Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [approximation]

For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

Approximations are representations of numbers that aren't exact, for example $\sqrt{2}\approx 1.41$. Such representations may be obtained using differentials (more generally, Taylor's formula), linear interpolation, etc.

4607 questions
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linear or quadratic approximation for $\exp(-x)$ for large $x$

Is there any linear or quadratic approximation of $\exp(-x)$ where $0
Pradipta
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How to include the fee of a sell order in itself.

Sorry for the poor title, I am sure there is a name for this problem (and an easy solution) I have an account balance of \$1000. When I want to buy some EUR, I need to sell USD with a fee of 1%. That means, if I want to sell \$1000 I can only sell…
Sandro
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Minimizing the $L^2$ error when approximating with trigonometric polynomials

I want to find approximations ${\rm g}_{n}\left(x\right) \in T_{n}$ of $\,\,{\rm f}\left(x\right)$, so that the error $$ \left\vert\left\vert\,{\rm f} - {\rm g}_{n}\,\right\vert\right\vert^{2} = \int_{0}^{2\pi} \left[{\rm f}\left(x\right) - {\rm…
mathNewbie
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Least squares problem linearization

Suppose we want to find the best coefficients $a$ and $b$ that fits the data we have according to a model of the form $$ y = a t e^{bt} \text{ or } y = a e^{bt} \text{ or } y = a \left( \frac{x}{b+x} \right) $$ for example. Then I know we can…
Guest
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Estimate a common formula

I know this should be easy, but I just can't find the proper search result. Thanks. $\left(1-\frac1n\right)^n$, what is the estimation value when $n$ is very large? Some follow-up, If $n = 100$, what is the formula to calculate this?
melodrama
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Absolute Error Confusion

I don't understand the following concept. It was given by the textbook. The dimensions of a cylinder are measured to the nearest millimeter using a measuring tape. The circumference is measured to be 22.0 cm and height measured to be 15.0 cm. Using…
Bobby
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Approximate expansion of Lorentz model near resonance frequency

I have a Lorentz model as $$\varepsilon_1 = \frac{\omega_p^2(\omega_0^2-\omega^2)}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2},$$ $$\varepsilon_2 = \frac{\omega_p^2 \gamma \omega}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}.$$ When the frequency…
Shang
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Regarding Padé Approximation of Neumann Series

I calculated the Padé Approximation of Neumann Series by hand, and then by Mathematica for different orders (from {0, 0} to some higher numbers), using the code below, in general: neu = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 +…
anon
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Best approximation of polynomial $f(x)$ by degree-one reduction

Given a polynomial of $n+1$-th. order $f(x)$ on $-1 < x < 1$, I want to find the polynomial of nth. order $p_n(x)$ that minimizes the maximum error. This is just minimax approximation, except that $f(x)$ is a one degree higher polynomial. If $f(x) =…
user89699
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Approximately solve the equation. Find the first two terms of the approximation.

Approximately solve the equation. Find the first two terms of the approximation. By $a>>1$ and $a<<1$ $\ln x=e^{-ax}$ $x=e^{e^{-ax}}$ $x_{0}\sim1$ $x_{1}=e^{e^{-a}}$ $\left|e^{e^{-a}}-1\right|<<1$ $x=1+e^{-a}+\frac{e^{-2a}}{2}$ Have I considered one…
Maxim
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$\frac{1}{1+x} -1 \approx -x$ is true?

First of all, I'm sorry to ask too simple math question. But I have little backgroud knowledge of mathmatics so it took so long time to me. In my lecture note it says $\frac{1}{1+x} -1 \approx -x$. How this is possible? Is it a kind of taylor…
user967536
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Handle zero in Inverse Distance Weighting

Inverse Distance Weighting is Weighted Average that uses Inverse Distances as Weights. We need to approximate function $z(x)$ and have bunch of sample points $(x_i,z_i)$. And for any $x$ we calculating approximation as: $$z(x) = {{\sum z_i w_i}…
Alex Craft
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approximaton to sin

If we have $s$ circle with the diameter $AB$ (with length $1$) and the center $O$, then we can approximate $\operatorname{chord} AC$ where $x$ represents the value of the $\angle AOC$ in degrees, and $t=90-\frac{x}{2}$.So formula…
Srbin
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Approximate $\sum_{i=1}^n e^{-\beta \ln(i)+\gamma {\ln}^2(i)}$ for large $n$

I need a numerical approximation with low computational complexity of $$f_n(\beta, \gamma) = \sum_{i=1}^n e^{-\beta \ln(i)+ \gamma{\ln}^2(i)}$$ for $n\approx10^6$, where $1\lt\beta\lt3$ and $0\leq\gamma<0.05$. For $\gamma=0$, it is straighforward…
Loic Thibaut
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How to derive the multivariate Pade Approximation for $\ln \left( {1 + \frac{x}{y}} \right)$?

How to derive the multivariate Pade Approximation for $\ln \left( {1 + \frac{x}{y}} \right)$ ? In this case, multivariate mean variable $x$ and variable $y$.
Tuong Nguyen Minh
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