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).

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1answer
29 views

Time series closed form similar to harmonic series

I want to find the closed form for the following: $$ S(k, \alpha) = \sum_{t=T-k}^T \frac{\alpha^{-t}}{t} $$ when $\alpha \in (0,1)$. For harmonic series there is an easy way to upperbound: $$ H(k) ...
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2answers
43 views

Converting 29^1312000 to base 10

I am trying to do some calculations with the number 291312000 and I find it would be much easier if I could convert it (approximately) to a base 10 number. The closest I could come was to start with ...
3
votes
1answer
88 views

How can I recursively approximate a moving average and standard deviation?

Consider a sequence of measurements $(x_1, x_2, ...)$. Let $\mu_n$ be the $p$-period moving average defined by $$\mu_n = \frac{1}{p}\sum_{i=n-p+1}^nx_i$$ and $\sigma_n$ be the $p$-period moving ...
0
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0answers
37 views

Polar coordinate for complicated curves

In general polar representation of a closed curve is done by coordinate $(\theta,r(\theta))$, $\theta\in (0,360)$. When working with real data, I got a closed curves whose graph looks like the below ...
0
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1answer
27 views

Series and quadratic approximation

Been working out series for the past day and have come across this question. I would kindly ask how I ought to tackle it, I've done Binomial and Maclauren's however this seems to require another ...
2
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0answers
39 views

Closed forms for two times series similar to geometric series, but with additional power

Does anyone know a close form solutions to any of the following time series? (approximate upper bounds might as well work). $$ \sum_{k=1}^T \frac{1}{2^{k^2}} $$ or $$ \sum_{k=1}^T k ...
4
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0answers
196 views

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
2
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0answers
32 views

Continued fraction approximation to a function and its derivative

I am recently working on fitting a model with incomplete beta function. In order to put it into my optimization algorithm, I must find out the derivatives of the incomplete beta function $B_p(x,y)$ ...
2
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1answer
50 views

Applying Newton-Raphson method to $a\cdot b^{-2}=c\cdot d^4+e\cdot f(d)$

I am familiar with the method and it's application in classic problems, but I have troubles tackling the function I need to solve with it. So, variables in problem: Real numbers, all are known ...
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0answers
31 views

Additive and Multiplicative Error in $n!$ Approximation

Let $S(n)=\sqrt{2\pi n}\big(\frac{n}{e}\big)^n$ be the approximation of interest to $n!$. What are good lower and upper bounds on the following two functions $$(1)\mbox{ }|S(n)-n!|?$$ $$(2)\mbox{ ...
1
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1answer
47 views

Derivative of $\|Ax-b\|_1$

Using least squares approximation $E^2 = \| Ax - b\|^2 = (a_1x - b_1)^2+...+(a_mx-b_m)^2$ The derivative of E^2 at the point $\hat{x}$ is zero if: $(a_1\hat{x}-b_1)a_1+...+(a_m\hat{x}-b_m)a_m=0$ ...
4
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2answers
96 views

Laplace's method with nontrivial parameter dependency

I need to approximate the following integral using Laplace's method: $$ \int_0^{\infty} \frac{x^{\lambda} \lambda^{-x}}{(1+x^2)^\lambda} dx \\ = \int_0^{\infty} \exp\left(\lambda \log(x) - ...
1
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1answer
33 views

Physics Approximations Quadratic Equation

I'm having a hard time following one of the solutions to this physics problem. In particular, the math. Consider, $$a\Omega ^2 + b\Omega + c = 0$$ The solutions to this quadratic equation are, ...
1
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1answer
60 views

Least absolute deviation for item prices

How would I calculate the values of $A,B,C$ using least absolute deviation? $R = 1$ $2A + B$ = $C + R$. $B + C$ = $5A$. $A + C + 2R$ = $B + 4R$. $A + B + C$ = $6.33R$. Using least squares ...
2
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4answers
92 views

How to extract fraction from a floating point number

I'm making some tests with float type (floating point number) with programming and in some of my tests I need to extract the fraction that originates the float value. Let $ x $ be a floating point ...
1
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2answers
40 views

Least Squares approximation for item prices

Let's say that $A$, $B$, $C$ are different items with different values. $R$ is a unit of currency, for simplicity I'll let it be $1$. Traders frequently trade these items on an open market. Price is ...
1
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1answer
36 views

Stirling formula on $\frac{\gamma(\frac{1}{2}s)}{\gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$

I tried to prove $\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$ using stirling formula when $s =\sigma+it$. However, since stirling formular for gamma function is ...
0
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3answers
48 views

Notation for asymptotic approximation

I was reading Stirling's approximation and got quite confused with the idea of asymptotic formula. So in Wikipedia it says that a function $F(n)$ of $n$ is asymptotic formula for $P(n)$ if $P(n)$ is ...
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0answers
34 views

$L^2$ inequality for derivatives of polynomials on triangles

I'm reading a paper which states the following inequality, but the (presumably) elementary proof is cited to be in a document, which is too old to get access to. Let $p: \mathbb{R}^2 \to \mathbb{R}$ ...
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4answers
288 views

Which expansion of $e$ is more accurate?

