# 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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### Implementation of Monotone Cubic Interpolation

I'm in need to implement Monotone Cubic Interpolation for interpolate a sequence of points. The information I have about the points are x,y and timestamp. I'm much more an IT guy rather than a ...
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### There is a unique polynomial interpolating $f$ and its derivatives

I have questions on a similar topic here, here, and here, but this is a different question. It is well-known that a Hermite interpolation polynomial (where we sample the function and its derivatives ...
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### Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.

I am trying to understand Hermite Interpolation. Here is my pedagogical example. I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation. I choose the Chebyshev zeros ...
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### Fritsch and Carlson is non-linear?

I was reading about this interpolation method and saw that it was mentioned that the algorithm is non-linear. What does that exactly mean? I am confused because I don't get what is "non-linearity" in ...
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### 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 2x2 ...
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### Profinite and p-adic interpolation of Fibonacci numbers

On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here) For each proﬁnite integer $s$, one can in a natural way deﬁne the $s$th Fibonacci ...
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### Uniform convergence of Lagrange polynomials

There is a well-known theorem that states that on a closed interval $[a,b]$ any continuous function is the limit of a uniformly convergent sequence of polynomials. Proofs for this theorem usually ...
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### The Weierstrass Approximation Theorem Vs The Runge's Phenomenon

I am learning about different interpolation methods in my internship. Today as I was looking this article on Wikipedia to learn about the Runge's Phenomenon exhibited by Polynomial Interpolation. I ...
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### 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 ...
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### 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 ...
When using cubic spline interpolation, we have to solve $n-1$ equations with $n+2$ unknowns. What we can do is set $z_0 = z_n = 0$, which gives the natural cubic spline. But could we also set some ...
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 $\textbf{x}=(x_0,\ldots,x_n)$...