For questions related to approximations

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37 views

Approximate an integral

In a physics textbook, I came across the integral $$I(r_1,r_0)=\int_{r_0}^{r_1}\frac{1}{1-2m/r}\left[1-\frac{r_0^2(1-2m/r)}{r^2(1-2m/r_0)}\right]^{-1/2}dr$$ The author said that the integrand can be ...
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1answer
21 views

travelling salesman problem approximation [on hold]

Consider the symmetric travelling salesman problem $\Delta TSP$: An instance consists of a complete undirected graph $G$ (all possible edges are present), a nonnegative integer cost $c_{ij}=c_{ji}$ ...
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14 views

numerial solution to fredholm integral equation

Consider the integral equation: $$ y(x) =1+\int_0^cK(x,t)\,y(t)\,dt, $$ where $x\ge0$ and $$ K(x,t) = \frac{\partial}{\partial ...
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2answers
26 views

Understanding an approximation equation

In this biology textbook I found the following approximation: $$\frac{1}{2N}\left( 1-\frac{1}{2N} \right)^t ≈ \frac{1}{2N}e^{\frac{-t}{2N}}$$ Can you help me to understand this approximation and ...
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2answers
32 views

Product approximation

In this biology textbok I found the following approximation: $$\prod_{i=1}^{k-1}1-\frac{i}{2N} ≈ 1-\frac{{k \choose 2}}{2N} $$ Can you help me to understand this approximation and help me to ...
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2answers
25 views

Approximation of $x\log(x/a)$ for $x$ near a

I'm trying to see where the approximation $$(x-a) + ((x-a)^2)/2a$$ of $x\log(x/a)$ comes from (for x near a). Might be missing something very trivial but I've already tried the usual expansions ...
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0answers
6 views

Approximate the local behavior of an unknown distribution with uniform distribution.

consider an arbitrarily smooth distribution function over support $S$. I am only interested in local behavior that happens in a very small area. To what extent can I approximate the local distribution ...
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1answer
25 views

division by sum of exponentials of large negative numbers

I need to evaluate the following numerically: $$ f = \frac{\exp(a)}{\exp(a)+\exp(b)+\exp(c) + \exp(d)} $$ $a,b,c$ and $d$ are large negative numbers, they are smaller than -1000. Numerically ...
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0answers
14 views

approximating a probability density

Let $f(x)$ be the probability density of a random variable $X$. Let the support of $f(x)$ be positive reals. If $f(x)$ is sufficiently smooth then one can approximate it with its Taylor series cut off ...
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1answer
22 views

Use tangent line to find approximation

$f(x) = x^2 - 3x + 5$, the tangent line to the graph of $f$ at $x = 3$ is used to approximate values of $f(x)$. Which of the following values $3.4$ $3.5$ $3.6$ $3.7$ $3.8$ is the greatest value of x ...
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1answer
46 views

Approximation of $\frac{1}{2^{N}} \binom{N}{\frac{1}{2}(N + s)} $ for big N

I have the following function that I have to approximate for big N and I really have no clue what to do - I hope that anyone can help me out: $$P_N(s) = \begin{cases} \frac{1}{2^{N}} ...
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2answers
62 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
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1answer
88 views

How to show this: $\sum_{k=2}^{n}\frac{\ln{k}}{k^2}\approx \ln{n}\cdot\left(\zeta_{n}{(2)}-\frac{\pi^2}{6}\right)+C$

Show that: $$\sum_{k=2}^{n}\dfrac{\ln{k}}{k^2}\approx \ln{n}\cdot\left(\zeta_{n}{(2)}-\dfrac{\pi^2}{6}\right)+C,n\to\infty$$ where $$\zeta_{n}{(k)}=\sum_{j=1}^{n}\dfrac{1}{j^k}$$and $C$ is ...
2
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1answer
22 views

How many bits of difference in a relative error?

I would like to know if there is a formula or any other way to find out how many bits of difference between two values given the relative error. For instance: $$\epsilon_{\text{rel}} = \frac{V - ...
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4answers
84 views

Demonstrate that $\displaystyle \frac{(2n - 2)!!}{(2n - 3)!!} \simeq 1.7 \sqrt{n}$

As in the title, I know that $\displaystyle \frac{(2n - 2)!!}{(2n - 3)!!} = \frac{(2n - 2)(2n - 4)\cdots 4 \cdot 2}{(2n - 3)(2n - 5) \cdots 3 \cdot 1} \simeq 1.7 \sqrt{n}$ Could you give some hint ...
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1answer
38 views

