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17
votes
1answer
340 views

How can I smooth a 3D surface generated by RegionBoundary?

I have a set of points that belong to the surface of an object. I would like to approximate a surface to which I can compute the distance. of other points. With the nice code of this post, I was able ...
11
votes
2answers
1k views

How to improve the accuracy of this Monte Carlo simulation

A friend of mine went on an interview and he was asked to calculate the angle between the hour hand and minute hand of an analog clock when the time was 3:15. Arguably, this is a trivial problem that ...
10
votes
3answers
485 views

Chebyshev Approximation

Is there functionality in Mathematica to expand a function into a series with Chebyshev polynomials? The Series function only approximates with Taylor series.
8
votes
3answers
314 views

Find parameters that minimize the distance between two curves in terms of the infinite norm

I am trying to implement a very simple NMinimize code that would search for parameters of a polynomial that minimize (globally) the distance between this polynomial ...
7
votes
4answers
806 views

Solving $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 \pi ^2 A^2+W^2}+6 \pi ^2 A^2+3 W^2}}{\sqrt{2}}+\frac{3 W}{2}$ for $W$

When I solve the aforementioned equation for $W$ or $A$ on Mathematica I get a long and ugly equation in return, namely one of the solutions for $W$ is: (attempt to read at your own health) ...
7
votes
2answers
395 views

Approximation to the prime counting function

Is there a function similar to PrimePi that gives approximate value for large numbers? In particular, I need a reasonably good approximation for $\pi(x)$, where $x\...
7
votes
1answer
115 views

Approximating for $a \gg b$

I'd like to know how I could go about making approximations where one quantity is much smaller or larger than another. For example, the expression $\frac{1}{b(a +b)}$ is approximately equal to $\frac{...
7
votes
2answers
237 views

Efficient Lloyd Sampling of Images

I want to sample a grayscale image so that every voronoi cell contains approximately the same total intensity using lloyd sampling. My current code is kind of slow and I was hoping for some advice to ...
6
votes
1answer
188 views

First order approximation

How can I neglect higher orders of approximation in Mathematica? Suppose I want to find the roots of a simple quadratic equation like ...
6
votes
3answers
208 views

Approximate analytic solution to a polynomial equation

The analytic solution of the equation $$ 1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\Lambda}{3}r^2 = 0 $$ obtained by using the code ...
6
votes
1answer
394 views

GeneralMiniMaxApproximation - Dividing out the zero

MiniMaxApproximation is used to generate minimax approximations. I'm interested in the "dividing out the zero" trick. The documentation discusses a nice example. Suppose we want to construct a ...
5
votes
5answers
186 views

Isolating cross terms

I have a set of expressions that contains terms like $\frac{1}{(x + a + b)(c+d)}$. I would like to simplify the denominator so that products of $a,b,c,d$ are dropped, but $x c$ and $x d$ are kept. ...
5
votes
1answer
1k views

Polynomial Approximation from Chebyshev coefficients

I would like to expand a function $f(r)$ in the domain $[0,R]$, around the points $r =0$, and $r = R$ in the following manner $f(r = 0) = \Sigma_{i=0,i = even}^{imax} f_i (r/R)^i$ and $f(r = R) = ...
5
votes
1answer
524 views

Time approximation of decrypting RSA algorithm

I've written a function that encrypts a text using the RSA algorithm. It then decrypts it using prime factorization, and takes the time it took to decrypt it and puts it in a vector together with the ...
4
votes
1answer
284 views

Series approximation to integral

I would like to approximate the integral $$ \int_0^\infty dy\,\frac{1}{\sqrt{2\pi y\sigma^2}}\exp\left(-\frac{(x-y)^2}{2y\sigma^2}\right)f(y), $$ as a series expansion in the limit $\sigma\rightarrow ...
4
votes
1answer
518 views

Small Issue with Chebyshev Derivative Approximation

I am trying to approximate the derivative of a function from its Chebyshev expansion. I start out with the following random function: ...
4
votes
2answers
127 views

Torus-geometry algebraic equations using Nsolve and Reduce

Somehow a set of naive-looking equations cannot be solved by using NSolve. Mathematica returns a message like this: ...
4
votes
0answers
51 views

Why does MiniMaxApproximation fail in this simple case?

