Questions tagged [approximation]
Questions on approximating functions (e.g. PadeApproximant), approximating integrals, working with approximate values (e.g. RootApproximant) etc.
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 (...
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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:
$$
2 (\gamma -1) \rho ^{\frac{\gamma }{\gamma -1}} \left(q^{\...
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Better Stirling Approximation Error
Stirling's Approximation is given by
$$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 ...
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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 ...
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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:
$$
\begin{equation}
...
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Using FindFit for Approximation Fourier Series
I am expecting to have a list of discrete points, which I would like to approximate using Fourier's series, then plot Fourier series approximation alongside with discrete points, then plot terms of ...
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Wolfram Mathematica Monte Carlo for integrals approximation
I wanted to implement the Monte Carlo method for multiple integrals approximation in Wolfram Mathematica. Namely I wanted to let the user insert as input the dimension of the integral and the number ...
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How can I calculate complicated infinite sums with FindIntegerNullVector (or related methods)?
I've recently been very interested in the wonderfully complex world of Euler sums, i.e. (convergent) infinite sums that, roughly speaking, consist of some rational polynomial combination of ...
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Polynomial approximation of max function
Let me just say upfront I'm not a mathematician, I'm rather looking for a practical answer to my question. I was wondering if there is a polynomial approximation for the function
$$\max(0,x)=\left\{\...
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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 ...
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Approximation in LinePlot from NIntegrate
I am facing an approximation issue during a plot from numerical integration through NIntegrate. The code is:
...
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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
$$A(...
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Can we approximate a matrix power series like NSum does?
Essentially, the following does not work, and I'm wondering if it can be made to:
NSum[ MatrixPower[B,n], {n,0,∞}]
(Here B is a ...
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Analytic continuation of a data set from the upper complex plane to the lower complex plane?
Context
I am interested in identifying damped modes such as those in self gravitating galaxies:
This requires extending to the lower complex plane a dispersion relation which is computed numerically ...
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Making algebraic substitutions with approximations
I'm struggling to make a substitution in a symbolic expression where I want to use an approximation. For example if we have the equation
$$\frac{1}{a}\frac{c}{a - b} = \frac{c}{a^{2} - a b} \approx \...
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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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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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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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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:
$$\min_{\gamma_j} \max_{...
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Finding a closed-form solution or a better approximation to a function in Mathematica
I have the following equation:
$$f(L) = \left\lfloor{\frac{1}{4}\sum_{n=1}^{L-1}\left\lfloor n+300\times2^{n/7}\right\rfloor}\right\rfloor$$ where $1\leq L \leq 99$
My goal is to find a closed-form ...
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Finding best polynomial approximation for function with complicated form and many terms [closed]
I have a function that contains 50 terms hence, it is a complicated-looking function. Now I expected it to be a polynomial function. How can I get this polynomial fit provided I don't have data, I ...
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plot mollifier of a function in mathematica
My goal is the compute and plot the mollifier of a function $f(x)=(1-x^2)^{-1/4}\chi_{(-1,1)}.$ Given an approximation of the identity $\rho$, the mollifier is defined as
$$f_\epsilon(x)=\int \rho_\...
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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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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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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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Multivariate Pade approximant
Does Mathematica generate multivariate or just two-variate Pade approximant? If so, what is the generating command?
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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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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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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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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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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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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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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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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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how to transform a decimal into a radical equivalent with minimal error,and find a correct syntax?
I have been solving this equation x^x^x == 36 , Mathematica does not give me a solution by applying traditional methods , neither with ...
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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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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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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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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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Numerically Approximating Solutions to Differential Equation
I'm trying to numerically approximate solutions to a messy differential equation, given below
$$(1-\alpha \frac{1}{\pi^{'}(s_2)})(s_2-\pi(s_2)+\frac{\beta}{2}\pi(s_2)-\frac{\alpha\beta}{2}s_2)+(p-\...
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How to approximate $PV\int_0^\infty \frac{\tan x}{x}\text{d}x?$
What's the mathematica command to get the numerical value of :
$$PV\int_0^\infty \frac{\tan x}{x}\text{d}x?$$
where $PV$ is the principal value.
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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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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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