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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Solving a low rank symmetric system

I'm working on a problem where I need to solve a large set of systems of equations, where each has a structure that looks like: $\left( M^\top_{n\times p}\Sigma_{p\times p} M_{p\times n} + \Lambda_{n\times n}\right) x_{n\times 1} = b_{n\times…
firdaus
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How to calculate the area of bizarre shapes

I'm looking for an algorithm to calculate the area of various shapes (created out of basic shapes such as circles, rectangles, etc...). There are various possibilities such as the area of 2 circles, 1 triangular and 1 square (intersections…
user769
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Montecarlo estimate of a integrand from 0 to $\infty$

I have a question about monte carlo estimation of integrals. Suppose I am told to estimate using monte carlo, the integral: $$f(y) = \int_{0}^{y}\frac{4}{1+x^{2}}dx$$ I want to estimate $f(\infty)$. I know that with some calculation, the exact…
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Confusion about principles of decimal approximation

I ran into this paragraph in the introductory chapter on Taylor series of Morris Kline's "Calculus: An Intuitive And Physical Approach": "Thus, if the value of $\sin(x)$ for a particular value of $x$ is needed to five decimal places, the…
jeremy radcliff
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Explain why $\exp(-7 \log_{10} n)$ approximates $1/n^3$ so well

I was graphing a few functions, and discovered that the graphs of $\exp(-7 \log_{10} n)$ approximates $1/n^3$ are almost the same. Can anyone explain why this is so? Is there a general result for this phenomenon?
user90593
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Why $(\alpha\frac{e}{t})^t e^{-\alpha}$ is an approximation for $P(X > t\alpha)$ for Poisson distribution $\frac{\alpha^ke^{k}}{k!}$?

I am reading Section 3.4 of Algorithms, 4th Edition. Page 466 is a proof of the following proposition: In a separate-chaining hash table with $M$ lists and $N$ keys, the probability (under Assumption J) that the number of keys in a list is within a…
Jingguo Yao
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Finding the closest distance between a point a curve for multiple Points (n>1000)

I am trying to compute the closest distance between a point a curve (polynom of 2rd degree) : $$f(x)=a*x^2+b*x+c$$ $a,b,c$ are established. So if we denote that D(x) is an distance from $(x,f(x))$ to $(p,q)$ so we get: $$ D(x)= \sqrt…
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Could anyone explain why this is a general case of Weierstrass Approximation?

Suppose $X_1, X_2 ...$ are independent Bernoulli random variables. with probability $p$ and $1-p$. Let $\bar{X}_n = \frac{1}{n} \sum\limits_{i=1}^nX_i$. If $U \in C^0([0,1],\mathbb{R})$, then $E(U(\bar{X_n}))$ converges uniformly to $…
user318352
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Approximating $\frac{a+\delta a}{b + \delta b}$

I have two quantities $a(t)$ and $b(t)$ that have a constant mean ($a$ and $b$) and some small fluctuating noise part with vanishing mean $\delta a(t)$ and $\delta b(t)$. I'll write them as $a(t) = a + \delta a(t)$ and the same for $b(t)$. I also…
user3183724
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Infinite products of even analytic functions - highly accurate approximation

I discovered a way to evaluate infinite products of even analytic functions with high accuracy. $$ \prod_{k=1}^{\infty} f(k^2) \approx \prod_{k=1}^{\infty} \left(1-\frac{A_1}{k^2}+\frac{A_2}{k^4}-\frac{A_3}{k^6}+\cdots+\frac{A_n}{k^{2n}} \right) =…
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Best Approximate Solution of Heat Equation (Diffusive Logistic Equation)

If $u(x,t) \ge 0$ in the domain $ (0 \times 1) \times (0,\infty)$, find a function that caps the value of $u(x,t)$ in the region $(0 \times 1) \times (0,T)$. $u(x,t)$ is a solution of the following system: \begin{cases} u_t = u_{xx} + u(1-u), & (0…
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Finding the degree of the Maclaurin polynomial approximation of cosine to approximate $\cos(1)$

I have a question where I am asked to find the amount of terms required in a Maclaurin polynomial to estimate $\cos(1)$ to be correct to two decimal places. So far what I have done is used Taylor's Theorem to get the follow: $$|R_n(x)| =…
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Simpler derivation to $\pi$

I'm an amateur in mathematics, being in 9th grade. I have been trying to derive $\pi$. During this I reached a limit to find the value of $\pi$. $$\lim_{x \to 0} \frac{180\sin x}{x}$$ Where $x$ is in degree measure. Isn't this better than the other…
N.S.JOHN
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Trying to rederive an exponential approximation

So I was reading a paper where the following approximation was made. Note that $p$ is small, $L$ is large, and $pL$ is $O(1)$: $$\left[1-e^{-4p(1+p)L}\right]^{L/2}=\textrm{exp}\left[\frac{2(pL)^2}{e^{4pL}-1}\right](1-e^{-4pL})^{L/2}+O(p^3L^2)$$ I…
user34364
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Integral of a Gaussian with Trigonometric functions Involved

I am having a difficult time evaluating an integral unlike any integral I have seen before. To get right into things here is the integral: $$\frac{A}{\sigma_o\sqrt{2\pi}}\int_{-\infty}^\infty…
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