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.

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Newton Raphson Method: approximating root

How do we start from approximating a root using this technique? I know of two, viz - a table of x vs f(x), and see where f(x) changes sign - plot a graph, and see where the graph cuts the x axis But both these methods are unsatisfactory and…
rolfk
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Approximation of logarithm of harmonic mean

Given a large $M\in\mathbb{N}$,if I could approximate $M\log\left(1+\frac{1}{\sum_{i=1}^M x_i^{-1}}\right)$ by $\sum_{i=1}^{M}\log(1+\frac{x_i}{M})$ in the case of $x_i\in\mathbb{R}$, $0
nnnmao
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Landau-Lifshitz light aberration formula order of accuracy

I'm reading this part of Landau, Lifshitz, The Classical Theory Of Fields: I'm able to derive the unnumbered formula for $\sin\theta-\sin\theta^\prime$, finding expansion to Taylor series, but what I fail to understand is how they derived $(5.7)$.…
Ruslan
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Why is $\frac{x}{\sqrt{1+\frac12x^2}}$ a good approximation for arctan?

I played around with the equation $$ f_{n}(x)=\frac{x}{\sqrt{1+nx^2}} $$ which is the $n$th composition of $$ f(x)=\frac{x}{\sqrt{1+x^2}} $$ Surprising to me, it is a good approximation for $arctan(x)$ when plugging in $n=\frac{1}{2}$. I tried…
Palehar
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Differences when using big-O and little-o as $x \rightarrow 0$ vs. $x \rightarrow \infty$

I am having trouble understanding the usage of big-O and little-o notation when $x \rightarrow 0$ rather than $x \rightarrow \infty$. Most definitions i have found online pertain to the case of $x \rightarrow \infty$. One definition given in a…
Jeppe Pilgaard
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Approximation for a Fractal Function

I have recently been working on a problem that requires division by the highest $n$ such that $n = 2^{k}$ and both $n$ and $k$ are integers. To find this $n$ value, I made the function: $$f(x)=\sum_{n=0}^{\text{log}_{2}x}2^{n}\left( \left\lfloor…
patrickbies
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How Can I Make This Approximation?

I've been tasked with showing that $\frac{\delta m}{m_0}$~$\frac{1}{2}(\frac{v}{c}))^2$ From $\frac{m}{m_0}$$=$$1$$-$$(-\frac{1}{2}(\frac{v}{c}))^2$ Assuming v is small. Im not sure what approximation I am supposed to make to show this. And that is…
Andrew Igdal
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piecewise linear approximation of $x^2$

Take the function $f(x) = x^2$. I read that it can be written as $$f(x) \approx k_1\cdotΔx_1 + k_2\cdotΔx_2 + \dots +k_n\cdot \Delta x_n,$$ where $k_n$ is the slope of the $n$-th piecewise linear block, and $\Delta x_n$ is the $n$-th linear…
spyimp
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Linear map of approximation

I need to answer this question, but I don't even clearly understand what is wanted: for approximation problem on $[a,b]$ show a linear map $x \rightarrow t$ that shows it is equivalent to same approximation problem on interval $[-1,1]$ or $[0,1]$…
abenol
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What are some alternatives to least squares method?

I am now working on sir model for epidemic modeling and I am trying to estimate the parameters of the system using available data,but I have to approximate the points representing the variables with respect to time. So apart from the least squares…
Eyeshield
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Can the "Babylon Method" be used Exponents with Decimal Arguments?

I was reading about the methods that the Ancient Babylonian Civilization used for approximately calculating square roots: Could this method be used for approximately calculating any "root" - for instance, could we use this method to approximately…
stats_noob
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Improve the numerator in the approximation by including a third-order term in it.

Improve the numerator in the approximation by including a third-order term in it. $\tan x\approx\frac{x}{\left(1-\frac{4x^{2}}{\pi^{2}}\right)\left(1+ax^{2}\right)}$ I can include a third-order term, but I don't know how to specify "a" $ \tan…
George
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Inverse of k = x/cosh(x)

Question inspired from Matt's video, The bubble that break maths Bubble shape is a cosh function: $\quad y = a \cosh((x-b)/a) $ With symmetry, we can reduce formula to: $ (R/r) = \cosh(H/r) $ where r = bubble minimum radius, R = bubble maximum…
albert chan
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Approximating an arc with a fixed number of line segments.

I'd like to approximate an arc with a fixed number of line segments in $L1$ or $L2$ sense. Let's define the problem as follows: Given a smooth function $f: [0,1]\to \mathbb{R}, f(0)=f(1)=0$ and $N\in \mathbb{N}$ find a sequence $\{(x_1,y_1), ...,…
Michael
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Padé approximation to a nonlinear first order complex ode

I wish to approximate the following ode with Padé: $$\dot \rho +i\frac{{\hat \sigma }}{2}\rho + i\alpha \rho \left[ { - {\gamma ^2} + \frac{{{\gamma ^3}}}{{\sqrt {{\gamma ^2} + 1} }}} \right] + i\frac{{\hat P}}{4}{\rho ^*} = 0$$ where $$\rho(t)$$…
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