Questions tagged [approximation]

Questions on approximating functions (e.g. PadeApproximant), approximating integrals, working with approximate values (e.g. RootApproximant) etc.

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Is function MiniMaxApproximation equivalent to Remez algorithm?

I'm looking for function calculating polynomial of best approximation (in sense of uniform norm) to given function $f(x)$ on interval $[a,b]$. I know Remez algorithm doing this. In ...
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126 views

What's the command for high approximation up to 50 digits?

I am trying to find the numerical value of $\displaystyle \sum_{n=1}^\infty \frac{e^{-n^2/\pi^2}}{n^2}$ up to 50 digits. I used ...
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4 votes
1 answer
203 views

Solving symbolically a trascendental equation containing an exponential

Is it possible to solve symbolically this equation for $x$: $$\exp \left(-x^2\right)=\frac{c_1}{\sqrt{c_2-c_3 x}}$$ Exp[-x^2] == c1/Sqrt[c2 - c3 x] $c_1$, $c_2$ ...
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  • 425
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2 answers
160 views

Minimax / Minmax optimization

For the complex polynomial $$P_n(z) := 1+z+\frac{z^2}{2} + \sum_{j=3}^n \gamma_j z^j,\quad z \in \mathbb C.$$ I want to solve the following minimax/minmax optimization problem: $$\min_{\gamma_j} \max_{...
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  • 113
1 vote
4 answers
349 views

Finding a closed-form solution or a better approximation to a function in Mathematica

I have the following equation: $$f(L) = \left\lfloor{\frac{1}{4}\sum_{n=1}^{L-1}\left\lfloor n+300\times2^{n/7}\right\rfloor}\right\rfloor$$ where $1\leq L \leq 99$ My goal is to find a closed-form ...
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  • 121
-1 votes
2 answers
86 views

Finding best polynomial approximation for function with complicated form and many terms [closed]

I have a function that contains 50 terms hence, it is a complicated-looking function. Now I expected it to be a polynomial function. How can I get this polynomial fit provided I don't have data, I ...
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0 votes
0 answers
46 views

How Mathematica searches prediction band lines?

After approximation, for example model=NonlinearModelFit[data, a + b*x + c*x^2, {a, b, c}, x] We can do this ...
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2 votes
1 answer
94 views

plot mollifier of a function in mathematica

My goal is the compute and plot the mollifier of a function $f(x)=(1-x^2)^{-1/4}\chi_{(-1,1)}.$ Given an approximation of the identity $\rho$, the mollifier is defined as $$f_\epsilon(x)=\int \rho_\...
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  • 91
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0 answers
44 views

approximating large numbers

I want to get the zeros of a function $f(w,a)$ with respect to $a$. Thats not the problem. But the problem is, that the zeros are very large numbers. And I want them to be displayed in a short ...
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1 vote
1 answer
153 views

Homework from studies! - module computation of an approximating function

Polynomial approximation The aim of the task: to write a program in the form of a module in the Mathematica® machine code computation of an approximating function F (·) for an approximated ...
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0 votes
1 answer
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Why do these two methods give different answers for this simple approximation?

I tried to find Padé approximant of the function below using two different methods, but the results were not equal. $f(x)=1+x+x^2+x^3+\cdots$ Method 1: Using the direct built in function of Padé ...
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0 answers
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Multivariate Pade approximant

Does Mathematica generate multivariate or just two-variate Pade approximant? If so, what is the generating command?
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  • 179
2 votes
1 answer
101 views

Setting Precision in For

How do you set the precision for a function where you substitute a number in a For loop? I mean I code something like this and I want Mathematica to set the precision for all the values ...
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1 vote
0 answers
54 views

What Algorithm Does EconomizedRationalApproximation Use?

I'm very interested in Approximation Theory, especially with Continued Fractions and Rational Functions. I really like the approximations that Mathimatica gives with EconomizedRationalApproximation[]...
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2 answers
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Nested NIntegrate over a region: Numerical Integration not converging

I need to computed two double nested numerical integrals, of which one is defined over a specific region (e.g. a pentagon). I've tried to use a single NIntegrate with four variables, but the result is ...
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  • 135
-1 votes
1 answer
68 views

Series expansion of a function up to linear terms [duplicate]

I have the following: \[CapitalSigma] = r^2 + a^2 Cos[\[Theta]]^2; \[CapitalDelta] = r^2 - 2 M r + a^2 - k/3 r^2 (r^2 + a^2); grr = \[CapitalSigma]/\[CapitalDelta]; ...
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0 answers
118 views

