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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How to approximate table-defined function using non-linear least squares

I read least squares method and haven`t found a good example of using non-linear least squares. Problem: I have an arbitrary values for x = 1, 2, 3, 4, 5, therefore, I have a table-defined function. I have to approximate this function using the…
not a Programmer
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Approximation - Weight function

Hi could anyone help with this task (point C): https://zapodaj.net/a079f713ac6c0.png.html As far I've got: $$Q(θ) = ∫_0^1(u^2−au_n)^2*(−2u+2)du$$ What next? Derivative by $$u$$ of this calculus function and then equate it to zero?
Charles
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Approximate relation between $k$ in $Y(t)=1-\exp(-kt)$ and $R=\frac{X(1-Y)}{Y(1-X)}$ where $d X/dt=15(Y-X)$ with $X(0)=0$?

How to get the approximation $$ \frac{15}{k}\approx\frac{4R}{(1-R)^2} $$ given that $$ \frac{15}{k}=\frac{X(1-X)}{Y(Y-X)} $$ and $$ R=\frac{X(1-Y)}{Y(1-X)} $$ where $k$ and $R$ are constants, and $Y(t)=1-\exp(-kt)$ with $t>0$. I was stuck with…
Highman
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How to find closest exponential approximation?

I have a bunch of data, and I'd like to find the closest exponential aproximation I can to fit the points. I'm guessing there's a (relatively) straightforward way to do this. For example, if I have the data points (5, 9, 17, 33), the equation I'd…
Beska
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square root of sum of squares approximated by sum times square root of 2 over pi

I found somebody using the following approximation for $\sqrt{x^2 + y^2}$: $\sqrt{x^2 + y^2} \approx (x+y) \sqrt{2/\pi},$ where $x$ and $y$ are positive numbers smaller than 1. It appears in this…
Christian
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Exponential Function Approximation

Let $n_1, n_2 \in \mathbb{N}$ be known constants such that $n_1 + n_2 = n$. Moreover, let $0 \leq p_1, p_2 \leq 1-1/n$. Assume an exponential model $f(k) = n_1 p_1^k + n_2 p_2^k,\ k\in \mathbb{N}$. I have access to points $y_k$ such that $|y_k -…
vkonton
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Approximating derivative as parementer decreases

Here is the question and below my understanding/attempt: Let $f(x) = \arctan (x)$. Use the derivative approximation: $f'(x) = \frac{8f(x+h) - 8f(x-h) - f(x+2h) +f(x-2h)}{12h} $ to approximate $f'(\frac14\pi)$ using $h^{-1}$ = 2, 4, 8 . Try to take…
steve
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Equality vs Approximation which one has the precedence?

Which one of these statements is true? 4/3 ≈ 1.3 = 8/6 (i.e. 4/3 ≈ 1.3 and 4/3 = 8/6) 4/3 ≈ 1.3 ≈ 8/6 Edit: Is this statement true or false? 4/3 ≈ 1.3 ≈ 8/6 = 12/9 = 16/12 ≈ 1 ≈ 1.3 ≈ 4/3 Edit: This sign ≈ denotes approximation. And this sign =…
user37421
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Approximation: (1 - 1/n) ^ m ~~ e^ (-m/n), why?

$\left(1 - \frac{1}{n}\right) ^ m \approx e^{-m/n}$ Can someone explain why the left hand side is approximated by $e ^{-m/n}$ ?
ajfbiw.s
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Long wave approximation

In the book "Sea Loads on Ships and Offshore Structures" by Faltinsen, page 60 there is an identity: $$ \sin(\omega t - kR \cos(\theta)) \cos(\theta) d\theta $$ They claim that under long wavelengths (much larger than the radius $R$, so $kR$ is…
user132716
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Estimating sums

Estimate following sums as the functions of variable $n$: a) $\displaystyle\sum_{i=1}^{n}e^i\ln i$ b) $\displaystyle\sum_{i=1}^{2n}(-1)^i\ln i$ c) $\displaystyle\sum_{i\ge 0}^{}\frac{\ln(n+i)}{e^i}$ d) $\displaystyle\sum_{i\ge…
ray
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How to use Taylor's Theorem to obtain an upper bound for an error approximation

$e \approx 1 + 1 + \frac{1^2}{2!} + \frac{1^3}{3!} + \frac{1^4}{4!} + \frac{1^5}{5!}$ must find upper bound for this but I don't see what I should be doing. The remainder/error is given by $\frac{f^{n+1}(z)(x-a)^{n+1}}{(n+1)!}$ but what is the…
Colour
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How can I compare two approximants to a bivariate function?

An application I'm working on has required me to find simple approximations to a rather complicated bivariate function $g(x,y)$ that also takes a long time to evaluate on the computer. Through sheer algebra and much sweat, I finally managed to…
Renata Porfirio
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Approximation when $|a(t)|\ll b$

If $|a(t)|\ll b$ is it alright to take $a\left({a\cdot \dot{a} \over b^2}\right)$ as $0$? Would the following argument make sense? I know that we can take $\left({a\cdot a \over b^2}\right)$ as $0$ and $0={d\over dt}\left({a\cdot a \over…
renne
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Method for finding square roots quickly (manual)

I was recently studying AC circuits and there I need to use Pythagoras theorem a lot.So I was looking for a method with which square roots can be calculated very fast,manually up to 1 decimal place of accuracy.Is there any such method? For…
user251680
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