Articles of interpolation

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 $$ u_i=u(t_i),~ \delta u_i=\frac{u_i-u_{i-1}}{\tau}, $$ define linear and piecewise constant interpolation functions of u as follows: \begin{gather} u_\tau = u_i+(t-t_{i-1})\delta u_i, & t\in(t_{i-1},t_i], \\ \bar{u}_\tau = u_i, & t\in(t_{i-1},t_i]. \end{gather} […]

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: $$c\cdot e_k =\dfrac{1}{V}\int_{\partial P}\dfrac{1}{2}\left(x\cdot e_k\right)^2\left(n\cdot e_k\right)=\dfrac{1}{2V}\sum\limits_{i=0}^{N-1}\int_{A_i}\left(x\cdot e_k\right)^2\left(n_i\cdot e_k\right), k=1,2,3$$ It remains to compute that: $$\int_{A_i}\left(x\cdot e_k\right)^2\left(n_i\cdot e_k\right)=\dfrac{1}{6}{\hat n}\cdot e_k\left(\left[\tfrac{1}{2}\left(a_i+b_i\right)\cdot e_k\right]^2+\left[\tfrac{1}{2}\left(b_i+c_i\right)\cdot e_k\right]^2+\left[\tfrac{1}{2}\left(c_i+a_i\right)\cdot […]

Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some $\textbf{x}=(x_0,\ldots,x_n)$ and this yields $\textbf{y}=(y_0,\ldots,y_n)$, where $y_i=f(x_i)$. We shuffle the elements in $\textbf{y}$. Then we give $\textbf{x}$ and the shuffled $\textbf{y}$ to an untrusted server. The […]

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 generate for me function formula y=x^2. I am aware its is not possible to find out exact function, but it would be […]

How to create a computationally cheap function passing through given points?

I am trying to develop a function which goes through the follow points. The function will be calculated on a microprocessor which has 20 mHz. List of given points: P1 = (36541 ,120) P2 = (37811 ,110) P3 = (39527 ,100) P4 = (41414 ,90) P5 = (44475 ,80) P6 = (48848 ,70) P7 = […]

Integrating over a specific vector field

I am trying to show that the solution of the following integral is as follows: Define the stopping time: $C(a) = \inf(u \ge 0 : H(\pi(0) |\mu)-H(\pi(u) | \mu) > a)$ Where $H(\pi(t)|\mu(t))=\sum^{n}_{i=1} \pi_i \log \frac{\pi_i}{\mu_i}$ is the ‘Relative Entropy’. Introduce the vector field: $U_{\mu}(\pi)=\textbf{$\pi$}-\textbf{$\mu$}$ and associated flow: $$ \frac{d}{du} \pi (u ) = \pi'(u) […]

Can Runge's approximating rat. fns. be required to take certain prescribed values?

Suppose $f$ is analytic on an open set $U$ containing the compact set $K$, and $\{r_n\}$ is a sequence of rational functions provided by Runge’s theorem (having poles in some prescribed set $A$). For a given finite set $\{x_1,\ldots,x_k\}\subset K$, can we additionally demand that for each $n$, $r_n(x_1)=f(x_1),r_n(x_2)=f(x_2),\ldots,r_n(x_k)=f(x_k)$? If yes, can we also require […]

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 construct a quadratic Bezier curve that passes through some given points. First, the standard approach: I would use three […]

Proof that lagrange and newton's interpolation are the same

Its known that newton’s interpolation and lagrange interpolation gives the same value All i need is to prove it

2D array downsampling and upsampling using bilinear interpolation

I am trying to understand how exactly the upsampling and downsampling of a 2D image I have, would happen using Bilinear interpolation. Now I am aware of how bilinear interpolation works using a 2×2 neighbourhood values to interpolate the data point inside this 2×2 area using weights. But what I am not aware of, is […]