Questions tagged [approximation]
A method of finding nearly-optimal solutions to a problem. Generally, this terminology is applied to algorithms and heuristics for solving NP-Hard problems in computer science.
87
questions
2
votes
1answer
65 views
Linearization of Remez algorithm rational case
In the rational case, we are interested to find polynomials $P(x)$ and $Q(x)$ s.t.
$f(x_k)-P(x_k)/Q(x_k)=(-1)^kE$ for $k=1,2,\ldots, N$ where $N=deg(P)+deg(Q)+2$
This can be rewritten as
$$
(1)~~~~~~(...
2
votes
1answer
2k views
imaginary time propagation to find ground state wavefunction
I understand the basic idea of imaginary time propagation method:
The wavefunction $\psi(x,t)$ as a superposition of energy eigenstates $\phi_m(x)$:
$$
\psi(x,t)=\sum_m \phi_m(x)e^{-iE_mt/\hbar}
$$
...
4
votes
1answer
115 views
Floating point and global error in Euler Method
Inspired by this answer, I tried to understand when floating point errors come into visibility and to check it also comparing the plot of the numerical solution with Explicit Euler with the analytical ...
3
votes
1answer
68 views
Is there a source/cookbook of equations that approximate geometric shapes?
I'm numerically modelling flows around various geometric 2D shapes. Is there a good source/cookbook of equations that approximate these? Some examples are
Rectangle:
$(x-a)^n+(y-b)^n < r^n$ where ...
4
votes
0answers
52 views
Continuous limit and nonlinear functional analysis
I have a kind of general question about approximation schemes in nonlinear functional analysis. Given a nonlinear map $\Phi$ from an open set (in an infinite dimensional Banach space) of functions to ...
1
vote
0answers
65 views
Error curve does not oscillate in between reference points using Remez
Using the Remez algorithm, implemented using multi-precision library, in certain functions that I want to approximate, the error curve does not oscillate in between reference points, and so no roots ...
1
vote
1answer
64 views
Questions on Daubechies wavelets
Is the refinement equation for the orthonormal Daubechies scaling function
$$\phi(x) = \sqrt{2} \sum_n h_n \phi(2x-n) \;?$$
The filter coefficients for Daubechies wavelets have been given e.g. in
this ...
2
votes
0answers
85 views
Remez algorithm convergence
I have implemented the Remez algorithm in Python where all calculations were done with the Python mpmath library. I have noticed that sometimes the $|E_{max}|$ and $|E_{min}|$ do not monotonically ...
17
votes
2answers
4k views
Drawbacks of Newton-Raphson approximation with approximate numerical derivative
Suppose I have some function $f$ and I want to find $x$ such that $f(x)\approx 0$. I might use the Newton-Raphson method. But this requires that I know the derivative function $f'(x)$. An analytic ...
3
votes
3answers
129 views
What are some algorithms to calculate the width of an arbitrary polygon when a bounding box approximation is inaccurate
What are some alternative algorithms to creating a bounding box for finding the max width of a concave, simple winding polygon, like the one in the below image? I prefer solutions that are more ...
5
votes
0answers
215 views
What algorithm does (or did?) Excel use for Bessel functions that is discontinuous at x=8?
Writing this comment reminded me of something I noticed years ago about evaluating Bessel functions of the first kind $J_n(x)$ in Excel. (BESSELJ)
I don't use Excel now but at the time I'd checked ...
1
vote
1answer
156 views
what is non-asymptotic convergence?
I guess convergence in general means it is in asymptotic sense but what does non-asymptotic convergence mean?. Can someone please explain with an example?
0
votes
0answers
27 views
Computing square root of rank-1 correction
In the post Computing square root of diag(u)-uu'?, both @YaroslavBulatov and @NickAlger suggested the factor $\frac{\sqrt{u^{T}D^{-1}u+1}-1}{u^{T}D^{-1}u}$ for their approximation.
Can somebody ...
2
votes
0answers
28 views
Fast approximate evaluation of Fourier-Legendre series
Suppose I know that a function from $[0,\pi] \to \mathbb{R}$ may be written as
$$
\sum_{k=0}^\infty A_l \frac{2l+1}{4\pi} P_l(r)
$$
where $A_l$ all are known. Is there a way in which I may very ...
7
votes
1answer
500 views
Problems Implementing the Remez Algorithm
So first off:
*** This code is not being used in production software.
It is a personal project of mine, trying to understand
approximation theory and advanced curve fitting.
In other words, I'm ...
0
votes
2answers
131 views
How can I calculate the exponential integral?
(I originally asked this in a different exchange.)
