# 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 ...
1 vote
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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] $c_1$, $c_2$ ...
1 vote
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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 ...
1 vote
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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### 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 ... 95 views

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 ...
1 vote
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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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### 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 ...
350 views

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 : ...
1 vote
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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: ...
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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### 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 ...
1 vote
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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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### 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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### Higher order Around, for large error propagation

TLDR question: How to redefine Around to work with higher order approximation. Motivation From the documentation Around ...
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 ...
1 vote
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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 ...
1 vote
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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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### 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 ...
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 == 1}, y, {t, 0, 10}, {a}] And I make a ...
1 vote
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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. ...
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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 ...
1 vote
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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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### 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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### 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 € ...
1 vote