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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345 views

Derive the error term of Basic Corrected trapezoidal Rule for a Cubic Hermite Polynomial

The basic trapezoidal rule for approximating $I_f = \int_{a}^{b}f(x)dx$ is based on linear interpolation of $f$ at $x_0=a$ and $x_1 = b$. The Simpson rule is likewise based on quadratic polynomial ...
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37 views

Spline interpolation problem akin to Bezier spline

Given three pairwise distinct points $p_1, p_2, p_3 \in \mathbb{R}^2$, I'd like to find a function $f: \mathbb{R} \to \mathbb{R}^2$ with at least $f \in C^1$ such that $f(0) = p_1, f(1) = p_3, f'(1) ...
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26 views

Parametric Cubic Spline interpolation

I'm trying trying to interpolate a hyper-surface from a n dimensional set of points using parametric interpolation . Now i see that there's a derivative term at each interior point of the curve that ...
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0answers
16 views

Interpolation of a function given two sets

$P$ and $Q$ are sets of $k+1$ points. $P\bigcap Q$ has $k$ points. $p(x)$ in $\mathbb{P}_k$ interpolations $f(x)$ at points of $P$. $q(x)$ in $\mathbb{P}_k$ interpolations $f(x)$ at points of $Q$. Let ...
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41 views

Tile a spline with bricks

I have a series of identically sized virtual Lego bricks. I need to "tile" these bricks along the length of a Hermite spline to create a curved road. The spline is two-dimensional. E.g. the curve only ...
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12 views

Show, that $ \displaystyle \sum_{k=0}^n \lambda_k(0)x_k^j = \begin{cases} 1 \ (j=0)\\0 \ (j=1,2,\ldots n)\end{cases} $

Show, that $ \displaystyle \sum_{k=0}^n \lambda_k(0)x_k^j = \begin{cases} 1 \ (j=0)\\0 \ (j=1,2,\ldots n)\end{cases}$, where $\lambda_{k}$ is the 'helper' polynomial from Langrange Interpolation ...
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15 views

Scaling range question

Let's say I have a random number $x$ between the values of $A$ and $B$. I now want to scale the value of $x$ so that it is within the range of $C$ and $D$ where $A \le C \le D \le B$. The new value of ...
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1answer
391 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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1answer
267 views

Newton's method for polynomial interpolation

I've seen that in Newton's method for interpolating polynomials, the coefficients can be found algorithmically using (in Python-ish): ...
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1answer
63 views

Error term for a cubic interpolation

I have a question on one interpolation problem. The problem is below. For the given points, $x_0 = -1, x_1 = 0, x_2 = 3$ and $x_3 = 4,$ find the error term $e_3(\bar{x}) = f(\bar{x}) - p_3(\bar{x})$ ...
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0answers
132 views

Interpolation between curves

I am trying to do a 2D interpolation between two curves as you can see from the attached picture. where $a$ is a varrying paramter. $f_{a_1}$ and $f_{a_2}$ are known, or at least I can perform ...
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0answers
393 views

How to calculate a frequency curve from a sample of values, and interpolate between curves

I have a set of about 500 values, from which I'm currently plotting a histogram. I'd like to plot a frequency curve, i.e. go from the left two graphs to the rightmost on the below image borrowed from ...
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2answers
84 views

Simple Polynomial Interpolation Problem

Simple polynomial interpolation in two dimensions is not always possible. For example, suppose that the following data are to be represented by a polynomial of first degree in $x$ and $y$, ...
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1answer
89 views

Cubic splines on a grid

I trying to work out how to interpolate on a grid with cubic splines. Let the point at which I'm trying to interpolate be at {xp,yp}. At the moment I am fitting splines across the rows and then ...
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1answer
43 views

Piecewise linear interpolation

A really simple question - how does it work? Linear interpolation is quite easy and understandable: ...
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1answer
37 views

Cant understand how we got this equation

I was going through a tutorial that introduces cubic splines. A snapshot of the tutorial is as follows : Image begins: Image End: Now I dont understand how we got: $y_1''=6a_1(x_1-x_1)+2b_1=0 + ...
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1answer
82 views

Catmull-Rom blending functions

I have a non-uniform Catmull-Rom spline (so the $t_i$ parameter values are not uniformly distributed). Is there a simple way to calculate the blending functions of the control points? So the spline ...
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34 views

Exercise 1.3.3(c) of GTM249(Classical Fourier Analysis)

Exercise 1.3.3(c) Let $0<p_0<p<p_1<\infty$ and let $T$ be an operator as in Theorem 1.3.2($\|T(f)\|_{L^{p_0,\infty}(Y)}\leq A_0\|f\|_{L^{p_0}(X)}$ for all $f\in L^{p_0}(X)$ and ...
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1answer
444 views

How to calculate cubic spline coefficients from end slopes

I want to know how to calculate cubic spline interpolation coefficients, which uses end point slope constraint. There are $N$ points $(x_0,y_0),(x_1,y_1),\dots,(x_{N-1},y_{N-1}) \in \mathbb{R}^2$ ...
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1answer
41 views

