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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A lemma for interpolation for propositional logic

I'm working on an exercise for William Craig's Interpolation Theorem for propositional logic, and I'm having troubles proving the following lemma: Let ϕ and ψ be sentences of propositional logic and ...
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1answer
35 views

Interpolation for $f(n),n\in\mathbb{Z}$: Does it converge?

Assume a function $f(n)$ which is defined for $n\in\mathbb{Z}$. For each period $[n,n+1]$ the function could be interpolated with a polynomial of degree $m$. The polynomials should be built in a way ...
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1answer
21 views

Interpolate a rectangular surface with given edges

I need to interpolate a surface by filling a rectangular hole. The height values of the edges are given. I would like to fill the rectangular surface patch by somehow interpolating the edge values. ...
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1answer
23 views

Create function from a list translated geo coordinates to points

I'm a software developer and I want to create a function from raw data which I collected. The data relates to a satellite image of Europe (Germany). I have a list of geo coordinates and the resulting ...
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17 views

Interpolating polynomial such that it is convex in specified region

The problem I have is that I have data at two points $x_1,x_2$ and $x_2>x_1>0$. At these two points, I know that the function $f$ has values $f(x_1)$ and $f(x_2)$ respectively. It is also ...
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12 views

Piecewise-linear (or otherwise monotonic) interpolation as a matrix problem

Background: I'm hoping to find (or write) an algorithm to piecewise linear-interpolate large sets of unevenly sampled functions (10s of thousands of arrays of a thousand or so $x$ and $y$ pairs, where ...
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1answer
26 views

Is a sigmoid function what I need to make this graph?

I'm designing a game where characters' speed starts slowing down after different distances. I'm not advanced in mathematics so I'm not sure if I'm on the right track. After researching on wikipedia I ...
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24 views

How can I visualize Quaternion Linear Interpolation?

It’s hard enough to visualize a quaternion, geometrically speaking. A complex number is simple: it’s a point in a plane. Suppose we had a number like this: a + bi + cj I supose you can visualize ...
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574 views

Spline with varing tension, selection of tension factor

I need to perform a special interpolation, using that kind of basis : $$\varphi_{i,j}(x) = a_i + b_ix + c_i(\cosh(\tau\ x) - 1) + d_i(\sinh(\tau\ x) - \tau\ x)$$ where the $a_i$, $b_i$, $c_i$ and $...
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30 views

Romberg, trapezoidal rule exact for polynomials

My question is, how can I proof that the rombergs method of the summed trapezoidal rule is exact for polynomials with degree $(2n+1)$ or less. Thanks for helping, one or two tips can help me here. ...
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31 views

CFD: finding the vorticity magnitude the streamwise direction of an airfoil

I am doing CFD and I have to find the magnitude of the vorticity vector in the streamwise direction of an airfoil in every mesh cell. The streamwise direction is defined as being parallel to the ...
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52 views

General issue when adding shocks on curves made of splines

Let us assume I have a "nice" curve and that I would like to introduce a small shock up/down of about 1% at a certain point along the curve. I am trying to find out what the best and most efficient ...
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38 views

Understanding divided difference table

To Construct a divided-difference table using the given value for x and f(x), This solution table seems so confusing to me i understood how the value of f1[] was calulated ...
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1answer
34 views

B-splines basis function

If the image domain is denoted as $\Omega_I=\{(x,y,z)| 0 \leq x<X,0\leq y<Y, 0 \leq z <Z \}$. Let $\Phi$ denote a $n_x \times n_y \times n_z$ mesh of control points points $\phi_{i,j,k}$ with ...
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36 views

Uniqueness of interpolation polynomial.

I am new to numerical analysis and this is the first thing I came across. It says on my textbook that interpolation polynomials are unique and to prove that it was assumed that let there be two such ...
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1answer
28 views

Is the sum of coefficients 2?