I'm writing a program that uses the prime-counting function. Right now, I'm using x/log(x), but I want to ...
4
votes
1answer
61 views
approximate function such that the inverse of the approximation is “simple”
I have a smooth enough injective function $f:[a, b]\to \mathbb{R}$ which I want to approximate by something that can be computed quickly, e.g., a Padé approximant of low degree,
$$
\frac{\sum_{j=0}^m ...
6
votes
1answer
278 views
Does this Algorithm (probably Fourier like) Exist for 2D Shapes? [closed]
Update: Someone changed the title to this post to a possible answer ("Fourier decomposition of parametric shapes") but I changed it to a different title as that makes it clear what I was asking. As I ...
6
votes
1answer
499 views
Fast algorithm for computing matrix square root using randomized linear algebra?
Is there a fast algorithm for computing the matrix square root of a real symmetric matrix using random matrices or randomized algorithms?
7
votes
2answers
936 views
Bin-packing: Maximise number of bins / “Fukubukuro” problem?
I recently encountered a problem that looks like a variation of bin packing or knapsack problem, but with the objective to maximise the number of bins/knapsacks:
Consider there is a list of M items ...
4
votes
0answers
71 views
Fast approximate solver for vehicle routing problem
I need to solve capacitated asymmetrical vehicle routing problem with time windows on ~30k points. Time limits for calculations are 2 hours. I've tried using Clarke and Wright savings algorithm, it is ...
4
votes
0answers
157 views
MATLAB: Compute the Schwarz-Christoffel transformation symbolically
Suppose we have a conformal mapping from the unit disk in the $\omega$ plane onto the exterior of a polygon in the $z$ plane.
The Schwarz-Christoffel mapping in this case is defined as:
$$f(u) = A - ...
7
votes
1answer
245 views
Size of jump for piecewise discontinuous approximations
If one has a sufficiently smooth function $u$ that is approximated by a piecewise constant function $u_h=\Pi^0_h u$ on a mesh of cell size $h$ (where $\Pi^0_h$ is the $L_2$ projection onto the ...
0
votes
2answers
409 views
closed form approximation of matrix inverse with special properties
I'm trying to find some theory to help me explicitly express the inverse of a matrix (or a close approximation of the inverse). My matrix has the following properties:
invertible
positive definite
...
6
votes
1answer
371 views
Matching/Assignment Problem
I'm not sure how I can represent and solve the following problem.
I have a list of sales (timestamp and quantity) and a list of corresponding inventory draws (timestamp and quantity). What I ...
0
votes
1answer
42 views
Slightly change two vectors to satisfy a constraint
$\vec{a}\cdot\vec{b} \approx c$
$\vec{\alpha} \cdot \vec{\beta} = c$
$\vec{\alpha}$ is close to $\vec{a}$ and $\vec{\beta}$ is close to $\vec{b}$
Given $\vec{a}$, $\vec{b}$ and c, how to find $\vec{\...
2
votes
0answers
58 views
Weighted Set Cover in practice, beyond the greedy algorithm
According to the wikipedia page for Set Cover, the greedy algorithm for weighted set cover achieves the polynomial-time approximation bound. There are other techniques for solving Set Cover, such as ...
14
votes
6answers
1k views
Approximate spectrum of a large matrix
I want to compute the spectrum (all the eigenvalues) of a large sparse matrix (hundreds of thousands of rows). This is hard.
I am willing to settle for an approximation. Are there approximation ...
1
vote
1answer
109 views
Approximation of a non linear problem with python
I need your help to solve a problem I'm working on for school.
My goal is to approximate the coefficients of a weight matrix so that they check particular properties.
So I have $W$ the weight matrix ...
3
votes
2answers
249 views
Image hash similarity matching possible?
I have the following question:
We have two face image files (JPEG), a Matrix of $128\times 128$ with values between 0-255.
We would like to hash both image files using a function $f(x, key)$. Where I ...
0
votes
1answer
78 views
Which are some good algorithms and heuristics to calculate the similarity between two matrices?
Say I have a matrix like this:
\begin{bmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 1 \\
0 & 1 & 1 & 1 \\
\end{bmatrix}
And this one:
\begin{bmatrix}
...
2
votes
1answer
108 views
Taylor expansion of error - Finite elements approximation
In some of my computations I calculate a scalar value $\lambda_h$ (in my case an eigenvalue) depending on a finite element discretization of the domain. Usually we can manage to find estimates of the ...
0
votes
1answer
212 views
Linear Least-Squares Point-to-Plane ICP degenerative case
I'm trying to implement Linear Least-Squares Optimization for Point-to-Plane ICP Surface Registration
Paper describes linear approximation for point to plane distance for rigid ICP. This approach is ...
