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

calculation wave function

I have a bunch of points from a segment (~1.5 periods) of a wave. The wave looks like a cosinus wave, but it isn't. The length between the left maximum and the minimum is shorter than the length ...
2
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1answer
218 views

Compound interest -like calculation, but with increasing rate

Let's say an organization has 100 employees in the beginning of 2013 and grows to 110 employees by the beginning of 2014. This implies a growth rate of 10% for 2013. Now, let's say a hiring manager ...
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1answer
66 views

Linear interpolation of rotations

To linearly interpolate between two points $p_1$ and $p_2$ in 3-space, I can calculate: $p_t = p_1 + t(p_2-p_1)$ where $t$ is a parameter $0 <= t <= 1$ Is there any representation of rotation ...
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1answer
68 views

Evaluation of a Function with Lagrangian Finite Element

Suppose we want to evaluate a function, derived through linear Lagrangian finite element, in a point which is not one of its nodes. Is a simple linear interpolation equal to the correct evaluation of ...
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0answers
67 views

What do quadratic smoothing splines minimize?

Cubic smoothing splines minimize a combination of Interpolation cost and Smoothness (roughness) cost: $\qquad$ min Icost + $\lambda$ Scost where $\qquad$ Icost $\equiv \sum (Y_i - \mu(x_i))^2$ ...
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2answers
451 views

Spline interpolation versus polynomial interpolation

What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?
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1answer
54 views

$C^2$ Smoothing of absolute value

I am looking for a function $f\colon \mathbb{R}\to\mathbb{R}$ such that $$g(x)=\begin{cases} f(x), &\text{if }| x|<k \\ | x|, &\text{otherwise}\end{cases}$$ is $C^2$ or $C^\infty$ (at ...
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1answer
103 views

How to find a surface from two lines?

sorry if this is a basic problem but I don't know where to start looking. Imagine two perpendicular lines ("profiles") in a "$T$" spatial arrangement. The lines are arbitrary (empirical functions ...
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0answers
249 views

What is the maximum overshoot of interpolating splines in $d$ dimensions?

Consider cubic splines $s( x, y )$ which interpolate values $y = \{ y_0, y_1, \dots,y_n \}$, on the uniform grid $\{ 0, 1,\dots, n \}$. Fix $s''(0) = s''(n) = 0$ (natural splines). How big can ...
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1answer
105 views

Standard Interpolation between Bochner spaces

I've read the following in a few papers: Given: Let $\Omega \subset R^d$ be a Lipschitz domain. A sequence $f_n$ converges strongly to $f$ in $L^2(0,T;L^2(\Omega))$ and weakly in ...
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0answers
61 views

How do you interpolate the local maxima of a set of points in more than 3 dimensions?

I have a set of about 400 points each with 6 coordinates and one scalar value. How can I find out where the local maxima are?
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1answer
212 views

Calculating Log-likelihood using Raphson and Jacobian matrices?

I am reading the following paper: http://www.ntuzov.com/Nik_Site/Niks_files/Research/papers/stat_arb/Ahmed_2009.pdf and in particular page 13. I want to try and calculate lambda_t(p) = exp^(Beta^T ...
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1answer
325 views

Spline Interpolation

I have four questions about splines. Any help would be greatly appreciated. 1) Boundary conditions for cubic spline interpolation to a set of data $a=x{}_{1}<x2<...<x_{m} , $ like for ...
3
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1answer
552 views

Is this algorithm an example of exponential interpolation?

We have an algorithm I'm trying to get my head around. The original author is gone and away and the whole thing seems to generally work, but I'd like to verify that it's working correctly. (And ...
1
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0answers
27 views

Interpolation of linear operators

If $T$ is a bounded linear operator from $L^{p_1}$ into a homogeneous Lipschitz space of order, say $\lambda.$ Further if $T$ is also bounded from $L^{p_2}$ into $L^{q}$ for some $p_1,p_2,$ and ...
2
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1answer
157 views

How do I apply the Fast Multipole Method to Thin Plate Spline interpolation?

