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
46 views

Optimal approximation of spline curve using linear interpolation

I have a parametric cubic spline which I need to draw in graphics. I am restricted to using a set of lines to draw this, and for performance reasons I need as few lines as possible. So I need an ...
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2answers
45 views

Approximation of $\ln(x+1)$ with $\Psi$ function

I found the following approximation for the function $$f=\ln(x+1)$$ $$f\simeq\Psi\left(x+\dfrac{3}{2}\right)-2+\gamma+\ln(2)$$ where $\Psi(x)$ is the 'Digamma' function: $\Psi(x)=\dfrac{\dfrac{d}{dx}\...
2
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1answer
63 views

A curious approximation to $\cos (\alpha/3)$

The following curious approximation $\cos\left ( \frac{\alpha}{3} \right ) \approx \frac{1}{2}\sqrt{\frac{2\cos\alpha}{\sqrt{\cos\alpha+3}}+3}$ is accurate for an angle $\alpha$ between $0^\circ$ ...
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0answers
22 views

Reduced Chebyshev approximation?

Few days ago my teacher mentioned a method of approximation for a function. I think it was called "reduced chebyshev approximation", where you find the taylor series of degree $N$ then subtract from ...
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1answer
37 views

What is an accurate approximation and asymptotic for this function?

$$f(n)=\Bigg(1-\Big(1-\frac{1}{2^{n/2}}\Big)^n\Bigg)^{n^7}$$ I am interested in large $n$. The number $7$ can be replaced by any fixed integer. I have $$f(n)\rightarrow\Bigg(1-e^{-n2^{-n/2}}\Bigg)^{...
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1answer
44 views

Upper bound for ratio of modified Bessel functions of second kind

I was wondering if someone has an idea if for $0 < x < y$, and $0< \nu \leq \frac{1}{2}$, one can obtain an upper bound for the ratio $$ \frac{K_{\nu}(x)}{K_{\nu}(y)} $$ Thanks.
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1answer
88 views

Alternate proof of the integral: $\int_0^1 x^x(1-x)^{2x}\,dx\neq3/8$

I am looking into the integral: $$I=\int_0^1 x^x(1-x)^{2x}\,dx\neq\frac{3}{8}$$ How might you prove this to be true? What's tough is that the integral $$3/8\lt I<0.37503$$ numerically. I managed ...
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1answer
55 views

Proof that Newton's Method gives better and better approximations with each iteration?

I've seen this question and answer: Why does Newton's method work? It gives some geometric intuition as to what is going on when applying Newton's method, but what I really need to know is why it ...
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1answer
56 views

How to prove that your approximation using Newton's Method is correct to $x$ decimal places?

I just watched a video about a problem stated as: "Find where $f(x)= x^7-1000$ intersects the $x$ axis. Find solution correct to 8 decimal places." The author uses Newton's Method, repeating the ...
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0answers
56 views

Ideas on how to simplify or approximate this nasty sum

I have a sum (let's call it $p$): $$p:= \frac{1}{n!}\sum_{i=2}^l (i-1)\frac{(n-k)!}{(n-k-i+2)!}(n-i)!$$ where $l, n, k$ are fixed positive integers, and $k \leq n$. I'd like to either simplify or ...
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1answer
32 views

Optimization algorithms for Distribution and Logistics scenario

I am looking for a way to express the following logistics/distribution problem as an equation that can be run thru a solver to find an approximate solution. The problems is described as follows: ...
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2answers
45 views

Binomial theorem estimate for very large samples

I have around $2^{105}$ balls, of which 1 in 20 is white. I expect that when I draw a random sampling of them, roughly 5% of all balls drawn would be white. What is the probability that, if I draw $...
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2answers
65 views

When is $\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$ a valid approximation?

It is often said that when the change in e.g. $\Delta x$ is small than we can make the approximation: $$\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$$ But it is not enough to say $\Delta x$ is small ...
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1answer
12 views

Approximately equal too symbol operations permitted

I have to doubt to clear if A is approximately to 2/3 B can I write it as A approximately equal to 4/6 B? Since it is approximate relation is it permitted to multiply both numerator and denominator as ...
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1answer
37 views

Why is $\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$?

I came across this approximation in the book Principles of Population Genetics by Hartl and Clark (page 130). $$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$ ...
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0answers
19 views

What is the name of this approximation?

I remember studying a while back about an approximation method where the error is calculated using $$ E_{n}=M_{n+1}-a_{n+1} \widetilde{T}_{n+1} $$ Where $\widetilde{T}_{n}=\frac{{T}_{n}}{2^{n-1}}$, ...
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1answer
29 views

Solve using succesive approximation methods

Consider the equation $f(x) = xe^x-1=0$. We wanna solve it using successive approx method, solving the fix point problem $x=e^{-x}$ I don't understand Banach's theorem, how can I solve this? Any idea,...
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2answers
44 views

Separation of integral by approximation

I'm working with the following integral $\displaystyle\int_0^y \frac{dx}{x \sqrt{1-ax-bx^2}}$ and would like to split it in something like $$\int_0^y\frac{dx}{x \sqrt{1-ax}}+\int_0^y\frac{dx}{x \sqrt{...
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0answers
17 views

methods function approximation

My goal is to modelize the temperature of a motor: I want to be able to calculate the temperature of a motor at a given moment, with a given torque and a given speed, in a constant environment. For ...
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3answers
50 views

Why transform degrees into radians when computing linear approximation to find $\tan{44^\circ}$?

