Questions tagged [approximation]
Questions on approximating functions (e.g. PadeApproximant), approximating integrals, working with approximate values (e.g. RootApproximant) etc.
179
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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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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:
$$\min_{\gamma_j} \max_{...
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4
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Finding a closed-form solution or a better approximation to a function in Mathematica
I have the following equation:
$$f(L) = \left\lfloor{\frac{1}{4}\sum_{n=1}^{L-1}\left\lfloor n+300\times2^{n/7}\right\rfloor}\right\rfloor$$ where $1\leq L \leq 99$
My goal is to find a closed-form ...
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2
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Finding best polynomial approximation for function with complicated form and many terms [closed]
I have a function that contains 50 terms hence, it is a complicated-looking function. Now I expected it to be a polynomial function. How can I get this polynomial fit provided I don't have data, I ...
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How Mathematica searches prediction band lines?
After approximation, for example
model=NonlinearModelFit[data, a + b*x + c*x^2, {a, b, c}, x]
We can do this
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2
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1
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plot mollifier of a function in mathematica
My goal is the compute and plot the mollifier of a function $f(x)=(1-x^2)^{-1/4}\chi_{(-1,1)}.$ Given an approximation of the identity $\rho$, the mollifier is defined as
$$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 ...
user82631
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95
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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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0
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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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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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1
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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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2
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1
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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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Nested optimization problem - Function approximation
I need to maximize the following function (the input to NMaximize below)
...
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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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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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1
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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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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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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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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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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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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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0
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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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Approximation of roots using Series
I am solving a fifth degree polynomial using Series. My equation looks like
...
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44
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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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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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1
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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.
...
2
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0
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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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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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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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73
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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(\...
0
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1
answer
76
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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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1
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143
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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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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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