0
votes
1answer
1k views
Derivatives of a Chebychev polynomial
I am using Chebychev collocation nodes for approximation, and my problem requires me to calculate derivatives of the polynomial. I have been reading from a few sources, but I am not sure I understand ...
0
votes
1answer
56 views
Chebychev Polynomial derivatives at zero points and extreme points
I was looking for some help with derivatives of Chebychev polynomials at zero points. The recursive expression,
$$
T_{(j+1)}(x) = 2xT_j(x) - T_{(j-1)}(x)
$$
has the derivative
$$
T'_{j+1}(x) = 2T_j(...
1
vote
0answers
55 views
Computing dilogarithm
I'm measuring the integral of a quantity which, mathematically, requires the computation of a dilogarithm function.
$$\operatorname{Li}_2(be^{ax})$$
where $b$ and $a$ (are real and) can be positive ...
0
votes
1answer
218 views
B-Splines Matlab Package
I'm looking for a good Matlab package/library for B-Splines approximation. Ideally, it would take knots $t_1, \ldots , t_n$, and data points $g(t_1),\ldots , g(t_n)$, and Produce $$Vg :\,= \sum\...
4
votes
0answers
83 views
Extrapolation after successive finite element refinement
I wish to compute a certain function $\lambda$ (in my case an eigenvalue of the Laplacian) to a certain accuracy, which I wish to guarantee, if possible. I started with a finite element method. I know ...
15
votes
4answers
1k views
Can the solution of a linear system of equations be approximated for only the first few variables?
I have a linear system of equations of size mxm, where m is large. However, the variables that I'm interested in are just the first n variables (n is small compared to m). Is there a way I can ...
4
votes
1answer
655 views
Chebyshev approximation by projection vs interpolation
Suppose we want to approximate a function $f: [a, b] \rightarrow \Re$ with a Chebyshev series:
$$ f(x) \approx \sum_{k=0}^n c_k \, T_k\left( \frac{2x-b-a}{b-a} \right) $$
where $T_k(x) = \cos(k\, \...
4
votes
1answer
184 views
Numerical evaluation of the Exponential Integral Ei by rational Chebyshev approximations fails
I am trying to evaluate the Exponential Integral $Ei(x)=-\int^{\infty}_{-x}\frac{e^{-t}}{t}dt$ for $x>0$ (interpreted as the Cauchy principal value) by using rational Chebyshev approximations, ...
1
vote
0answers
37 views
Find alternative path closest to original
I have a set of points in a 2D space. I want to connect the outer points so I get the convex hull. The problem here is that there is a limit to the distance between two points. Let me clarify that ...
6
votes
2answers
3k views
Solving a system of linear equations with only an approximate solution
I have a system of linear equations that is derived partially from experimental data. Theoretically, the system should have a single, exact solution; however, experimental error causes it to not have ...
5
votes
1answer
416 views
Polynomial order of an approximation of a section of sine and numerical accuracy
I was playing with the idea that a sine function is periodic. But even within one period there are symmetries, namely the second fourth of a period is the mirror image of the first fourth and the ...
1
vote
1answer
51 views
Finding optimal point distance to get desired number of random points in an area
I have a random point generator which takes a distance $d$ and fills an area with points such that distance between any two points is no less that $d$:
I need to control the number of points in the ...
4
votes
2answers
392 views
Interpolation with the roots of orthogonal polynomials & Spectral expansion
I'm a bit confused about the relationships between these two approximation methods mentioned in the title.
Does this kind of interpolation also belongs to the field of spectral methods?
Are the ...
6
votes
3answers
3k views
Fast algorithm for Polar Decomposition
As it known, according to the Polar Decomposition, square matrix can be expressed in the next form $$M=QR$$ ($Q$ - orthogonal matrix $R$ - positive-semidefinite Hermitian matrix)
I need to find this ...
3
votes
2answers
2k views
Methods for fast approximation of convolution
What are the state of the art methods for fast 2D convolution approximation?
I'm familiar with SVD based multiplication and cross approximation approaches, but would be thankful to get additional ...
1
vote
0answers
45 views
Alternative to two “for” loops in finding best neighborhoods for TSP?
I am trying to solve Travelling Salesman Problems using tabu search. I have been able to successfully find "near enough" optimal solutions (as well as one optimal, yay!).
For the moment I am using ...
5
votes
0answers
127 views
Are there any benefits of computable analysis to numerical algorithms
Computers can work only with computable numbers, while most of the algorithms are based on analysis of real numbers (real analysis).
When I heard of the existence of computable analysis I ...
