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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149 views

Is there any reason why $4-\pi$ is quite close to $\frac{\sqrt{3}}{2}$?

In this question obviously the error of our "approximation" is $4-\pi=0.858...$ . I tried to reconstruct the false argument with $\tau=2\pi$, and the error in that case would be $8-\tau=1.716...$, ...
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1answer
28 views

How can missing data be organised or classified (Interpolation vs Approximation)?

I'm looking for a way to distinguish between the various types of missing data techniques? Can someone help to clarify or organize these categories in sub-sections or indicate similarities or ...
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0answers
34 views

Series representation for $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 \pi ^2 A^2+W^2}+6 \pi ^2 A^2+3 W^2}}{\sqrt{2}}$

My question is, is there a series representation or other function of $L$ and $A$ I can use when I solve the following equation for $W$? $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 ...
1
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1answer
27 views

Curve approximation by some known points on the curve

I want to approximate a curve by some known points on the curve. I can choose these point. My curve is shown as below: I have to use such a equation: ...
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0answers
48 views

Determining when the approximation fails

Context Suppose we have a grid-based game where a unit has a range parameter that serves as the upperbound of the sum of the costs of his movements in a single turn. Moving orthogonally to an adjacent ...
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1answer
37 views

Closed-form term for this expression

I have a normal Distribution $X \sim N(\mu, \sigma)$. Is there an easy way to give an asymptotic estimate with small error (I would prefer with relative error $\rightarrow 0$) for $P[X \geq k]$? We ...
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1answer
22 views

Bound on the derivative of a cut-off function

Let $\rho$ be a smooth function in $\mathbb R^n$ such that $0 \leq \rho \leq 1$ and $\rho$ is supported in the unit disk and let $\rho_\epsilon(x) = \epsilon^{-n}\rho(\epsilon^{-1}\|x\|)$. If $f$ is ...
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1answer
51 views

Approximating a Continued Fraction

From a paper I was reading, If: $$w=\frac {1}{3}\left\{ \frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac ...
1
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2answers
43 views

When to Taylor expand in a differential equation

I've come across a differential equation: $\dfrac{d\theta}{dx} = f(\theta)$, whose analytic solution is very complicated, but in this situation it is valid to Taylor expand functions around $\theta ...
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0answers
19 views

Represent this differential equation as a set of n+1 equations w n+1 unknowns

Given the following differential equation: $$s''w'' + 2s'w''' + sw'''' = q$$ We use these approximations: $$w''''(x_i) \approx \frac { { w }_{ i+2 }-4{ w }_{ i+1 }+6{ w }_{ i }-4{ w }_{ i-1 }+{ w ...
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0answers
8 views

Vertex Cover Approximation

Is there any Vertex Cover approximation algorithm that gives the optimal solution for some graphs but otherwise near-optimal solutions for other graphs? Would an algorithm like that be useful? From ...
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2answers
12 views

Vertex Cover - Understanding the bounds

I was reading on wikipedia about the approximations of the Vertex Cover problem and saw that an approximation algorithm with an approximation factor of $\displaystyle 2 - \Theta \left( ...
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0answers
28 views

extrapolate boundary data of a 2d function

I have a function defined over a 2d domain. I want to keep all the internal data, but the information near the boundaries are less accurate and I want to extrapolate them from the inner points. Is ...
2
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1answer
41 views

How do you derive the secant method formula from the equation below?

The Secant Method forumula is; $$ x_{i+1}=x_i - \frac{f(x_i)(x_i-x_{i-1})}{f(x_i)-f(x_{i-1})}.$$ Derive the formula from the equation below; ...
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0answers
39 views

Archimedes' Apprxomation of Square Roots

Supposing a square root $\sqrt{X}$, let $x$ be the approximation of $\sqrt{X}$, then we get these 2 formulas to estimate $\sqrt{X}$: $x_{n+1}=\frac{x_n+\frac{X}{x_n}}{2}$ and ...
1
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2answers
54 views

I do not quite understand what destructive cancellation is, can someone explain it please?

