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Questions tagged [approximation]

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

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Solving a complicated equation for approximate analytical solution

I am trying to solve the Following equation for r, 2*a*Q^4 + 5*r^4*(-4*Q^2 - 2*r*(-3*M + r) + 3*c*r^(1 - 3*\[Omega])*(1 + \[Omega])) ==0 Clearly this is unsolvable....
Debojyoti Mondal's user avatar
9 votes
1 answer
409 views

Calculating relative error of Ramanujan formula for ellipse perimeter

On this page, they present the Ramajujan's second formula for the perimeter of an ellipse: $$P \approx \pi (a+b) \left(1+ \frac{3 h}{\sqrt{4-3 h}+10}\right),$$ where $h=(a-b)^2/(a+b)^2$. They expand ...
Ytrewq's user avatar
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0 answers
172 views

How can I solve $\sum_{i=1}^{M-1} (M+i)^{M+i+1/2}/i^{i+1/2}$? [migrated]

I am trying to solve an equation in Mathematica: $$ \sum_{i=1}^{M-1} \frac{(M+i)^{M+i+\frac{1}{2}}}{i^{i+\frac{1}{2}}} $$ Does a general solution exist for this expression? And if $M \to \infty$, can ...
No Yeah's user avatar
  • 53
-1 votes
1 answer
91 views

Discrepancy between Mathematically Calculated Maximum Error and Maximum Error Graphically Observed in Absolute Difference Plot [closed]

Question I am encountering a discrepancy between the maximum error calculated mathematically using the NMaximize command in Mathematica and the maximum error observed graphically in the absolute ...
Alvira Yawar's user avatar
3 votes
2 answers
329 views

Is there a way in Mathematica to simplify expressions based on inexact assumptions? [closed]

Is there a way in Mathematica to simplify expressions based on inexact or "relative" assumptions such as $x<<y$ or $x\approx y$? I see ...
Andrew Cheong's user avatar
0 votes
0 answers
169 views

How to get an easy to compute representation defined by plot of involved function?

I am doing a windmill vortex sheet calculation where I have to guess its shape and then vorticity distribution and then make sure that the net flow is along the sheet. The problem is adjusting my ...
simon's user avatar
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0 answers
81 views

How can I approximate a function with composite polynomials?

I know that to approximate a function with for example $f(x) = \sin(x)$ using polynomials with degrees up to 4, I can use the Fit function : ...
何子钦's user avatar
10 votes
5 answers
1k views

How can I get samples of f(x) that are roughly evenly spaced?

I am looking for an efficient way of getting 60 to 80 samples of an arbitrary f(x) such that the distance between adjacent samples are approximately equal. My first attempt is based on a first order ...
Ted Ersek's user avatar
  • 7,134
0 votes
1 answer
43 views

How can we obtain an approximate simplified equation (QsNr) from a set of given equations using Mathematica?

I am trying to find an approximate equation for a complicated expression (QsNr) using Mathematica. The equation involves several parameters and expressions, and I want to simplify it by considering ...
Sumit Sagar's user avatar
1 vote
1 answer
117 views

How to invert this function when the argument e is small?

I need to find an analytical solution the following equation for e in terms of n and p ...
Dotman's user avatar
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1 vote
1 answer
55 views

Graph lines do not show dashes [closed]

I have the following code of an iteration: ...
Junaid Ahmad's user avatar
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0 answers
38 views

How to treat small numbers in order to gain efficiency and precision in neural network algorithm?

Unfortunately I won't be able to provide a MWE, as it would be too big and complex, and this question will be based on pure semantics, basically (and I understand if this is impossible to answer in ...
GaloisFan's user avatar
  • 405
3 votes
0 answers
70 views

How to ask Mathematica to give me possible closed forms from a given numerical approximation

Imagine I am solving a some Infinite Series and I cannot directly compute the value using Mathematica. The value of this series approximates $ 0.785398....$ (Actually is $\frac{\pi}{4}$). I want to be ...
No-one Important's user avatar
0 votes
2 answers
86 views

Calculation of Approximate Equation with Mathematica

I am trying to find an approximate equation for a complicated expression (L1, L2) using Mathematica. The equation involves several parameters and expressions, and I want to simplify it by considering ...
Sumit Sagar's user avatar
1 vote
1 answer
104 views

How to obtain approximate form for a calculation using Mathematica with specific approximations?

I am currently working on a calculation in Mathematica that involves various parameters, including Nb, k, and Ns. I want to obtain an approximate form of the calculation by considering the following ...
Sagar's user avatar
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1 answer
81 views

Approximate an expression with condition

I do the following computation: ...
JieJiang's user avatar
-2 votes
1 answer
120 views

How to find an approximate equation from a given equation using Mathematica? [closed]

I am interested in finding an approximate equation based on a given equation using Mathematica. Specifically, I am working with the quantum illumination transmitter's quantum Chernoff bound (...
Sumit Sagar's user avatar
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0 answers
41 views

Coaxing DSolve: Second-order PDE

I am trying to solve the following PDE for a function $\Gamma(q, r)$, which I have derived in the context of an optimal control problem: $$ 2 (\gamma -1) \rho ^{\frac{\gamma }{\gamma -1}} \left(q^{\...
Anthony's user avatar
  • 231
5 votes
3 answers
431 views

