All Questions
Tagged with approximation-theory approximation-algorithms
15 questions
9
votes
6
answers
8k
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How to approximate a solution to a matrix equation? [closed]
Suppose a matrix equation $Ax = b$ has no solution ($b$ is not in the column space of $A$)
How can I find a vector $x^\prime$ so that $Ax^\prime$ is the closest possible vector to $b$?
- 461
9
votes
1
answer
553
views
Polynomial approximations of curves
This is the 3D version of this question. The responses to that question contained a lot of complaints about fuzzy definition of the problem, so I made this new question very narrow and explicit.
For ...
- 649
7
votes
1
answer
725
views
Norms of B-spline coefficients
In Shumaker's book (Spline Functions: Basic Theory), we know that the $l^\infty$-norm of B-spline coefficients is bounded above and below by the $L^\infty$-norm of the spline itself. Are there similar ...
- 176
6
votes
3
answers
502
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Approximating derivatives between gridpoints
Suppose we have a grid (possibly irregular) of $N$ function/value pairs, $(x_i, f_i)$, $i=1...N$. The function is differentiable everywhere at least twice (perhaps more).
What would be a good way to ...
- 177
5
votes
4
answers
4k
views
Construct the best piece-wise linear continuous function fitting given curve
How to construct the optimal piece-wise linear continuous function fitting given curve and given number of knots (optimal knots positions also must be determined by this method)?
- 161
5
votes
2
answers
350
views
How expressive is $e^A$ in the sense of universal approximation?
For any real matrix $B\in\mathbb{R}^{n\times n},n\ge 2$ and precision $\varepsilon$, is there a real matrix $A\in\mathbb{R}^{n\times n}$ such that $\|e^A-B\|_F<\varepsilon$? ($F$ refers to ...
- 109
4
votes
4
answers
3k
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When we use Bernstein polynomials in application
When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", &...
3
votes
2
answers
756
views
Approximation of curves
When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
- 649
3
votes
0
answers
38
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Do higher-order splines with Lipschitz derivatives exist on finite sets?
Fix $k\in \mathbb{N}^+$ and let $E=(e_i,f_i)_{i=1}^I\subset \mathbb{R}^n\times \mathbb{R}^m$ be a non-empty finite set with $e_i\neq e_j$ whenever $i\neq j$.
If $n=m=1$ then it's easy to see that:
$$
...
- 5,405
3
votes
0
answers
180
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A.G. Vitushkin's "Easily representable families of functions" - can it be generalized?
Background
In his monograph "Estimation of the complexity of the tabulation problem" (translated into English as "Theory of the Transmission and Processing of Information") Vitushkin studies ...
- 959
2
votes
2
answers
456
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Approximation of a given function by rational functions
Given a function $1/\sqrt{x^2 -k^2}$ where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the L_p (p=2 or $\infty$) norm of ...
- 21
1
vote
2
answers
401
views
Bivariate Function Approximation
I am working on a nonlinear control design and having difficulty in finding approximation of bivariate functions. Are there papers or methods discussing the following question:
For any bivariate ...
- 11
1
vote
0
answers
59
views
Functional approximation with derivatives
I am trying to solve a functional approximation problem.
Consider a set of measurements of a d-dimensional state $\mathrm x \in \mathbb{R}^d$, together with velocities $\dot{\mathrm x}$ and ...
1
vote
0
answers
315
views
Can Carlsons's iterative algorithm for $\arctan x$ be inverted to get one for $\tan x$?
In the article An algorithm for computing logarithms and arctangents, by B. C. Carlson, the following iterative algorithm for arctangents is given:
The algorithm uses that $2^n\tan(2^{-n}\arctan(x))=\...
0
votes
0
answers
50
views
Can we talk about approximation when the decision problem for solution existence is NP-Hard
I am wishing to design an approximation algorithm for an optimization problem where the existence of solution for corresponding decision problem is not guaranteed. Is it wise to find an approximation ...
