# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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) ...
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) $... 1answer 22 views ### How to show a piecewise quadratic interpolant is$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 ...
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 ...