Approximations. Newton's method - composite Simpson's rule

Can you help me please to solve this problems and if you can give me some helpful information. Thanks!
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0answers
27 views

Approximating piecewise linear function

I'm trying to derive an analytic approximation to the following piecewise linear function: $$ f(x) = \left\{ \begin{eqnarray} \frac{x}{x_s} & & \text{if} & x < x_s \\ ...
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0answers
29 views

Approximating $C^2$ functions with compactly supported $C^2$ functions

Let $C^2$ be the space of twice-continuously differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$ and $C^2_K$ be the subset of functions in $C^2$ with compact support (that are zero outside ...
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1answer
22 views

Approximation to the square root

I was reading an article that approximated a square root operator as follows $\sqrt{1+x+y} \cong \sqrt{1+x} + \frac{1}{2}y + O(xy,y^2) $ At first glance that looks like a Taylor series expansion, ...
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2answers
85 views

Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
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5answers
319 views

Why is $e^\pi - \pi$ so close to $20$?

$e^\pi-\pi\approx 19.99909998$ Why is this so close to $20$?
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0answers
18 views

Padé approximant of transfer function with gain and time delay.

$$ H(\omega) = A e^{-j \omega \tau} $$ I'm trying to use Padé approximation to generate a numerator and denominator polynomial for the above transfer function but genuinely struggling with how to ...
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1answer
10 views

approximation for formula — relation to exp function

Let $n>0$ be a natural number. I am looking for an M(n) of the following function: $$ f(n) = (1 - n^{-1/4})^n < M(n) $$ I know that if $n$ goes to infinity $f(n)$ goes to $0$. Now, I wonder if ...
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1answer
67 views

Simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and a natural ...
0
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1answer
25 views

Approximating an interval [duplicate]

Corollary C is derived from $E_(x) = \frac{f''(x)}{2} \cdot (x-a)^2$ I'm having serious issues understanding this problem and I'm just not getting it right. I'm preparing for a test so this isn't ...
2
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1answer
28 views

Error formula for linearization

Can anyone shed some light on this formula? I can't find any information on it. It has three corollaries that I also need to understand:
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0answers
24 views

Best Fit Curve and Function for 4D Data?

I have experimental data of 4 dimensions, and I want to computer-generate an approximated function from that. If it helps to clarify, I'm testing a pendulum's period based on variable length, starting ...
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0answers
36 views

Expanding the Exponential at Infinity

Can anyone give me a good approximation to $e^{-ax}$ for large $a$? This is based on trying to find a good approximation to $$ \int\frac{dx}{\sqrt{x}}e^{-\left(x^{2}+a x\right)} $$ when $a$ is ...
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0answers
24 views

Approximation of sum with binomial summands

I am new here, so hopefully my question will be understood correctly. I have a function (originating from expected untility theory in economics) that looks the following: ...
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1answer
46 views

Approximating hypergeometric function F(1,1+a,2+a,z) for z->1

in my studies a normalization constant for a pmf includes the hypergeometric function ${}_2F_1(1,1+a,2+a,z)$ The parameters are in the range $0.99<z<1$ and $0<a<5$. I have tried some ...
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0answers
21 views

Approximation of eigenvalue

I'm having trouble with the following problem: Let $A\in M_n(\mathbb{C})$ be a Hermitian matrix, $\mu\in\mathbb{C}$, $\epsilon>0$ and $x\in\mathbb{C}^n$ such that $\mid\mid x\mid\mid=1$ and ...
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3answers
31 views

An approximation: 2*number of events = twice the probability

My statistical mechanics textbook uses an approximation to derive a well-known result. The approximation is: Suppose for an event, the probability of an outcome is P. For n events, the probability of ...
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2answers
147 views

Find the integral part of $\sum_{i=2}^{10000}\frac1{\sqrt{i}}$

$$A = \frac1{\sqrt{2}}+\frac1{\sqrt{3}}+\cdots+\frac{1}{\sqrt{10000}}$$ Find $\lfloor A\rfloor$ where $\lfloor x\rfloor$ is the greatest integer less than, or equal to $x$ I got stuck on this, so ...
2
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1answer
35 views

Approximation of a factorial

In books I can find a lot of approximations of factorial, which doesn't require the a lot of multiplications, like for example Stirling's approximation. Very popular is also the Euler's solution which ...
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1answer
37 views

Forcing coefficient of best fit quadratic

So I have a question regarding best-fit polynomials. Currently, using matrices and then Gaussian elimination, I come up with f(x) = Ax^2 + Bx + C and use the derivative of that fit to come up with the ...
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0answers
21 views