Inverse Functions

I'm new to Mathematica, so I fell sorry for posting relativily naive/dumb questions. I have these two functions that represent X coordinate distortion of an SLA 3D printer. ...
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Numerical Integration of Data in Sensitivity Analysis

I'm attempting to do some Sensitivity Analysis on the following function of interest $f(\pmb x) = x_1 + x_2 x_3^2 $. The variables have the ranges of {1,1000}, {1,100}, {1,10}. The first step is to ...
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2 answers
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Approximating giant polynomial expression

Suppose I have a giant polynomial expression analogous to $$ F=a_1xX^6+a_2x^2X^5+a_3x^3X^4+a_4x^4X^3+a_5x^5X^2+a_6x^6X $$ and it is true that $X \gg x$. Let it be enough for me to approximate this ...
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  • 353
2 votes
2 answers
350 views

Two-Points Padé Approximant

My Question is about two points Pade approximant From Mathematica References. I just find the Pade approximant for a real function in one point Example : ...
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1 vote
2 answers
87 views

Function decomposition to Fourier series using first impulse function

I have periodic function with certain first impulse function and period value. My task is to decompose it to approximate Fourier series (k_max = 10) using Laplace transform of first impulse function: ...
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2 votes
1 answer
116 views

Comparing the plots of two functions in number theory

Definition. For $x>1$, let $$R(x):=\sum_{n\ge1}\frac{\mu(n)}{n}\,\operatorname{li}(x^{1/n})$$ denote the Riemann prime counting function. If you are not familiar with the mathematical expressions ...
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4 votes
2 answers
351 views

Suggest an irrational number from decimal one

I want to know is there any function in Mathematica that suggests a simple irrational number combination for decimal one? For example, if I give 0.804738 then I get ...
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0 votes
1 answer
49 views

3D plot using Table & FindRoot

I have two expressions involving terms $S_1$ and $S_2$, call the expressions $F1, F2$. I cannot solve $F_i=1$ for $S_i$ so instead I numerically approximate using ...
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1 vote
3 answers
174 views

How to approximate the harmonic sum $\sum_{n=1}^\infty\frac{{4n\choose 2n}\overline{H}_{2n}}{n 2^{4n}} ?$

How to approximate $$\sum_{n=1}^\infty\frac{{4n\choose 2n}\overline{H}_{2n}}{n 2^{4n}} ?$$ Where $\overline{H}_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k}$ is the skew harmonic number. The mathematica ...
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-1 votes
1 answer
83 views

Numerically Approximating Solutions to Differential Equation

I'm trying to numerically approximate solutions to a messy differential equation, given below $$(1-\alpha \frac{1}{\pi^{'}(s_2)})(s_2-\pi(s_2)+\frac{\beta}{2}\pi(s_2)-\frac{\alpha\beta}{2}s_2)+(p-\...
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7 votes
3 answers
247 views

How to approximate $PV\int_0^\infty \frac{\tan x}{x}\text{d}x?$

What's the mathematica command to get the numerical value of : $$PV\int_0^\infty \frac{\tan x}{x}\text{d}x?$$ where $PV$ is the principal value.
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0 votes
1 answer
66 views

Plotting function and its approximation function

I have a problem which I have not been able to solve. I want to plot a function and and operator which approximates it when you let w to infinity. I will give all ...
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0 votes
1 answer
82 views

Finding an approximate solution to this integral: $\alpha(\phi,r,p,d)=\int_0^\infty w(z,r,p,d)Q(z,r,\phi)dz$

I'm working on a physics problem and encountered a rather complex integral for which I'm trying to find an approximate solution. The integral is of the following form: $\alpha(\phi,r,p,d)=\int_0^\...
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3 votes
4 answers
449 views

Nested optimization problem - Function approximation

I need to maximize the following function (the input to NMaximize below) ...
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1 vote
1 answer
150 views

Optimizing my code for Broyden's method

The following code is my attempt of employing Broyden's method for root finding on the function $f(x,y)=(e^{xy}-y^2-2,\cos(x+y)+\frac{1}{2})$. Where the first matrix is the Jacobian, then it gets ...
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1 vote
2 answers
186 views

Minimizing simple function of three variables fails

I need to minimize simple function with all variables are positive integers, but the out is the same as the input. No solution ...
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0 votes
1 answer
57 views

Expansion of nonlinear functions with damping properties in exponential series

I am working on solving nonlinear differential equations and found such a solution with exponential properties. $\frac{dx}{dt}=\frac{d}{dx}(sech(x)^2)$ The solution of which is: $x(t) = \sinh ^{-1}\...
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6 votes
3 answers
221 views

Higher order Around, for large error propagation

TLDR question: How to redefine Around to work with higher order approximation. Motivation From the documentation Around ...
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8 votes
1 answer
180 views

How to find a numerical antiderivative with NIntegrate methods?

