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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inequality about linear and piecewise constant interpolation?

$\Omega\subset\mathbb{R}^3$ is a bounded, and $u(\mathbf{x},t) \in C\big(0,T,L^2(\Omega)\big)$. We divide the interval $[0,T]$ in $N$ equal subintervals with the time step $\tau$. With the notaion $$ ...
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20 views

Interpolation between curve and line

I have two functions with known coefficients as follows: $$ f_1(x) = x^2 -2x + 2 $$ $$f_2(x) = -4x + 1 $$ and I know that $$ (1-z) f1(x) + z f2(x) = 0 $$ $$ (1-z) f1'(x) + z f2'(x) = 0$$ then I ...
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341 views

Cubic spline interpolation - how to calculate second derivative

I ask this qeustion on stackexchange sites: stackoverflow, codereview, and signal processing and no one can help and they send me here :) So I implement cubic spilne interpolation in Java base on ...
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65 views

Use Lagrange interpolation to prove $\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$

Suppose $f\in C^2([a,b])$ and $f(a)=f(b)=0$,use Lagrange interpolation to prove $$\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$$ I tried to use the theoretic error to prove ...
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445 views

How to evaluate Newton's Divided Difference Polynomial in MatLab with an unknown degree?

I already have the code that finds the coefficients for the polynomial, but how do you find a value for the polynomial if given an x coordinate in MatLab code?
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106 views

How to calculate a spline for points in general position?

I want to find a curve passing through (or near) $n$ points in the plane. The catch is that the curve need not be a function. That is, a vertical line might pass through the curve in more than one ...
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46 views

Other way to write Lagrange's form (with derivative)

Prove that we can write polynomial $L_{n}\in\Pi_{n}$ which is interpolating function $f(x)$ in $n+1$ nodes $x_{0},\,\ldots,\, x_{n}$ in following form: ...
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298 views

Bilinear interpolation formula derivation

I am basically following two wikipedia articles, linear interpolation and bilinear interpolation I am having problems getting the same formulas presented for bilinear interpolation but I cannot see ...
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2k views

How to calculate interpolating splines in 3D space?

I'm trying to model a smooth path between several control points in three dimensions, the problem is that there doesn't appear to be an explanation on how to use splines to achieve this. Are splines a ...
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96 views

Degrees of interpolating polynomials

Given a collection of $m+1$ points $\{(x_0,y_0), (x_1,y_1), ..., (x_m,y_m)\}$, we can form the interpolating Lagrange polynomial $L(x)$: $$ L(x) = \sum_{i = 0}^{m} y_i l_i(x) \\ l_i(x) = \prod_{0 \le ...
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155 views

Hermite-Birkhoff interpolation for finding a polynomial

So I have to consider $p \in\mathscr{P}_3$,where $\mathscr{P}_3$ is the set of polynomials of degree $3$, such that $p(0)=1, p'(0)=1, p(2)=1, p'(1)=2$ using the Hermite-Birkhoff interpolation with the ...
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1answer
23 views

The universal function for the class of functions defined on a finite set is computable; (Lagrange interpolation polynomials).

Theorem: A computable universal function for the class of functions of $n$ variables exists that are defined on a finite subset of $\mathbb N^n$. Attempt at proof: Each such function is completely ...
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97 views

How to calculate a trigonometric interpolation polynomial

I have the following $2 \pi$-period function f: $$ f(x) = \left \{ \begin{array}{l l l} x: & 0 < x < 2 \pi \\ \pi: & x = 0 \end{array} ...
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242 views

Wolfram Mathematica - Newton Backward Interpolation?

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
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59 views

Differentiation by interpolation.

I am asked to show that the formula: $$ f'(x)\sim \sum_{i=0}^n A_i f(x_i) $$ which is derived from differentiating the interpolation polynomial is similar to that derived from checking/evaluating the ...
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64 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}] = ...
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186 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 ...
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1answer
91 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 ...
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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 ...
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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 $x,x_1,...,x_k$. ...
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44 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 $q_1$) ...
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75 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 ...
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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 ...
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1answer
387 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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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
692 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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393 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
78 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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226 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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489 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
96 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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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 ...
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1answer
46 views

Piecewise linear interpolation

A really simple question - how does it work? Linear interpolation is quite easy and understandable: ...
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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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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 ...
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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
579 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
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 $(1, P(1)), (2, P(2)), ..., (11, P(11))$. ...
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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 ...
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1answer
396 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 ...