I am asked to find the linear approximation of $\tan{44^\circ}$. Why should I transform degrees into radians to do that? I understand that using degrees would give me a wrong solution (which would be ...
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1answer
33 views

Linear approximation to find $\frac{1}{4.002}$

I am asked to find $\frac{1}{4.002}$ using linear approximation. The way I proceeded was: $$ f(x) = \frac{1}{x} \quad a = 4 \quad f(a) = f(4) = 0.25\\ f'(x) = - \frac{1}{x^2} \quad f'(a) = f'(4) = - ...
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1answer
47 views

Is there a simple way of computing when $a^n=b^m$

I don't want exact equality just close enough to be useful in approximation. i.e. $2^{10 }= 10^3$ is very useful and used daily for an approximation. Is there a do this efficiently? Is there a way ...
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1answer
36 views

Proportion in sets

We have $3$ sets of positive integers. $$A = \{x_1,y_2,z_3\},\quad{} B = \{x_2,y_2,z_2\}, \quad{} C= \{x,y,z\}$$ Which proportion do we use for adding $A$ and $B$ ($x_1+x_2$ and so on), so the ...
2
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1answer
29 views

Local quadratic approximation

I wanted to implement some penalized regression parameter estimation algorithm by Fan&Li (http://sites.stat.psu.edu/~rli/research/penlike.pdf, section 3.3, [1]), but cannot catch the idea of some ...
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4answers
143 views

Approximation of $\sqrt{2}$

I got the following problem in a chapter of approximations: If $\frac{m}{n}$ is an approximation to $\sqrt{2}$ then prove that $\frac{m}{2n}+\frac{n}{m}$ is a better approximation to $\sqrt{2}.$(...
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1answer
19 views

Inequality insde algorithm design (kleinberg), Disjoint Path

I am reading the "Algorithm Design" (Kleinberg et al). Inside this book, at the chapter 11.6 there is an inequality I have failed to resolve. Considering $\beta=m^{(1/3)}$ and $|I|>=1$ (it is a ...
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0answers
13 views

How do I show I've rounded to a certain decimal?

I noticed that we could say $\sqrt{0.9}\approx0.9$ if we round to one decimal, or $\approx0.95$ if we round to two decimals. But what is the correct way to show how many decimals we rounded to? ...
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9answers
3k views

How is the derivative truly, literally the “best linear approximation” near a point?

I've read many times that the derivative of a function $f(x)$ for a certain $x$ is the best linear approximation of the function for values near $x$. I always thought it was meant in a hand-waving ...
0
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1answer
72 views

Approximation of a polynomial with fractional power

I have a polynomial I need to find the roots of, the major difficulty is that this polynomial has fractional exponents. I have made an approximation and I would like to have some idea of the error I ...
3
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1answer
31 views

How do I calculate this limit when two terms tend to infinity at similar rates

In a particular problem that I am currently trying to solve, I have the following expression (this is not the entire expression, I have included only the terms involving $a_1$ and $b_1$), $\lim_{(...
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0answers
16 views

Computing the partition-function of an exponential family member

I am working on an Monte Carlo Expectation Propagation problem. In that context I got the following property: $ I = \sum\limits_i w^{(i)} \log p_\eta(x^{(i)}) $ where $\{w^{(i)}\}_i$ are weights, $...
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0answers
63 views

What's about $\sum_{n=1}^{\infty} \frac{e^{H_n}\log H_n}{n^3}$, where $H_n$ is the nth harmonic number?

I would like to do a toy verification of the Riemann hypothesis exploiting theLagarias theorem (see the section Applications in the following link) and the fact that we know a lot of decimals for ...
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1answer
50 views

B-Spline approximation deviates a lot while increasing the number of control points???

I'm dealing with a problem to approximate some data points with B-Spline. I follow the method and implemented the algorithm from this site: Curve Global Approximation. 1) The first step is to ...
3
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3answers
53 views

Question for the estimation of $\sum_{i=1}^x \frac{1}{w+i}$ as $x \to \infty$.

I have a question of the estimation of this summation: $$ \frac{1}{w+1}+\frac{1}{w+2}+\cdots+\frac{1}{w+x}$$ Which is: $$\sum_{i=1}^x \frac{1}{w+i}$$ What I have tried: applying limit to the ...
0
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1answer
56 views

Approximation of product of Bernoulli with different proportions

I want to update a variable $Y$ with Beta (uniform for simplicity, $Y \sim U(0, 1)$) distribution, with Bernoulli information each period... But each period the proportion parameter of the Bernoulli ...
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3answers
116 views

Approximating the normal CDF for $0 \leq x \leq 7$

In answer http://stackoverflow.com/a/23119456/2421256, an approximation of the complementary normal CDF (ie $\frac{1}{\sqrt{2 \pi}} \int_x^{+\infty} e^{-\frac{t^2}{2}} dt$) was given for $0 \leq x \...
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0answers
6 views

Other types of mean error

Let $\tilde f$ be an approximation of the function $f(x) = \arccos(x)$. I'm using MATLAB to figure out how good this approximation is by calculating a mean error. My first idea was to use this formula ...
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0answers
5 views

Equi-oscillation condition $EOC(n)$ for best polynomial approximations.