I have n scattered measurements of elevation, z, as a function of x and y coordinate: $ \{(x_i,y_i,z_i)\}_{i=1..n}$ that I want to interpolate so that I find z(x,y) for all x and y. Using Thin Plate ...
3
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2answers
130 views

How to fit a function that depends on several nominal and one real variable?

I have data that map several nominal variables and one real parameter into a real value. For example: ...
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1answer
186 views

Interpolating missing points in 3D data-set

Given the following x,y,z points (z is actually a signal strength indicator in dBm): ...
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1answer
119 views

How can I add a curve to my otherwise linear values?

I have built an interactive map for the Web that transitions smoothly from lon/lat point to lon/lat point. The duration of the transition is calculated dynamically and depends on the distance between ...
2
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0answers
54 views

Smooth detour from one function to another

Suppose I am given two smooth functions $f$ and $g$ on the real line and real numbers $a<b$ such that $f(a)<g(b)$ and $f'(a),g'(b)\ge0$ I want to get a smooth $H:\mathbb R\rightarrow\mathbb R$ ...
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2answers
44 views

Interpolation in 2-D co-ordinate system

Suppose there is a function f defined on (x, y) such that x, y $\in$ (-$\infty$, $\infty$), but function is not known. Let n data points are given f($X_0$, $Y_0$) = $Z_0$ f($X_1$, $Y_1$) = $Z_1$ ...
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3answers
191 views

How to understand and create quaternions?

I have to multiply two quaternions to calculate a so called spherical linear interpolation between two $R^3$ coordinate systems within the interval $t = [0, 1]$. I understand how to do the ...
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1answer
844 views

Minimum surface Curvature Interpolation Method

In this paper about Interpolation Methods, I am trying to learn Minimum curvature method. I have not done partial differential equations before; hence I am finding it tough to penetrate through this ...
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2answers
143 views

What's the best way to calculate all of the points for a curve given only a few points?

I've been reading up on curves, polynomials, splines, knots, etc., and I could definitely use some help. (I'm writing open source code, if that makes a difference.) Given two end points and any ...
2
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0answers
20 views

Analogue of Helly’s theorem for non-exact interpolation

Let $\overrightarrow{x}=(x_1,x_2, \ldots ,x_n),\overrightarrow{a}=(a_1,a_2, \ldots ,a_n)$ and $\overrightarrow{b}=(b_1,b_2, \ldots ,b_n)$ be vectors in ${\mathbb R}^n$, with $a_k \leq b_k$ for every ...
2
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2answers
255 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
45 views

interpolation and Vandermonde

Looking at a problem of interpolation, I find a Vandermonde type matrix. To be precise I consider the following, let $$A(z)= \sum_{i=1}^p \sum_{j=2}^{n_i+1}\frac{a_{i}^j}{(z-z_i)^j}$$ where the $z_i$ ...
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0answers
172 views

Best method of interpolation?

I am learning different interpolation methods, and their pros and cons. Which interpolation method do you think is the best for practical use? If you can give me links to research papers about various ...
2
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2answers
55 views

Converting data to a specified range

I am trying to convert data to a particular range of 0-10. Actual Data may vary from - 50000 - 26214400. I have broken this down in to 4 parts as follows - ...
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0answers
75 views

Quaternion barycentric interpolation

Let's say that i have a set of quaternions, each representing a 3-angle orientation. And with each quaternion is associated a real value (let's say a speed value for explanation's sake). Now with an ...
3
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0answers
108 views

Interpolation method

I am researching about different interpolation methods, and their pros and cons. Please help me to understand ideas behind Gaussian interpolation method. Although, I looked at different papers for ...
4
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1answer
84 views

Interpolation between iterated logarithms

I am investigating the family of functions $$\log_{(n)}(x):=\log\circ \cdots \circ \log(x)$$ Is there a known smooth interpolation function $H(\alpha, x)$ such that $H(n,x)=\log_{(n)}(x)$ for ...
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1answer
385 views