Is the sum of the coefficients of the polynomial interpolation of the data $(1,p_1),(2,p_2),...,(n,p_n)$ for some positive integer $n$ (where $p_n$ is the $n$th prime) always equal to two? I've ...
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1answer
51 views

Precision in Cubic spline interpolation

I am working on cubic spline interpolation with set of data points from CAD with following steps: Form piecewise spline equations between points. cubic equation : $ ax^3 + bx^2 +cx + d = P(x) $ ...
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1answer
26 views

Interpolation of Symmetric Data

For symmetric data $(x_i,y_i), i=-n,-n+1,..., n-1, n$ such that $x_{-i}=-x_i$ and $y_{-i}=-y_i, i=0,1,...n$ what is the required degree for an interpolating polynomial $p$? Since there are $2n+...
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1answer
37 views

interpolation polynomial error

We have points $x_0=a \lt x_1 \lt x_2 ....x_n=b $ and $\;w_{n+1}(x)=\prod_{k=0}^{n}{(x-x_k)}$. Let $h=max_{j=0...n}|x_j-x_{j-1}|$ Let $f \in C^{n+1}[a;b]$ and $p_n\in \mathbb P_n$ be the ...
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33 views

Interpolation and Interpolationerror - how to compute ?

I want to compute the greatest $a>0$ for given $\epsilon>0$ such that $$max_{x\in [-a,a]}|f(x)-p_2(x)| < \epsilon$$ where $a$ is the distance between two grid points and the maximum is the ...
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31 views

Lagrange Interpolation - two approaches

I want to ask something about Lagrange-Interpolation Polynomials: Given the following pairs of values: $p(x_i,y_i): p_0(0, 1), p_1(1.5, 2), p_2(2.5, 2)$ I found two ways of calculating the ...
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2answers
294 views

Accurate floating-point linear interpolation

I want to perform a simple linear interpolation between $A$ and $B$ (which are binary floating-point values) using floating-point math with IEEE-754 round-to-nearest-or-even rounding rules, as ...
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1answer
539 views

Newton backward interpolation in Mathematica

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 + $i$$\...
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1answer
906 views

What is a cardinal basis spline?

Wikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as "Bk". Meanwhile, cnx.org Signal Reconstruction says: The basis splines Bn are shown ... ...
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2answers
32 views

How do I compare rates of error between two different sample sizes?

I'm unsure on how to normalize for two different variables. Person A makes 20 pastries total, whereas Person B makes 50. 5 of those pastries, so 25%, are sampled from Person A; 10 for Person B, for ...
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42 views

What is the simplest way to estimate future values of two variables?

I have two variables such as x and y. The data is not increasing linearly. Also, I have restricted boundry [10;20]; x y 10 15 11 15,5 12 17 13 19 14 19,5 15 20 16 20,2 17 20,8 18 ...
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1k views

Plane fitting using svd

I am trying to get a best fit plane in a 3d space of points. I am using an svd as described in http://stackoverflow.com/questions/10900141/fast-plane-fitting-to-many-points. If I use the data provided ...
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28 views

How to recognize function by its points? [closed]

I have collection of points $(p_1, p_2,\dots, p_N)$ of some function $f(x)$, where point $p_j = (x_j, f(x_j))$. Function $f(x)$ can be only of given type $(f(x) = x, f(x) = x^2, f(x) = x^3, f(x) = c, ...
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7 views

interpolation terminology

I am trying to write a discussion on various interpolation methods, and I need a systematic terminology for interpolation, or the article will have an unnecessary digression to define terms, which won'...
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21 views

Did I interpret this wiki article on spherical interpolation correctly?

In Lua pseudocode, I believe the wikipedia article here is saying that the formula is used in the following way: ...
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22 views

Interpolating a vector about an arc (Slerp)

In the following image, how can I solve for $k_0$? I know that $\mathbf v_1$ is a unit vector and $k_1 = \sin tω/\sin ω$.
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14 views

Linear interpolation on Plane (Marching Cubes)

Let's assume I have the following cube. Let's assume the isovalue = 0. I would like to draw the resulting triangles of the isosurface. I know that first I define which values are inside or outside ...
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47 views

Tricubic Interpolation

I am currently writing a plugin for 3D analysis software and I am working with a data grid where certain values are stored at XYZ coordinates, and I need to find an estimated value of a point that ...
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31 views

A simple Lagrange interpolation-type identity

I am unable to prove an identity that looks very much like the Lagrange interpolation identity, Problem: Given $f(x)$ is a monic, $n-1$ degree polynomial and $a_1, a_2, \cdots a_n$ distinct real ...
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1answer
26 views

Knowing two Vectors, and the distance to a 3rd, how to get the 3rd

If I know the two Vectors $v_1$ and $v_2$, which describe points in a 2D space, and I also know that a vector $v_3$ is on the line segment between $v_1$ and $v_2$, how can I get the $x$ and $y$ ...
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1answer
37 views

