dfrac{k^x}{(k!)^x}$

I found that this series: $$S(x)=\sum_{k=1}^{+\infty}\dfrac{k^x}{(k!)^x}$$ can be very well approximated in this way: $$S(x)=\dfrac{1}{\left(a+b\exp(cx)\right)^d}$$ with: $a=0.1876$, $b=-0.1895$, ...
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0answers
29 views

Derivation of composite Gaussian quadrature error formula

I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following: Derive the error term for the composite Gaussian quadrature rule with ...
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0answers
14 views

How to find the tangents for a spiral geometry at each point accurately

I have been working on a progam that generates spirals from contours that have been formed by slicing a surface by various planes along its height.The contours are a collection of linear line ...
0
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1answer
26 views

The cubic interpolation

I try to understand the cubic interpolation for my studies. The following website says " (1) The four equations above can be rewritten to this (2):" but how? Can anyone explain me the the necessary ...
3
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1answer
28 views

Optimal way to find derivative - numerically

Suppose we are given points $x_0,x_1,x_2$ evenly spaced points $(x_0-x_1=x_1-x_2)$, and $u(x_1),u(x_2),u(x_3)$ Where $u$ is some function. Find the best way to approximate $u''(x)$ using only the ...
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1answer
32 views

Integration Rule Exact Degree

Given the integration rule $Q(x) = \alpha_1f(0)+\alpha_2f(1)+\alpha_3f'(0)$ for interpolating the integral $\int_0^1f(x) dx$ , I need to find $\alpha_1,\alpha_2,\alpha_3$ values s.t Q has exact degree ...