Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points.

learn more… | top users | synonyms

3
votes
1answer
63 views

DPLL Algorithm $ \rightarrow $ Resolution proof $ \rightarrow $ Craig Interpolation

I really need help here for an exam that I got tomorrow .. Let's say I got a bunch of constraints: $ c1 = { \lnot a \lor \lnot b } \\ c2 = { a \lor c } \\ c3 = { b \lor \lnot c } \\ c4 = { \lnot b ...
1
vote
0answers
23 views

$M(x)$ and $L(x)$ interpolate $f(x)$ on $n+1$ points. Show, that $f(x)$ lies between $L(x)$ and $M(x)$

We have $n+2$ points $x_0 \lt x_1 \lt x_2 ... \lt x_{n+1}$. We have two polynomials - $L$ and $M$. $L(x)$ interpolates $f(x)$ on points $x_0,...,x_n$ and $M(x)$ does so on $x_1,...,x_{n+1}$. The ...
0
votes
1answer
53 views

Intuitive proof of interpolation polynomial existence

Problem: Given a set of $n+1$ data points ($x_i, y_i$) where no two $x_i$ are the same, one is looking for a polynomial $p$ of degree at most $n$ with the property $p(x_i) = y_i$ for all $i∈ [0, n ...
0
votes
0answers
14 views

Parametric Surface Interpolation

I compute the set of states that can be achieved by a system defined as xdot=f(x(t),u(t)) with given initial states x(0)=x0, u is element of U and t element of [0,tf]. Result is a ...
0
votes
1answer
26 views

Show $\Delta^mp(x) = 0$ when $p(x) \in P_n$

Show that $\Delta^mp(x)= 0 $ when $p(x) \in \mathscr P_n$ and $m\ge n+1$, where $\mathscr P_n$ is the set of polynomial of degree $n$ and $\Delta^m$ is the operator for the $m$-th forward ...
0
votes
1answer
32 views

Proof that $f[x_{i0},…,x_{ik}] = f[x_0,…,x_k]$

i try to show, that $f[x_0,...x_k]$ is a symmetric function of $x_i$. What means, that for a permutation $x_{i0},...,x_{ik}$ of numbers $x_0, ...,x_k$ applies: $$f[x_{i0},...,x_{ik}] = ...
0
votes
1answer
176 views

How do I interpolate the derivative of a catmull-rom spline?

I am creating an implementation of a cubic hermite spline in Python. One feature I would like to add is a method to compute the slope (IE the derivative) for a given T value. Currently, I can do it ...
1
vote
1answer
90 views

Interpolation- Barycentric coefficients for nodes that are Chebyshev points of the second kind.

So I came across the following theorem: If the interpolation node are Chebyshev points of the second kind given by : $$ x_k=\cos \left( \frac{j\pi}{n}\right) \qquad ( 0 \leq j \leq 0) $$ Then the ...
0
votes
0answers
26 views

Polynomials of the form $g(x)=f[x,x_1,x_2,…,x_m]$.

Can anybody point me to some materials about polynomials of the form $g(x)=f[x,x_1,x_2,...,x_m]$, meaning that for given $x$ they will give back a leading coefficient of function $f$ interpolated at ...
1
vote
1answer
49 views

$p$-polynomial of $n$'th degree, $q(x)=p[x,x_1,x_2,…,x_k]$, prove that q has the same leading coefficient.

So I have a polynomial $p$ of $n$'th degree and q given by $q(x)=p[x,x_1,x_2,...,x_k]$, meaning that for $x$ it gives back the leading coefficient in interpolation of $p$ on points $x,x_1,...,x_k$. ...
1
vote
0answers
43 views

How stable is my quaternion interpolation?

after some experimentation to optimize slerp I found that finding the middle between quaternion is rather cheap (for $t=0.5$) in particular: (with $\theta$ the angle between $q_1$ and $q_1$) ...
1
vote
0answers
70 views

Successive parabolic interpolator for sub pixel interpolation

Is it possible to use successive parabolic interpolator for doing sub pixel interpolation. In the case of non sub pixel interpolation it is very easy to apply successive parabolic interpolation as ...
1
vote
1answer
111 views

how covert joysitck (x, y) coordinates to robot motor speed?

I am trying to formulate an equation to calculate left and right motor speeds of a robot. The robot is four wheeled drive with independent motors connected to tank-like best on each side. see this ...
1
vote
1answer
377 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 ...
0
votes
0answers
38 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) ...
0
votes
0answers
27 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 ...
0
votes
0answers
17 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 ...
0
votes
0answers
42 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 ...
1
vote
1answer
23 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 ...
0
votes
0answers
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 ...
2
votes
1answer
636 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? ...
1
vote
2answers
390 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): ...
1
vote
1answer
77 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 $e_3(\bar{x}) = f(\bar{x}) - p_3(\bar{x})$ ...
2
votes
0answers
211 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 ...
1
vote
0answers
485 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 ...
1
vote
2answers
95 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 $y$, ...
0
votes
1answer
101 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 ...
0
votes
1answer
46 views

Piecewise linear interpolation

A really simple question - how does it work? Linear interpolation is quite easy and understandable: ...
-1
votes
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
vote
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
votes
0answers
37 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
votes
1answer
562 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
votes
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
votes
1answer
82 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
votes
1answer
384 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
votes
1answer
110 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
votes
1answer
113 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
vote
0answers
42 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
votes
0answers
96 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
vote
1answer
157 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
214 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
323 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
votes
1answer
223 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
vote
4answers
591 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
148 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
vote
1answer
319 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
votes
0answers
66 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
votes
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
vote
0answers
213 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 ...