Why does MiniMaxApproximation[Erfc[t], {t, {0, 2}, 0, 1}] give the following error? ...
3
votes
2answers
493 views

Assigning an analytical approximation to the error function erf(x)

Working with some iterative integral equations, I have Gaussian density functions involved therein. Integrating the Gaussian function I obtain the error function. When I take the second integration, ...
3
votes
3answers
197 views

FindFit with a sophisticated function (integral)

I am trying to find a fit to the distribution function (empiricial data) in terms of a function which is itself an integral of a product of two simpler functions (two polynomials), that is the model. ...
3
votes
1answer
180 views

Find root approximants for low-precision numbers

I want to be able to find low-complexity algebraic approximants to decimal numbers. For example, 1.41 can be approximated as a solution to the polynomial x^2==2. ...
3
votes
1answer
61 views

Looking for a convenient way to replace a large data sample with an analytic approximation

Suppose that data is a large data sample, and that Histogram[data] produces a very smooth-looking distribution. Since ...
3
votes
0answers
98 views

Writing a function to set up and solve the least squares problem

I have setup the explicit line equation for a given set of points using least-squares approximation and the knots computed below. I have tried extending the same code for use with an arbitrary degree, ...
2
votes
2answers
131 views

Approximation of $f(x)$ at large values of $x$?

Suppose you a function $f(x)=1/x+x^2$. At large values of $x$, it is closely approximated by $f(x)=x^2$. Can Mathematica do this for me for a more complicated function? How would I input the ...
2
votes
1answer
118 views

Approximating Magnitude of Solutions

I was recently lecturing on the hierarchy of functions in a calculus class. Discussing the fact that eventually any exponential growth function will overcome any polynomial. As an example, I tried ...
2
votes
2answers
349 views

Perturbation theory with Mathematica: Definite integral of polynomial times exponential times hypergeometric function of imaginary argument

I would like to ask also Mathematica users about my question from the math forum. To expand, I'm adding the code which calculates the full double integral for $n=0$ and $\mu=0$ (the second in the post)...
2
votes
1answer
66 views

Finding input of the function $f(a,b,c) = (d,e)$ such that $|d-e|$ = constant for $d\in[d_0,d_1]$

The actual question contains a few more parameters than mentioned in the title, but the idea will remain very similar. Lets begin abstractly; I have a certain function $f(a,b,c) = (d,e,g,h)$. I am ...
2
votes
1answer
171 views

Seeking simple function in interval

I am a bit rusty, and struggling to solve this problem. Looking for a simple, well behaved parametric function f[t], from [0,1] to [0,1] such that: f[0] = f0; f[1] = f1; f'[1] = ff1; where -1<= ...
2
votes
3answers
172 views

Use NMinimize instead of FindFit for constrained search (of coefficients)

(My problem is more complex, but let us formulate it through this example) I am trying to find the best polynomial approximation to the following function ...
2
votes
0answers
25 views

Approximating Pi, fraction in another [duplicate]

In a previous exam, this question was asked: We want to approximate Pi using this formula Write a function that calculates the right hand side of this equation truncated at the nth term. For ...
2
votes
0answers
47 views

Computing an inexact derivative with some terms preserved in exact form

I am beginning to learn Mathematica and have the following question: How can I return a derivative with exact numbers instead of one involving numerical approximations? My original, single variable ...
2
votes
0answers
242 views

How do I estimate the Pareto front in my multiobjective optimization problem?