Better Stirling Approximation Error

Stirling's Approximation is given by $$n! \sim \sqrt {2\pi n} \left ( \frac{n}{e}\right)^n$$ From a substantial improvement of the Stirling formula, we have an elegant approximation given by $$n! \sim ...
Moo's user avatar
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6 votes
4 answers
443 views

Approximation of the Fabius function with a quotient of exponentials

Approximation of the Fabius function $f(x) = \text{FabiusF}[x+1]\cdot \text{HeavisideTheta}[1-x^2]$ - FabiusF[x] doesn't work in Wolfram-Alpha I am looking to figure out how well the displaced version ...
Joako's user avatar
  • 163
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0 answers
69 views

Approximating Exp[-x] in partial fraction form [duplicate]

I'm looking to obtain order-$k$ approximation of $\exp(-z)$ for real-valued $z$. $$R_k(z)\approx \exp(-z)$$ The constraint is that I need the result in partial fraction form, ie: $$ \begin{equation} ...
Yaroslav Bulatov's user avatar
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0 answers
41 views

Using FindFit for Approximation Fourier Series

I am expecting to have a list of discrete points, which I would like to approximate using Fourier's series, then plot Fourier series approximation alongside with discrete points, then plot terms of ...
Akhvlediani's user avatar
1 vote
1 answer
139 views

Wolfram Mathematica Monte Carlo for integrals approximation

I wanted to implement the Monte Carlo method for multiple integrals approximation in Wolfram Mathematica. Namely I wanted to let the user insert as input the dimension of the integral and the number ...
Greg's user avatar
  • 11
2 votes
2 answers
138 views

How can I calculate complicated infinite sums with FindIntegerNullVector (or related methods)?

I've recently been very interested in the wonderfully complex world of Euler sums, i.e. (convergent) infinite sums that, roughly speaking, consist of some rational polynomial combination of ...
TheOutZ's user avatar
  • 215
3 votes
3 answers
653 views

Polynomial approximation of max function

Let me just say upfront I'm not a mathematician, I'm rather looking for a practical answer to my question. I was wondering if there is a polynomial approximation for the function $$\max(0,x)=\left\{\...
Jimakos's user avatar
  • 169
1 vote
1 answer
65 views

Representing a number in r0 + r1 E + r2 E^2 form

Let E be the base of natural logarithm 2.71... A Sequence S[n] is believed to converge to a ...
imida k's user avatar
  • 4,295
6 votes
2 answers
200 views

Approximation in LinePlot from NIntegrate

I am facing an approximation issue during a plot from numerical integration through NIntegrate. The code is: ...
A_Dutta's user avatar
  • 115
1 vote
2 answers
133 views

Fundamental matrix solution of a differential equation $x'=A(t)x$

In this question about Floquet theory the author asked about the fundamental matrix solution $X(t)$ of the following $2\pi$-periodic differential equation $${\displaystyle {\dot {x}}=A(t)x}$$ with $$A(...
user50618's user avatar
  • 123
2 votes
0 answers
32 views

Can we approximate a matrix power series like NSum does?

Essentially, the following does not work, and I'm wondering if it can be made to: NSum[ MatrixPower[B,n], {n,0,∞}] (Here B is a ...
Good Boy's user avatar
  • 121
1 vote
1 answer
135 views

Analytic continuation of a data set from the upper complex plane to the lower complex plane?

Context I am interested in identifying damped modes such as those in self gravitating galaxies: This requires extending to the lower complex plane a dispersion relation which is computed numerically ...
chris's user avatar
  • 23k
5 votes
3 answers
285 views

Making algebraic substitutions with approximations

I'm struggling to make a substitution in a symbolic expression where I want to use an approximation. For example if we have the equation $$\frac{1}{a}\frac{c}{a - b} = \frac{c}{a^{2} - a b} \approx \...
user27119's user avatar
  • 2,520
3 votes
2 answers
356 views

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 ...
Maciess's user avatar
  • 31
1 vote
0 answers
135 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 ...
Ali Shadhar's user avatar
4 votes
1 answer
212 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] $c_1$, $c_2$ ...
umby's user avatar
  • 585
1 vote
2 answers
304 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_{...
Dan Doe's user avatar
  • 113
1 vote
4 answers
458 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 ...
Suic's user avatar
  • 121
0 votes
2 answers
751 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 ...
RISHAV SAGAR's user avatar
3 votes
1 answer
197 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_\...
Student's user avatar
  • 113
0 votes
0 answers
74 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 ...
peter mafai's user avatar
1 vote
1 answer
161 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 ...
KMbuuble's user avatar
0 votes
1 answer
73 views

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 ...
user avatar
0 votes
0 answers
190 views

Multivariate Pade approximant

Does Mathematica generate multivariate or just two-variate Pade approximant? If so, what is the generating command?
Hans's user avatar
  • 179
2 votes
1 answer
152 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 ...
Aji Wibowo's user avatar
1 vote
0 answers
75 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[]...
jedi945's user avatar
  • 11
0 votes
2 answers
123 views

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 ...
MariNala's user avatar
  • 135
-1 votes
1 answer
129 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]; ...
user583893's user avatar
0 votes
0 answers
154 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. ...
Anton's user avatar
  • 345
0 votes
2 answers
58 views

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 ...
GaloisFan's user avatar
  • 405
2 votes
2 answers
971 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 : ...
user745750's user avatar
1 vote
2 answers
154 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: ...
Cpp Nosavvier's user avatar

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