# 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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### What is this polynomial zero everywhere but $n!$ at point $n$?

For an integer $n \geq 2$, define a set of $n$ points $(p_{x,1},p_{y,1}),\ldots,(p_{x,n},p_{y,n})$ as follows. For $1 \leq i \leq n-1$, let $(p_{x,i},p_{y,i}) = (i,0)$, and $(p_{x,n},p_{y,n}) = (n,n!)$...
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### How to choose new points after grouping/resampling?

I'm resampling a signal (which takes values [0,1]) of N samples (blu points) to one with N/5 samples, where (for each group of 5 samples) I store in two arrays the max and the min values of the ...
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### Best 2D interpolation for a set of points (x/y)

Given a set of X/Y points in a bounded area ( 0<=X<=Xmax and 0<=Y<=Ymax ), what are the best 'interpolation' ...
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### Lagrange interpolation not working?

I have been trying to use Lagrange Interpolation on some data to find the equation of a curve that fits the following points (x,y): (0,5.2) (1,3.8) (2,4.5) (3,5.5) (4,6) (5,6.25) (6,6.25) (7,6) (8,5....
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### Interpolation of rounded data

I would like to interpolate (not fit) a data set whose points have been rounded. Lets say I have some observations $y_i$ of a function sampled at $x_i$. The sample locations $x_i$ may be considered ...
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### Find $p(x)=a_1 x+a_2 x^2$ with $p(x_i)=f_i, i=0,1$

Given $x_0,x_1,x_0\neq x_1, x_i\neq 0, i=0,1$, and $f_0,f_1$, find $p(x)=a_1 x+a_2 x^2$ with $p(x_i)=f_i, i=0,1$. Can you say if the interpolating polynomial in this case is unique? Why? Can you write ...
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### How can I derive the second degree equation for a curve if I know the slope at two points and the y-intercept = 0?

How can I derive the second degree equation for a curve if I know the slope at two points and the x and y-intercepts = 0?
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### Temerature and Strain images comparison & interpolation: finding offsets in X,Y coordinates systems.

I have two kind of 2D images from a Strain test, which show temperature and strain distributions in X and Y cordinate systems. One is thermal image, which gives temperature T values in a 200x300 ...
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I am just wondering if anyone has encountered a similar problem before. Let's consider a differentiable function in 1 dimension for now. Suppose that this function is expensive to compute. In addition,...
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### Error of derivatives of polynomial interpolant and best nodes with lobatto constraint

I know when interpolating a function $f(x)$ by a degree $n$ polynomial $P(x)$, the error is $$f(x) - P(x) = \frac{f^{n+1}(\xi)}{(n+1)!}\prod_{i=0}^n(x-x_i)$$ What does the right side hand look like ...
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### Is the following described $p(x), q(x)$ the same interpolation polynomial?

Suppose we are given an odd number of data points $x_i$ and the corresponding values $f_i=f(x_i),i=1,...,n+1$($n$ is even), which are symmetric about the origin, i.e for each $x_i$ there is a $j$ such ...
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### How many multiplications and additions will be needed to evaluate the interpolating polynomial at a point using the Lagrange formulas?

Consider the interpolating problem with $n+1$ points. How many multiplications and additions will be needed to evaluate the interpolating polynomial at a point using the Lagrange formulas? I think it ...
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### C2 continuous interpolation on a 4-dimensional dataset

I am currently coding up a project where interpolations must be performed such that C2 continuity be preserved along the length of the whole set. The end result ought to look like a line (which will ...
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### Parabolic interpolation in $k$ dimensions

I know the values of a smooth function of $k$ variables at $3^k$ points on a cube in $k$ dimensions (where $k=2,3$ or $4$). The central value is known to be the largest. I want to estimate the ...
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### Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
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### Terminology: Spline interpolation

I have read two different definitions of Splines: A differentiable piecewise polynomial. A piecewise polynomial. If I build a piecewise polynomial using cubic polynomials, it's ...
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### What is this graphed function with asymptotes at π/2 and -π/2?

I've come across this function with asymptotes at π/2 and -π/2, which crosses the axis at y=1. It doesn't seem to be polynomial or exponential—can anyone figure out what it is? The asymptotes and y=...
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### Need some help with applying specific boundary conditions to b-spline system of equations

I'm building a package for B-spline interpolation in Julia, and I've come across a boundary condition that I want to implement but can't wrap my head around how to do it (mathematically). Basically, ...
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### Non-Piecewise interpolation over 3 points

I am trying to write an algorithm that interpolates between 3 values. The interpolation will be over the interval [0,1]. What I would like to do is: (Hopefully this makes sense) at x = 0, y = [...
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### Should an interpolation coincide the original function on the given data points?

Suppose having a model $f(x)=y$ where $f$ is unkown. Moreover, suppose you have some data points for this model i.e. $(x_1,y_1), (x_2,y_2), \dots , (x_n,y_n)$. If one can find an approximate of $f$ ...
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### Calculating B-Splines and dimension of spline space

I've got the following assignment: Let $S$ be the space of piecewise polynomials of degree $3$ on the intervall $[-1;1]$ with knots $x_i = -1+\frac{i}{2}, 0 \leq i \leq 4$. (a) Calculate a basis of ...
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### Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is defined....
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### Finding an Entire function with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$

I am really stuck on a homework problem, which boils down to the following: We need to exhibit an entire function $f$ with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$. The only sorts of functions ...
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### Bernstein Interpolation 2D

I am well aware of the equations for 1D Bernstein Interpolation. But I do not understand how to extend it to 2D. I am guessing that the equations in the following image would do for 2D Bernstein ...
I'm currently trying to explain divided differences with a view to defining the Newton form of the interpolating polynomial. I'm using the definition: The $k$th divided difference of a function $f$ ...