@JimBelk asked in Interpolating an Antiderivative how to find a numerical antiderivative. I gave an answer that uses NDSolve with the default method for integrating ...
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  • 216k
1 vote
2 answers
140 views

Evaluate function at "good" points used by Plot in numerical problems

Plot function with option Mesh->All shows how mathematica evaluates function to make it most optimal for plotting. I'd like to evaluate some physical function in ...
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5 votes
1 answer
155 views

Series function not expanding an expression

I have the following code: FDE[d_, η_] := η^(d + 1)/Gamma[d + 2] + π^2/(6*Gamma[d])*η^(d - 1); Series[FDE[d/t, 1/η]/FDE[d/t - 1, 1/η], {η, 0, 3}] The series ...
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2 votes
1 answer
84 views

FindFit low speed when fitting polynomial coefficients: is it possible to significantly increase the speed?

I work in Mathematica with a Butterworth filter, the transfer function of which depends on the selected order and cutoff frequency. I want to evaluate the change in the location of the roots on the ...
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1 vote
0 answers
127 views

Approximation of a function by a polynomial (Chebyshev First Kind, Bernstein, etc...) containing only even degrees and constants in a given Range[a,b]

In Mathematica, how can I create a polynomial function containing only even degrees and constants? That is, I have a function: $f(x)=\frac{\pi ^2}{\left(\frac{\pi }{2}-\tan ^{-1}(k (x-1))\right)^2}$ ...
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2 votes
0 answers
56 views

Approximation of roots using Series

I am solving a fifth degree polynomial using Series. My equation looks like ...
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0 votes
0 answers
44 views

Nintegrate providing results much smaller than what is expected

This is the first time I am posting something here. So, apologies if I make any mistake. I am dealing with a numerical integration in Mathematica-11.0 that has a Bessel function in it. In order to ...
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7 votes
3 answers
825 views

Extracting a function from a Contour Plot

Context to understand the question Suppose that I have the next equation sol = ParametricNDSolve[{y'[t] == a y[t], y[0] == 1}, y, {t, 0, 10}, {a}] And I make a ...
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  • 585
1 vote
1 answer
162 views

How do you find the Inverse of Elliptic Integral of Second Kind when modulus is large

So I tried to take the inverse of EllipticE when modulus is large, in Mathematica, but the solution gives wrong answer. ...
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2 votes
0 answers
36 views

Setting the tolerance for `Equal[]` [duplicate]

I would like to control how Equal[] works and allow a certain error associated with it. I would like numerical values which are, say, within one-thousandth of one ...
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  • 3,862
3 votes
1 answer
34 views

Estimating the range of values of x for which the absolute error in the approximation of $\cos[x]$ is accurate to within 0.08

The approximation is for $\cos[x] \approx 1 - \frac{x^2}{2}+ \frac{x^4}{24}$. I've tried a combination of many different things, but can't figure it out. My error always ends up as {0.00136436}. Any ...
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  • 141
1 vote
1 answer
45 views

How does the Bias parameter adjusts Chebyshev nodes in RationalInterpolation of the Function Approximation Package?

How does the Bias parameter adjusts Chebyshev nodes in RationalInterpolation of the Function Approximation Package? What is the mathematical mapping/formula applied in the Bias adjustment from -1 1 ...
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2 votes
1 answer
73 views

Numerical solution of a differential equation with a condition on a parameter

I'm considering the Mazenko equation as it's written in https://doi.org/10.1103/PhysRevB.46.10594 (eq. 7) \begin{equation} f''+\left(\frac{1}{x}+\frac x 4 \right)f'+\frac \lambda \pi \,\tan\left(\...
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0 votes
1 answer
76 views

QR-Decomposition [closed]

I should make a program in which with help of QR-decomposition find approximation of x^sinx shaped a+bLnx+c*e^x for a values x € ...
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  • 101
1 vote
1 answer
143 views

Is it feasible to curve fit to a staircase function? [closed]

Taking as a starting point the advice given in this post I am trying to fit data to a staircase function as follows: ...
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0 votes
0 answers
43 views

What is the limit of digits to show in the conversion of an exact number to an Arbitrary Precision Number?

By executing something like N[π, c] what is the maximum positive integer value c can have until the output is compromised by ...
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