Thin Plate Spline interpolation of scattered $z(x,y)$ data

I am trying to understand Thin Plate Spline interpolation of scattered data. As I understand it TPS is just a special case of Radial Basis Function interpolation: $$ z(x,y) = p(x,y) + \sum_i ...
2
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0answers
104 views

monotonic smoothing fit to be implemented (in python or other language)

In a post that already exists, implementation-of-monotone-cubic-interpolation, there is a good method for fitting data which necessarily includes all of the given points. But, what if I need to ...
2
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4answers
109 views

Function generation by input $y$ and $x$ values

I wonder if there are such tools, that can output function formulas that match input conditions. Lets say I will make input like that: $y=0, x=0$ $y=1, x=1$ $y=2, x=4$ and tool should ...
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2answers
193 views

Why are there problems when interpolating $f(x)=\arctan(10x)$?

Given $f(x)=\arctan(10x)$, there would be a problem when we interpolate it by using Lagrange's method. This would have something to do with the derivatives of $f(x)$. I plotted some derivatives of $f$ ...
2
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0answers
73 views

Minimal surface representation from a 3D contour

I have a set of 3D points defining a 3D contour, as shown below. The points in this contour lie in their best-fit plane and I want to obtain a 3D triangular mesh representation of the surface inside ...
2
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0answers
102 views

Ellipse radius interpolation with different radiuses

I am writing a library for graphical LCDs and I want to incorporate a function to draw a circle on the screen. I have already succeeded in drawing simple circles, however, I want to be able to pass a ...
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1answer
235 views

Coefficients of Newton interpolation polynomial

Given distinct $y_0,...,y_m$ in $\mathbb R$, let $N_m(x)$ be the Newton interpolation polynomial of degree $m$. That is, $N_m(x) = \sum_{n=0}^{m}a_nw_n(x)$ where $w_0 = 1$, $w_n(x) = ...
3
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1answer
197 views

Curve through four points — simple algebra??

The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms. Suppose I want to ...
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2answers
499 views

Forcing Bezier Interpolation

I found this very informative site that discusses forcing bezier interpolation and the site gives formulae for calculating the control points so that the curve goes through a set of four points, y0, ...
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4answers
398 views

Proof of Schur's test via Young's inequality

I am able to prove the following generalization of Schur's test using the Riesz-Thorin interpolation theorem, however I have been stuck for days now trying to prove it using Young's inequality: Let ...
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0answers
195 views

Multigrid Interpolation and Restriction operators

I have a question about the restriction and the interpolation operators of a Multigrid algorithm. Let those be given: The full weighting restriction stencil (in 2D): $\frac{1}{16} \left[ ...
5
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0answers
218 views

Runge's phenomen: interpolation error using Chebyshev nodes oscillates

We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
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0answers
159 views

Non-linear interpolation. (1D Perlin Noise)

In this document (http://webstaff.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf) about Perlin (and Simplex) Noise you can find an explanation about 1D Perlin Noise (at the top). ...
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0answers
157 views

Interpolate to 3D functions

Hy, In math classes, I've learned that given some points in 2D space: a(1,2), b(7,3), c(8,5),... You can find an equation that goes through these points (using interpolation). Now I was wondering if ...
2
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1answer
61 views

Interpolation with degree restriction

(using $f[x_1, ... , x_n]$ to denote the forward difference operator) I have a polynomial $P(x)$ interpolating $5$ points $x_0, ... , x_4$ and $2$ derivative values $x_0, x_3$ across an evenly spaced ...
1
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1answer
61 views

Extension of Fourier Transform

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
2
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1answer
103 views

Nontrivial expansion of a multivariate power series in form of a single variable series?

I am trying to interpolate a function defined over a three-dimensional real space: $$f: R^3\rightarrow R\\(x,y,z)\rightarrow f(x,y,z)$$ Let assume I have $N_1 N_2 N_3$ points in the space which form ...
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0answers
22 views

Is there a way to estimate the range of fitting coefficients from only the data?

Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain $f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$ where ...