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
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 + ...
1
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1answer
92 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 ...
2
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0answers
39 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 ...
0
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1answer
603 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 $(x_0,y_0),(x_1,y_1),\dots,(x_{N-1},y_{N-1}) \in \mathbb{R}^2$ ...
0
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1answer
45 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 $(1, P(1)), (2, P(2)), ..., (11, P(11))$. ...
2
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1answer
84 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 ...
0
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1answer
417 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 ...
0
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1answer
118 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 ...
0
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1answer
118 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 ...
1
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0answers
44 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 ...
0
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0answers
100 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 ...
1
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1answer
161 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, ...
0
votes
1answer
221 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 ...
1
vote
2answers
348 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 ...
0
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1answer
238 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 ...
1
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4answers
709 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 ...
8
votes
6answers
2k views

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 ...
0
votes
1answer
149 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 ...
1
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1answer
408 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 ...
0
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0answers
70 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 ...
0
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1answer
45 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 ...
1
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0answers
222 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 ...
2
votes
1answer
141 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 ...
0
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0answers
35 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
votes
1answer
237 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 ...
0
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1answer
69 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 ...
0
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1answer
69 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 ...
1
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0answers
71 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 $\equiv \sum (Y_i - \mu(x_i))^2$ ...
3
votes
2answers
491 views

Spline interpolation versus polynomial interpolation

What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?
0
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1answer
55 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 ...
1
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1answer
107 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 ...
4
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0answers
272 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 ...
1
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1answer
109 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 ...
1
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0answers
64 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?
0
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1answer
222 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 ...
0
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1answer
333 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
votes
1answer
629 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 ...
1
vote
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 ...
2
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1answer
171 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
votes
2answers
132 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: ...
1
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1answer
200 views

Interpolating missing points in 3D data-set

Given the following x,y,z points (z is actually a signal strength indicator in dBm): ...
3
votes
1answer
123 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
votes
0answers
55 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 $H:\mathbb R\rightarrow\mathbb R$ ...
1
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2answers
47 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($X_1$, $Y_1$) = $Z_1$ ...
5
votes
3answers
200 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 ...
0
votes
1answer
925 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 ...
1
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2answers
147 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
261 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 ...