# 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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### construct the forward differencce table from the following data. Evaluate delta(y^3) suffix 1 and y suffix 5? [on hold]

I have constructed the forward difference table. I got $\Delta^3(y_1) = 0.4$ but I am unable to get the value of $y_5$. Can anyone help me? ...
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### Interpolating data points using a closed B-Spline with nonuniform knot vector

I want to create B-Splines which interpolate a set of given data points (knot points). The data points either describe a closed curve or an open & clamped curve. My main source is this website. ...
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### Existence of biholomorphic map from unit disk to itself that interpolates one set of points

How do you prove that given two points $z_{1}, w_{1} \in D = \{z: |z|<1\}$, there exists a biholomorphic (bijective and analytic) function $f: D \to D$ such that $f(z_{1}) = w_{1}$? Perhaps using ...
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### How to calculate error bounds/ accuracy of b-spline interpolation

I am interpolating a set of time values "t" to a set of population "x". I have tried polynomial interpolation (polyfit) using matlab. But the accuracy measured using coefficient of determination, R^2 ...
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### Getting the DFT of irregularly spaced points

I am trying to estimate the discrete Fourier transform of a discrete surface, $x:\{1,\dots,N\}\times \{1,\dots,N\} \to\mathbf{R}$, given a sparse set of samples on the grid. If we had all the ...
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### Detecting highly oscillatory polynomials

Given a set of known values $\phi_1 \ldots \phi_n$ located at points $(x_1, y_1) \ldots (x_n, y_n)$, I want to approximate the value $\phi_0$ at the origin $(0,0)$. To do this, I am using a least ...
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### Absolute value of an RBF distance is less than the absolute value of an actual distance

I have a radial basis function with a linear kernel f(r)=r in 3D. I constructed the surface based on this RBF and noticed that the absolute value of actual distance from any point to the constructed ...
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### Bicubic spline interpolation solver

I created a small delphi application to solver bicubic spline interpolation problem. I have a grid of 4x4 functions value and I want use bicubic spline interpolation to get value for x, y point. ...
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### contours of 2D functions with missing information

If it is known how a function $f(x,y)$ varies say in x for fixed $y=y_0$ and how it varies in $y$ for fixed $x=x_0$, does a method exist to compute the contours for $f(x,y)$ in the entire range of $x$...
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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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### 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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### 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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...