I have been given this example 2.0013−2.0005=0.0008. The destructive cancellation is the large common value, here the 2, that disappears.
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0answers
11 views

Firstly what does relative tolerance mean?

Apply the Jacobi iteration method to the system Ax=b with $$ A= \begin{pmatrix} 3 & -1 & 1 \\ 3 & 6 & 2 \\ 3 & 3 & 7 \\ ...
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1answer
62 views

Is this possible or hopeless to try to prove?

If I have $x_1, ..., x_k=o(n)$ and $j=O(1)$. Is it possible to prove something like: $$\sum_{i=1}^k {n \choose j} \left(\frac{x_i}{n}\right)^j \left(1-\frac{x_i}{n}\right)^{k-j} \sim {n \choose j} ...
6
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1answer
367 views

Approximating the exponential function

I have found experimentally something that seems graphically like an approximation of the exponential function. However, it is totally experimental and I have no idea whether it really converges ...
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0answers
60 views

Rational numbers imply reals?

I was solving an inequality today and I proved it for rational numbers (it was easier because I was able to "strengthen" by doing things like "$\frac{p}{q}>\frac{r}{s}\implies ps\ge qr+1$ since ...
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1answer
31 views

Can someone explain what destructive cancellation is as well as how to answer the question?

Let $f(x)= \sqrt{x^2 + 1} - 1$ (taking the positive real square root, as usual). When $a = 10^{−3}$, compute $f(a)$, working to $5$ significant figures at every stage of the calculation. Also it can ...
1
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1answer
81 views

Why is $\tan 54^\circ\approx \frac{\sin24^\circ}{1-\sqrt{3}\sin24^\circ}$

This question was asked as an equality on MSE and I am quite surprised to find that its strictly false However I would like to see why is their difference of the order $10^{-15}$? $$\tan ...
27
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4answers
3k views

How much gas does a car use to carry its own gas?

I have always been curious about this one. Since the gas has some weight, the car will have to burn some extra gas to carry it's own fuel around. How can I calculate how much that extra gas is? ...
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0answers
23 views

How to get a nice approximation of $f(N,s)=\sum_{k=0}^{N}{N \choose k}{k \choose s-k+N}$ when $N>>1$ and $|s|<<N$?

I need to approximate the above sum in order to calculate $\mathbb{E}(s^2)$, which is the expectation value determined by the probability density function $f$ and the position $s$. Any idea?
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0answers
12 views

Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ I can approximate this to first order as: $$\tilde ...
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1answer
33 views

I am not sure how to use the secant method formula without a function being given?

Calculate an approximation value for $4^{\frac34}$ using four steps of the secant method with the starting values of $x_0=3$ and $x_1=2$.
2
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1answer
132 views

Upper and lower bound of an approximate computation

I'm currently working on an exercise in The Art Of Computer Programming, Vol. 1 relating the algorithm for computing $\log_{10} x$ presented in the Mathematical Preliminaries. The algorithm is ...
0
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1answer
33 views

Show that the approximation to $f$'($x_0$) has discretization error $O$($h^2$)

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h$ and $x_{1} = x_0 + h$ with $h$ > 0. Given a smooth function f, show that the approximation to $f'(x_0)$ given by the centered ...
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1answer
63 views

Approximation for lower incomplete gamma function

Does any one know approximations for the lower incomplete gamma function $\gamma(a,bx)$. The problem is this: I want to find the quantile function for the CDF of the gamma distribution. The CDF of the ...
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2answers
44 views

Firstly what is an $O(h^3)$ formula? Also I am not quite sure how to answer the question?