I'm given the funciton $f(x) = x^3$ on $[-1, 1]$ and am told to find the best order one polynomial approximation to this. The solution says that $p(x) = \frac 34 x$ is this polynomial (and I verified ...
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2answers
32 views

Is it true that $ \sum_{i=1}^m \frac{1}{\sqrt{i}} = O \left( \sqrt{ m-1 } \right) $?

Is it true that?: $$ \sum_{i=n}^m \frac{1}{\sqrt{i}} = O \left( \sqrt{ \frac{m-n}{n}} \right) $$ In special case if we have $n = 1$, is it true that?: $$ \sum_{i=1}^m \frac{1}{\sqrt{i}} = O \left( \...
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3answers
65 views

Probability of alternating sequence from uniform distribution

Say I sample a discrete uniform distribution $U$ (say $U$ has a support of $N$ elements, and there is a total order on the elements) a number $K$ times, resulting in a random sequence $A_{i}$. What ...
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38 views

Upper-bounding $\sum_{i=1}^n \sum_{j = i}^{i+a_i} \frac{1}{\sqrt{j}}$?

Suppose $a_1, ..., a_n \in \mathbb{N}$ are arbitrary integers. Is it possible to bound $$ A =\sum_{i=1}^n \sum_{j = i}^{i+a_i} \frac{1}{\sqrt{j}} $$ with either of the following: $$ B = c\sqrt{\sum_{...
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0answers
23 views

Bernstein polynomial in Banach

For a continuous function $f : [0,1] \to R$, there exists a sequence of polynomial functions: $P_n(x)=\sum_{k=0}^n C^k_n x^k(1-x)^{n-k} f(\frac{k}{n})$ (Bernstein's polynomes) which converges ...
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0answers
52 views

Understanding a medieval approximation

A medieval text (Maimonides's commentary to chapter 2 of Eruvin in my retranslation from the Hebrew) discusses a rectangle whose area is $5000$ square cubits. It reads in relevant part: … that the ...
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2answers
77 views

Arriving at the asymptotic $\int \limits_\lambda^\infty e^{-t^2/2}dt \sim \frac{e^{-\lambda^2/2}}{\lambda}$

In the book "The Probabilistic Method", the integral $\int_\lambda^\infty e^{-t^2/2}dt$ is said to be "approximately equal" to $\frac{e^{-\lambda^2/2}}{\lambda}$ for large $\lambda$. I assume what is ...
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1answer
35 views

How can I express this in terms of Gauss-Hermite Quadrature?

I am having the following expression. This is the PDF of Nakagami-Lognormal Distribution. I want to express in terms of Gauss-Hermite abscissas and weights. How can I do it? $$f_Z(z)=\int_0^{\infty}\...
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0answers
31 views

Approximate $e^{0.01} \sin(0.02)$ by using its linearization

$f(x,y) = e^x \sin(y)$ $f_x = e^x \sin(y)$ $f_y = e^x \cos(y)$ $L(x,y) = f(0,0) + f_x(0,0)x +f_y(0,0)y$ Solving: $f(0,0) = e^0 \sin(0) = 0$ $f_x(0,0)x = e^0 \sin(0) \cdot x = 0$ $f_y(0,0)y = e^...
2
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0answers
45 views

Uniform approximation. Two boundary layers?

Find uniform approximation up to order $O(\epsilon)$: $$ \begin{cases} \epsilon y''+\epsilon y' - y^2=-1-x^2 \\ y(0)=2 \\ y(1)=2 \end{cases} $$ At $\epsilon=0$ solutions $\pm \sqrt{1+x^2}$ don't ...
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0answers
10 views

Accuracy Rebonato Swaption Approximation Formula among Different Strikes

Can somebody explain me if the Rebonato swaption volatility approximation formula is accurate for only ATM strikes, and if yes why? Can it also be used for ITM and OTM strikes? My foundings: Let $0 &...
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0answers
11 views

Error bound for function limit with arbitrary $\Delta h$

Say we have some function $f\in C^1$. I would like to somehow bound the error that is made when the limit is approximated using some arbirary $h$, that is to find $E(x,h)$ such that: $$|f'(x)-\frac{f(...
0
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1answer
21 views

For $f(x)=x^4$, find its projection $f(x)^*\in P^2(-1,1)$ onto $W$

Consider the vector space $V=C[-1,1]$ and $W=P^2[-1,1]$. $V$ is an inner product space withe inner product $\langle f, g\rangle=\int_{-1}^1f(x)g(x)dx$. Consider a function $f(x)=x^4$ whcih is in $V$ ...