# Tagged Questions

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### How to obtain Padé approximant of $\log(x)$? [on hold]

Having read this answer on Math.SE, I wanted to try seeing how Padé approximations converge to $\log(x)$. But my first attempt after reading the documentation failed: ...
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### Integral evalutation

I am trying to integrate the following expression over $L$. E^(-2 L n) (1 - L/(2 s))^(-1 + 4 n µ) (L/s)^(-1 + 4 n v) I did... ...
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### 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 ...
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### 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) ...
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### How to generate specific function using BSpline command/

I want to generate an approximation function that fits a curve to points. My goal is to obtain an actual formula. Is this possible with the BSpline function? Using the following code: ...
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### Approximate a function of four variables with a rational polynomial

I have a function that consists of four variables. Currently, I want to view it as function of one variable over the interval (0, 1) with the other three variables as parameters. I will plug values ...
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### 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 ...
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### How to approximate a computationally expensive 5-dimensional manifold?

I have a 4-parameter family of functions, which I'll refer to as y[a, b, c, d][x]. In this notation, a, b, c, d are the 4 ...
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### Solving an Integral Numerically

I have been trying to solve the integral equation below, but cant seem to find a way out of this. Can someone please help me out with suggestion? $f(t)=\int_0^{\infty}\frac{K_1a(t)}{a(t)+K_2}\,dt$ ...
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### Trying to find the best Pade approximant for a given numer of terms

Let suppose that I have a function $f[x]$ I want to approximate using a Pade expansion and that I decide what would be maximum number of terms to be used. Is there any way with Mathematica to find ...
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### Formulas for combining ChebyshevT coefficients?

I have two 5th-order (6th-order?) Chebyshev approximations: f1[x_] = Sum[a[n]*ChebyshevT[n,x],{n,0,5}] f2[x_] = Sum[b[n]*ChebyshevT[n,x],{n,0,5}] and f1[1] == ...
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### 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: ...
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### 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 obrain the error function. When I take the second integration, ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### Transforming Sin into Abs[Sin] dynamically

Is it possible to interpolate a polynomial that approximates Sin, and then be able to manipulate the polynomial or a sample of its points to make sections of the ...
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### 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: ...
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### Formatting results of integral approximation methods comparison.

The integral is Integrate[Sqrt[1+x^3], {x, -1, 1}]. I want to create a Table with three columns and four rows that shows the ...
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### 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. ...
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### 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. ...
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### 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 ...
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### Multivariate Function Approximation With a Large Dataset

I have a nice amount of data from a trading strategy I am working on, where I have two different liquidity parameters as x, y variables. Before entering a trade I am taking the moving average of ...
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### Calculating error of the approximate formula in calculations

I have an exact equation: $$a = \sqrt{1 + x^2 - \sqrt{(1 + x^2)^2 - 2 x^2 \cos^2\theta}}\tag{1}$$ and its approximation: a = \dfrac{x ...
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### 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 ...
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### 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 ...
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) = ...