I need to minimize a function of two parameters that gives two "stress" outputs {A,B} = f[tau,mu]. How do I estimate (even roughly), the Pareto front for this ...
2
votes
0answers
133 views

Smooth Max and Abs

I'm trying to implement smooth approximations of Max and Abs functions. Moreover I want the functions to map element-wise on tensors. Here's my code. ...
1
vote
2answers
629 views

How to visualise an iterative approximation to pi

I'm experimenting with different algorithms that approximate pi via iteration and comparing the result to pi. I want to both visualise and perhaps know the function (if any) that describes the ...
1
vote
1answer
258 views

Showing separate surfaces

The XYZPipeJunction is removed using circular trig functions from Both. ...
1
vote
2answers
475 views

Approximate solution to system of polynomial equations

I'd like to solve approximately the following system of equations for $a_1$, $a_2$, $\alpha_1$ and $\alpha_2$ all real and nonzero: $a_1(1+4\alpha_2)^3+8a_2(1+4\alpha_1)^2+\frac{\sqrt{\pi}}{64}(1+4\...
1
vote
2answers
514 views

Calculating error of the approximate formula in calculations

I have an exact equation: \begin{equation} a = \sqrt{1 + x^2 - \sqrt{(1 + x^2)^2 - 2 x^2 \cos^2\theta}}\tag{1} \end{equation} and its approximation: \begin{equation} a = \dfrac{x \cos^2\...
1
vote
1answer
51 views

Approximating exponentials in a nice to read format

I need to make some approximations, basically I have something like $$ e^{i*a} = -0.735145 + 0.67791*I $$ and I want to approximate this to something that is easily readable, like $$ e^{\frac{3*\...
1
vote
2answers
314 views

How can I get an approximate contour plot from a 3D plot?

I have some larger square, Hermitian matrices, M, (dimension 20, say, much more than 4) with 2 independent variables, call them x and y. I can plot the eigenvalues as functions of x and y with: <...
1
vote
1answer
496 views

Relative error plot [duplicate]

I was plotting the relative error of the $e^{11/12 -n}n^{n+1/2}$ approximation to $n!$ as $n$ gets larger and larger, and at some very large value of $n$ Mathematica gives this plot: Can somebody ...
1
vote
1answer
171 views

Approaches to solving non-linear systems of equation with assumptions

The specific type of system looks as follows: \begin{align} (r-r')x+4d_1 x^3+4d_{12}xy^2 &= 0 \\ (r+r')y+4d_2 y^3+4d_{12}yx^2 &= 0 \\ \end{align} Where $r,r',d_1, d_2, d_{12}$ are all real ...
1
vote
1answer
279 views

Trying to find the best Padé approximant for a given number of terms

Let suppose that I have a function $f[x]$ I want to approximate using a Padé expansion and that I decide what would be maximum number of terms to be used. Is there any way with Mathematica to find ...
1
vote
0answers
33 views

How does WolframAlpha calculate closed form approximations of irrational numbers? [closed]

How does WolframAlpha do this: I'm aware of services such as RIES, which is an inverse equation solver, and ISC, the Inverse Symbolic Calculator. But WA goes much further. I've tried doing a ...
1
vote
0answers
111 views

How to derive finite-difference scheme automatically on a quite general stencil

Info This question is a generalization of the following one Derivation of numerical scheme for linear transport equation on a variable stencil. Statement of a problem Linear scalar hyperbolic (...
1
vote
0answers
59 views

FindFit with a sophisticated function (2), with corrected question and code

I am trying to find a fit to the distribution function (empiricial data) in terms of a function which is itself an integral of a product of two simplier functions. In particular, I observe T(x) (this ...
1
vote
0answers
220 views

Compute coefficient of Generating function

Given a generating function $F(z)$, is there any way to compute the coefficient $a_k$ of $z^k$ by Mathematica? One method is to calculate by the equation $\frac{F^{(k)}(0)}{k!}$. However, this ...
1
vote
0answers
71 views

Decimal representations of analytic values in an expansion

I have a solution to an ODE (which I call sol), which I would like to expand in terms of Log[1+x] around x=-1. I thought that: ...
0
votes
1answer
51 views

Two Variable First Order Approximation [duplicate]

I found that using Series and Normal command, I am able to approximate the first order of x or y, but If I also get rid of x*y (cross term), Series command does not work any more. Let's define some ...
0
votes
1answer
80 views

The 1st approximation of multi-viables which contains the differential [closed]

When I input to Mathematica as following, ...
0
votes
4answers
114 views

I want to cut off secondary approximate parts

When I input to Mathematica as follows, ...