The forward-difference formula can be expressed as $$f'(x_0)=\frac{1}{h}(f(x_0 +h)- f(x_0))-\frac{h}{2}f''(x_0) - \frac{h^2}{6}f'''(x_0) + O(h^3).$$ Use Richardson's extrapolation to derive an ...
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2answers
64 views

show that $\lim_{k\to\infty}\frac{ k!} { k^k}= 0$ using stirling's formula [closed]

Prove that $\lim _{k\to \infty \:}\left(\frac{k!}{k^k}\right)$ = 0:
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2answers
33 views

What does “symbolically tractable” mean?

What does "symbolically tractable" mean in the following quote? "Traditional treatments of mechanics concentrate most of their effort on the extremely small class of symbolically tractable dynamical ...
1
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0answers
26 views

Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
1
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1answer
44 views

How large should n be to guarantee that the Simpson's Rule approximation on the Integral (from 0 to 1) 19e^x^2 dx is accurate to within 0.0001?

I'm very lost on the following problem and will appreciate your help very much. How large should n be to guarantee that the Simpson's Rule approximation on the Integral (from 0 to 1) 19e^x^2 dx is ...
1
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0answers
43 views

Which (approximative) methods are there to compute the inverse of a complicated function?

I have a complicated function $f(x)$ for which I want to compute the inverse $f^{-1}$ over a certain range $R(f): a \leq f(x) \leq b$. The only way to find the inverse I can think of is power series ...
2
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1answer
50 views

approximation of $\log(1+z)=z$ as $z\to 0$

This is new to me and I have not done any asymptotic approximation. I don't understand how they get that $\frac{n}{N}$ stays close to $\frac{2}{3}$ as N goes to infinity. Also how do they do get ...
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0answers
25 views

How do you solve part (b) to this polynomial interpolation question?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the polynomial interpolation method. (b) Assuming $f(x)\in C^3$, evaluate ...
0
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2answers
46 views

How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
3
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0answers
138 views

What domains of $\mathbb{R}^n$ have the property that $H^1(\Omega)=H^1_0(\Omega)$?

i wonder what are sufficient conditions on an unbounded domain of $R^n$ called $\Omega$ to get : $C_c^\infty (\Omega)$ dense in $H^1 (\Omega)$ ? where $C_c^\infty$ stands for the set of functions with ...
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0answers
26 views

solve equation involving digamma function

I have the following equations that I need to solve. $$ \psi(\alpha)-\psi(\alpha+\beta)=X_0 \\ \psi(\beta)-\psi(\alpha+\beta)=Y_0 $$ $X_0$ and $Y_0$ are known constants. Is there a way to atleast ...
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1answer
20 views

can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that ...
12
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3answers
209 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
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0answers
64 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
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2answers
37 views

Does $E[X]\gg E[Y]$ for independent RV imply that $Pr[X+Y \geq x] \sim Pr[ X \geq x]$?

We have two independent random variables $X$ and $Y$, where we know that $E[X]\gg E[Y]$, thus $\frac{E[Y]}{E[X]}\rightarrow 0$. I am now interested in $Pr[X+Y \geq x]$ and would like to show that ...
0
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1answer
38 views

Newton-Cotes Quadrature formula

Im trying to find more information about numerical integration methods. When is a Newton-Cotes Quadrature formula on n nodes exact?
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1answer
42 views

Poisson approximation of $X$ by $Poisson(E[X])$

I've tried to find something, but couldn't find anything about the following question. Is it possible to approximate any random variable $X$ with $E[X]=o(1)$ by a Poisson random variable ...
0
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0answers
24 views

Spatial Basis Functions…any practical hints on choosing them?

I am not coming from a pure mathematical background, and now in my daily life as an engineering student, I am getting faced to spatial basis functions to approximates variables. The problem is: Are ...
3
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1answer
28 views

Find a maximum triangle that lies on a polyline (with constraints)

If there's a polyline (a GPS track, actually) with a lot of points (could be several thousand), that looks like this 1) How can I find such a triangle with the biggest possible perimeter, that its ...
7
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3answers
890 views

Is it possible to find square root using only rational numbers and elementary arithmetic operators

Suppose I have a number a How can I find it's square root using only +, -, /, ...