We have two forms of $e^x$ $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+....$$ and $$e^x=\frac{1}{\displaystyle 1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+....}$$ The second form comes from ...
1
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2answers
209 views

How come $\pi$ is usually approximated as 3.14 or 22/7?

I've heard that $\pi$ is usually approximated as 3.14, but it can also be approximated as 22/7, which is equal to 3.142857142857142857.... Guess what? $\pi$ can also be approximated as 355/113, ...
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0answers
58 views

Approximation for uniform load on parabolic cable along its arc length

I am doing analysis for cable structures. Let's say that the cable stretches from point A to point B and carries a vertical ...
1
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1answer
78 views

Prove this inequality $ \sqrt{5} > \frac {13 + 4\pi}{24 - 4\pi} $ [closed]

$$ \sqrt{5} > \frac {13 + 4\pi}{24 - 4\pi} $$
1
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1answer
74 views

Approximating a binomial coefficient using Stirling's formula

I am working on a problem of modelling a rubber molecule as a one-dimensional chain consisting of $N=N_{+}+N_{-}$ links, where $N_{+}$ points in the positive $x$-direction a distance $a$ and $N_{-}$ ...
0
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0answers
15 views

Approximation technique of common probability distributions that can be convoluted and integrated fast

I am looking for a approximation technique of functions with two conditions: It is possible to perform a fast approximate convolution with the approximate functions. It is possible to numerically ...
0
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0answers
42 views

Lebesgue integration of $f(x)=\frac{1}{x}$ where $x\in[0,3]$

We have the function $f(x)=\infty$ if $x=0$ and $f(x)=\frac{1}{x}$ if $x$ otherwise. So, in this two values of function, I made simple approximation of $f(x)$ by the help of simple function : ...
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4answers
51 views

What is the approximation of trigonometric function by simple function

for $f(x)=\sin x$, $g(x)=\cos x$, $h(x)=\tan x$, What is the approximation of each function by using simple function?
0
votes
2answers
18 views

How is this approximation related?

I'm in the dark about the following approximation. Given that $r >> (l_1 + l_2)$ $\phi(P) \approx \dfrac{\lambda}{4\pi\epsilon_0}ln(\dfrac{r+l_2}{r-l_1}) \approx \dfrac{\lambda(l_1 + ...
0
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1answer
51 views

Expansion in large and small limits

Let $$f(x) = \frac{1}{\log(\frac{x}{c})}$$ where $c$ is some constant number. Consider the variable $x$ in the large regime where $x \gg c$ and small regime where $x \ll c$. How would $f(x)$ depend on ...
1
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1answer
88 views

Comparison between lebesgue integral and riemann integral of $f(x)=x^2$ in $[0,2]$

If we have an example $f(x)=x^2$ let's say for $[0,2]$. In lebesgue integral, I already use a sequence of function $f_n(x)$ as approximation to $f(x)$ ($f_n(x)$ converges to $f(x)$) which is stated ...
2
votes
1answer
57 views

Numerical approximation, or is there a better way? $x=1+\ln(1+\ln(1+\ln(x)))$

Don't have enough reputation to post on this but is there any way of calculating the following? $x=1+\ln(1+\ln(1+\ln(x)))$
4
votes
1answer
47 views

The Cosine and an Enigmatic Parabola

In the interval $[-\frac{\pi}{2},\frac{\pi}{2}]$, the cosine function superficially resembles an inverted parabola of the form $-ax^2+1$: I wanted to know more and computed the $L^2$ norm ...
3
votes
2answers
91 views

Can one apply a WKB method to an inhomogeneous first order differential equation in order to find the asymptotic expansion of the solution?

Consider \begin{equation} \varepsilon \frac{dy}{dx} = Q(x)y + R(x) \end{equation} where $\varepsilon$ is a small parameter. Can one apply a WKB method to find an asymptotic expansion for the ...
0
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0answers
26 views

Probability - Characterizing goodness of moment matching method.