Matching First and Second Derivatives: Taylor Series

I have $$ f(x) = \sqrt{\ln\left(a\cosh^{2}(mx)(1+bx^{2})\right)} $$ If I expand this as a series I should get something of the form $$ \sqrt{\ln a}+gx^{2}+\mathcal{O}(x^{4}) $$ but I'm having ...
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0answers
32 views

Approximation for the logarithm of a summatory

I would like to find an approximation for: $$ \log \left(\sum_{i=1}^{N} a_i\exp(-b_i^2)\exp(-c_i^2)\right) $$ with $$ a_i = \frac{1}{\sqrt{(e^2 + e_i^2)(g^2 + g_i^2)}} \\ b_i = \frac{b-d_i}{2(e^2 + ...
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0answers
41 views

Does a complex polynomial can be approximated by a Euler-type exponential function?

Can one uniformly approximate a complex polynomial with a Euler-type exponential function? I mean: let $E\subset\Bbb C$ be a compact set with the connected complement $\Omega:= \bar {\Bbb C}\setminus ...
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1answer
23 views

MatLab: estimate number of iterations

I want automatically estimate iterations number in matlab. Suppose we have for(int i = 1; i < N; i++). It's clear that for-loop prodices ...
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0answers
11 views

approximation by superpositions of a sigmoidal function

I have a question about superposition of sigmoidal function with limited bound. Let $f$ be a signmoidal function with amplitude $\alpha$. For example, let's assume we have tanh(): $f_i=\alpha \cdot ...
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0answers
10 views

Correlation 4-point

I need to calculate $\langle x_{i}x_{j}x_{k}x_{l}\rangle $, where $$ \langle f(x) \rangle = \int e^{-\frac{1}{2}A_{ij}x^{i}x^{j} - \frac{\lambda }{4!}\sum_{i}x_{i}^{4}} f(x)d^{n}\mathbf x , $$ for ...
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0answers
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Copying the Curvature of One Function onto Another: Approximation

I have a polar function $$ r(\theta)=\left(r+\epsilon\right)\cos(\theta)-\sqrt{r^{2}-\left(r+\epsilon\right)^{2}\sin^{2}(\theta)} $$ Is it possible to methodically conjure another polar function ...
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0answers
30 views

Using Polars to Approximate a Cartesian line: Approximating an Integral

I have the equation of the lower semicircle of radius $r$ centred at a distance $a+r$ above the x-axis $$ f(x)=r+a-\sqrt{r^{2}-x^{2}} $$ which I can approximate (for small $x$) as $$ f(x)\approx ...
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2answers
18 views

Divide $n$ seats by a list of $\mathbf{w}$ weights proportionally.

I have $n$ number of seats and I have list of weights $\mathbf{w} \in \mathbb{R}^{k}$ which is a probability distribution with $k$ possible values, $\sum_{i=1}^k{w_i}=1$. I want to divide the $n$ ...
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2answers
35 views

How to use $t(29/\sqrt{2})<0$ where $t(x)=x^2-41x+420$ to prove that $41/29<\sqrt{2}<42/29$??

So I was investigating different ways to approximate $\sqrt{2}$. Here's my latest: $$Let:t(x)=x^2-41x+420$$ then the roots of $t(x)$ are $20$ and $21$. I showed that then $t(x)=(x-20)(x-21)$ and ...
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2answers
167 views

Evaluate $\sum_0^\infty \frac{1}{n^n}$

Courtesy of this xkcd comic I now know that $$ \sum_{n=1}^\infty \frac{1}{n^n} \approx \ln^e(3) $$ Echoing the views of the comic itself, if I ever find myself taking $\ln^e(x)$ then something has ...
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0answers
26 views

How to evaluate the derivate of a hypergeometric function w.r.t. one of its parameters?

I have to numerically evaluate the derivative of the hypergeometric function w.r.t. its first and second parameters $\large\frac{\partial}{\partial a}{_2F_1}\left(a , b ,c;z\right)$ and ...
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1answer
30 views

Approximations to the Roots of a Function

I want to find approximations to the root of a function in two variables using the Newton-Raphson method. I can use the method on a function in a single variable but I'm lost as to how you can use it ...
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1answer
68 views

Weierstrass Approximation Theorem for $\Bbb C$

The Weierstrass approximation theorem states that any continuous function $ f : I \rightarrow \Bbb R $ on a closed, bounded, connected subset $ I \subseteq \Bbb R $ can be uniformly approximated by ...
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4answers
391 views

Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$

Just curious, is there a geometry picture explanation to show that $\sqrt 2 + \sqrt 3 $ is close to $ \pi $?