Cubic spline interpolation results

I have a set of data points on which i am trying to do cubic spline interpolation. Below is the snapshot of the curve with the input data points marked in green color. And the red color marked point ...
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Strict order on propositions and interpolation

We can define a strict order on the set of propositions in countably many propositional letters in the following way: $$\varphi\sqsubset\psi \iff (\models \varphi\rightarrow\psi)\, \land (\not\models ...
2
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1answer
619 views

Hermite Polynomials Triple Product

Similar to the question Legendre Polynomials Triple Product, I would like to ask whether there are any explicit formulas for the inner product of the Hermite polynomial triple product \begin{align} \...
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21 views

Find the smallest maximized absolute error in polynomial interpolation

Given $$ f(x)=\begin{cases}1&,0\le x \le1 \\2x&,1<x\le2 \end{cases} $$ I found that the interpolating polynomial $p \in \mathbb{P}_{2} $ at $x_{0}=0,x_{1}=1,x_{2}=2$ is$$p(x)=\frac{3}{2}x^2-...
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Interpolate from a point on a sphere to a point on a another sphere?

I am at the moment trying to come up with an solution which is capable of interpolating between a point on a sphere A to a point on a sphere b. The interpolation should both provide me with minimal ...
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1answer
13 views

find estimation of interpolation error for non differential function

Given $f(x)=|x|^{1/2}$ , $-1\le x\le 1$ , I have found the interpolating polynomial $ p(x)=x^2$ for $x_{0}=-1,x_{1}=0,x_{2}=1$. How to estimate $$\max_{-1\le x\le 1}|f(x)-p(x)|$$ now that $f$ is not $...
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1answer
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interpolation error using higher derivatives

Given $x_{0},x_{1},x_{2}\in[a,b] $ each one different from the others,$f \in C^{4}[a,b]$ and $p\in\mathbb{P}_{3}$ so that $$p(x_{i})=f(x_{i}), i=0,1,2 $$ and $$p'(x_{1})=f'(x_{1})$$ prove that: $$\...
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28 views

Function that satisfies the given (x,y) values

I am trying to come up with a function that (approximately) satisfies these (x,y) values. (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 6), (9, 5), (10, 4), (11, 3), (12, 2), (13, 1), (14, 2), (...
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1answer
8 views

Given this discrete non-linear set of values how can I get an equation for it?

I want to generate a equation in the form f(x) = {...} for this discrete data below. As X doubles Y halves but its a bit more complicated. Using an online Polynomial Interpolation calculator I got: ...
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27 views

Interpolating discrete data with completely monotone analytic functions

Suppose we have a positive integer $n$ and a finite list of real numbers $\{a_1,\,a_2,\,\dots,\,a_n\}$. We want to find a real-analytic function $f:[1,n]\to\mathbb R$ such that $f(m)=a_m$ for all $m\...
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1answer
19 views

Conditions under which a discretely defined function can be extended convexly

Suppose we have a set of points $u_1,\ldots, u_m \in \mathbb{R}^d$. Suppose $F$ is a function into the reals defined at each of the points $u_i$. My question is how do we know when $F$ is really ...
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1answer
72 views

Do nth degree polynomials derived using Least Squares Interpolation always have n+1 intersections with the function?

I have recently studied Interpolation Techniques in my College Numerical Methods class and I have this question: If we have a function $f(x)$ and we are asked to use Least Squares Interpolation(LSI) ...
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1answer
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Discrete surface interpolation

I'm working on implementation of a Fast surace interpolation using hierarchial basis functions (Szeliski et al) algorithm. The idea is: given a discrete function measurements of its values (depths) $...
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1answer
22 views

How to show a piecewise quadratic interpolant is $H^1$

I am preparing for a final exam and came across this question: Suppose that $\Omega\subset\mathbb{R}^2$ is an open bounded domain with triangulation $\mathscr{T}$. Suppose that $v_h$ is a ...
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1answer
60 views

Cubic Spline Interpolation

My problem is to find a interpolating cubic spline to the points $$\left\{(0,0), \left(\frac{\pi}{2}, 1\right), \left(\pi,0\right), \left(\frac{3\pi}{2}, -1\right),(2\pi,0)\right\}$$ I did as ...