I have a question about how to characterize the goodness of approximating a distribution using its moments. Suppose I have a probability density function $p(x)$ (e.g., normal distribution), and I am ...
1
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0answers
25 views

Fitted function - Which is better to use?

So I have some data for program running time, that follows a power law relation aN^b. I log-log plotted the data and saw that it became a straight line, so I calculated the slope of this line to get ...
0
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1answer
48 views

How to write approximations of a sequence $x_n = {1/3^n}$

Write three approximations of the sequence ${x_n} = {1/ 3^n}$, using the following scheme - $P_0= 1, P_1 = 0.33332$ and $P_n = (6/5)P_{n-1} - (1/5)P_{n-2}$ for $n = 2, 3,\dots$ Further, make a ...
0
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0answers
72 views

Does a greedy task selection algorithm find a c-approximate solution?

A scheduling problem can be stated as: Given a set $\{(s_i,f_i)\}_{1\le i\le n}\}$ of tasks identified by their start and end times, choose the maximum size subset of non-overlapping tasks. ...
0
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0answers
21 views

Finding when $(x^3 + …)/(x^4 + …)$ reaches $10^{-n}$

A problem I’m working on throws up equations like: $$ \frac{1}{4k + 9} + 3\frac{k+1}{(4k + 9)^2} + \frac{2k+1}{(4k + 9)^3} + \frac{k}{(4k + 9)^4} $$ I need to know the value of $k$ for which this ...
0
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1answer
30 views

Error evalution for Newton-Raphson method

I'm supposed to approximate a solution of an equation using the Newton-Raphson method, knowing one real solution to this , namely $x \approx 0.61803$. $$x^4 + 3x - 2 = 0 $$ Therefore I start by ...
2
votes
1answer
33 views

How to interpolate multidimensional functions?

I'm learning about interpolation and I wanted to ask if there's a "good" method to interpolate multidimensional functions (when the dimension can be even a few thousands)? Is there a theoretic limit ...
3
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0answers
69 views

Taking a stationary phase approximation of a multidimensional integral

I'm looking for a way to take a stationary phase approximation of an integral of the following form: $$ \int_{-\infty}^\infty d\vec{q} \exp\left(2 \pi i N \left(S(q_{n+1}, \vec{q}, q_1) - ...
1
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1answer
32 views

Question on probability and approximation

Okay I think you are all familiar to YouTube videos and some facts are: to comment, like and dislike on a video you need a Google account. when someone views the video the view count of the video ...
0
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1answer
31 views

Binomial cumulative probability

Here is the question I need help on : Let $X$ be a binomial random variable with p = 0.5 and n = 100. Give $P(X \geq 60)$ rounded to two decimal places without using a calculator (by using ...
2
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0answers
97 views

Lower bound on a polynomial far from its zeros

Let $p(x) = \sum_{i=0}^{d}c_{i}x^{i} \in \mathbb{R}[x]$ and assume that all its zeros are real and in $[-1,1]$. I am interested in lower bounding the value of $|p(a)|$ in case $a \in [-1,1]$ is far ...
0
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0answers
9 views

Root finding: Distance b/w 2 objects = 0. (and other examples of finding roots?)

Can someone explain general uses of finding roots? I understand you can find roots to help manually graph a function, but there's gotta be more. For example, in video games, I recall something about ...
0
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1answer
45 views

Approximating a power of a root of unity to within $\delta$

I have an estimate of $\omega$, a root of unity. I'm really wondering how small the error (in the estimate), which I give as $\epsilon$, has to be, so that when I take my estimate of omega to the ...
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0answers
41 views

Univariate Polynomial Approximation

I'm working on an algorithm in which I need to approximate the behavior of a polynomial by computing its roots to some $\epsilon$ precision. The problem can be defined as follows: Let $f(x) = x^n + ...
2
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1answer
63 views

How to approximate this nasty exponential function with an integral?

What is the best way to approximate a function of the following form, $$ \text{exp}\left(-\int_{y}^{+\infty} f(x)\ dx \right)$$ Any approximation to this, does taylor series work? The reason I am ...
14
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2answers
170 views

A Mathematical Coincidence, or more?

According to the paper "Ten Problems in Experimental Mathematics", $$\int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)dx \quad = \quad \frac{\pi}{8}\color{blue}{-7.407 \times ...
1
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1answer
31 views

Angle measurement

Assume I want to compute one of the angles of a right triangle doing $n$ measurements of the sides with a ruler. In order to increase the precision I make several measurements. After that I compute ...