Lagrange Interpolation — Challenge Problem

Say you're performing Lagrange Interpolation on a function $P(x)$ and you've found that $$ P(x) = \sum_{i = 1}^{11} \Delta_i(x) $$ given the eleven points $(1, P(1)), (2, P(2)), ..., (11, P(11))$. ...
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1answer
77 views

contour integral representation of lagrange interpolating formula

how can i show that if the interpolation nodes are complex numbers $a_1,a_2,...,a_n$ and lie in some domain $G$, bounded by a piecewise-smooth contour $\gamma$, and if $f$ is a single-valued analytic ...
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1answer
283 views

What is linear interpolation?

I am learning about linear interpolation however, we were not taught how to formally solve a problem using linear interpolation. A practice problem involving is the following: Find how long it will ...
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1answer
100 views

Spline interpolation on $n$ dimension

I'm trying to interpolate an $n$-dimensional function $f(x)$ where $x$ is a vector . Can I use spline interpolation for this interpolation using $x$ as an $n$-dimensional variable (vector) ? and ...
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1answer
89 views

Interpolating geographic coordinates

I have two geographic coordinates. Let's call them $A$ and $B$: A = latitude 41.34759, longitude -75.77415 B = latitude 41.34769, longitude -75.77404 My unknown ...
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39 views

Given a finite set of points construct a polynomial that meets the points.

Say I have a set of points in $\mathbb{Z}^3 \times \mathbb{Z}_2$ each of which represent part of a mapping $(z_1, z_2, z_3) \mapsto z_4 \in \mathbb{Z}_2$. How do I find the the simplest polynomial ...
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72 views

Notion of Linear Interpolation in Language Model

I am curios about Linear Interpolation (LI) in a sense of Evaluating Language Model in the course of Natural Language Processing (NLP). I review the material from the slides Natural Language ...
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1answer
146 views

Trigonometric interpolation

From http://en.wikipedia.org/wiki/Trigonometric_interpolation trigonometric interpolation can be calculated as follows: Now assume we have 6 data points (0, 0.1), (1, 0.3), (2, 0.4), (5, 0.3), (6, ...
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1answer
191 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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2answers
265 views

Piecewise interpolation with derivatives that is also twice differentiable

This question regards the issue of interpolation of one dimension real functions. If one has a finite set of function values and its corresponding derivatives, one could find unique continuous ...
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1answer
172 views

Implemented Cubic Spline is not smooth

I want to implement cubic spline, so I found and implemented this discription and this method for solving tridiagonal system. I'd like to draw without using any math libraries, so cubic spline is ...
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4answers
374 views

How do I move through an arc between two specific points?

I've found many answers to similar questions here, but I'm still stuck. I want to move an object from point sx,sy to point dx,dy through an arc that bulges by distance b from the line straight ...
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6answers
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What is the pattern to this sequence?

$$0, 1, 3, 13, 51, 205$$ More specifically, $$(0,0)\quad(1,1)\quad (2,3)\quad (3,13) \quad(4,51)\quad (5,205)$$ I have tried using the interpolation feature in Grapher.app and Wolfram Alpha, but ...
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1answer
133 views

1-D Interpolation between any number of points

I'm looking for an algorithm which will allow me to calculate outputs for any given input instead of just the few ones I'm given in advance. The input is a number from interval $0 - 100$ representing ...
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1answer
158 views

How can I find which function corresponds to a set of data points?

Suppose I have a set of data points like this: 1;1 2;4 3;9 4;16 5;25 6;36 ... The first column is the input of the function and the second one is the result. I ...
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0answers
61 views

Rational interpolation with integer coefficients

Given a function $f$ on an interval $(a,b)$ (or alternatively a set of data points ${(x_i,y_i)}$), is there an efficient algorithm to find a rational function of type $(m,n)$ with integer coefficients ...
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1answer
43 views

Rotation matrix for non-isometry transformation

Imagine that you have a sphere in $\mathbb{R}^3$ and a plane (that is parallel to the x,y plane) through the sphere. Now you want to have a clockwise rotation in the x/y plane that does the ...
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192 views

Shannon vs dirichlet kernel interpolation method for signal reconstruction

I am currently studying fourier transform, and especially the way that the signal could be reconstructed from its spectrum. In many lectures, I have seen the shannon interpolation method to ...
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1answer
120 views

Spline interpolation that is non-decreasing when given non-decreasing sequence

How can I achieve a spline interpolation such that when given non-decreasing sequence of points the resulting spline will also be a non-decreasing function (and vice-versa, when given non-increasing ...
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0answers
31 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 ...
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1answer
177 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
60 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
61 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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60 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
338 views

Spline interpolation versus polynomial interpolation

What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?
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46 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
98 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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187 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
91 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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57 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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176 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 ...