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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4
votes
1answer
103 views

Proof about binomial coefficient

I today see a approximated equation, when $n \ll u $: $$\log {u \choose n} \approx n \Big(\log \frac{u}{n} + 1.44\Big)$$ I would like to know how to prove it.
6
votes
2answers
438 views

Elementary bound of binomial coefficient

I'm working my way through an Erdős paper from the sixties and some of the elementary bounds he claims seem to be just beyond my reach. The expression looks horrendous but maybe there is a clever ...
3
votes
0answers
84 views

Numerically estimate $a^b$ [duplicate]

Possible Duplicate: How can I calculate non-integer exponents? What is the most efficient way to estimate $a^b$ ($a > 0$) numerically? My goal is not to use built-in math functions (like ...
4
votes
1answer
2k views

Approximation of a bounded measurable function with step functions?

I'm having trouble judging whether this statement is correct: For an arbitrary bounded measurable function $f$ defined on $[0,1]$, $\exists{}\ $a sequence of step functions $\{\phi_n\}$, such that ...
2
votes
2answers
222 views

How many significant figures are needed in base 2?

$x \in \mathbb{R}$ $2^{500}<x<2^{501} $ How many significant figures are needed in base 2, to know in high approximation whether $2^x$ is integer?
1
vote
0answers
293 views

Calculation of a 'double' sum

Let $n \in N$ and $q\geq 2$. I am trying to calculate the following sum: $$ \sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!} $$ Any help will be ...
4
votes
1answer
214 views

Evaluating a limit involving binomial coefficients.

If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer: $$\lim_{n\rightarrow ...
1
vote
1answer
104 views

Limit of a sum for which the upper limit is also in the argument of the sum - Taylor series of $e^x$

A book I'm reading claims that $\frac{1}{2}(k-1)!\sum \limits_{j=0}^{k-3} \frac{k^j}{j!} \sim (\pi / 8)^{1/2}k^{k-\frac{1}{2}}$ as $k \to \infty$. I can get most of the expression to work out nicely ...
1
vote
1answer
75 views

Bound for sum with geometric progression

Let $n_i$, $i=1,\ldots,m+1$ be nonnegative natural numbers, sum of which $\sum_{i=1}^{m+1}n_i=N$. I woul like to find an upper bound for the following$$ \sum_{i=1}^{m+1}\frac{\sqrt n_i}{2^{i-1}}$$
10
votes
4answers
3k views

Find bound for sum of square roots

Let $a_1,...,a_n$ be real numbers, such that $a_1+...+a_n=A$. What can we say about $\sqrt{a_1}+...+\sqrt{a_n}$? I would like to bound from above thus sum in terms of $A$.
2
votes
1answer
261 views

Best and most efficient way to numerically compute $e$?

There are many well-known methods for efficiently numerically computing $\pi$, such as Chudnovsky's Method or perhaps Gauss-Legendre's algorithm. I was wondering what the best method for computing $e$ ...
4
votes
2answers
382 views

computation of the sum

I am having trouble to compute the following sum: $$ \sum_{k=0}^n(n-2k)^p \frac{{n \choose k}{2m-n \choose m-k}}{{2m \choose m}} $$ Here $p\geq 2$. To simplify the question, we can even assume that ...
4
votes
1answer
1k views

Method for estimating the $n^{th}$ derivative?

When using numerical analysis, I often find that I am required to estimate a derivative (e.g. when using Newton Iteration for finding roots). To estimate the first derivative of a function $f(x)$ at ...
1
vote
1answer
238 views

Approximation of Integration by parts

I'm trying to approximate a integral of the form: $$\int_V{g({\bf x})f({\bf x})} \; d^3x$$ Where the functions $f(\bf{x})$ and $g(\bf{x})$ are positive functions, but where only $f(\bf{x})$ is known ...
2
votes
1answer
308 views

General bound on a polynomial's root with the largest norm

Is there a general bound on a polynomial's root with the largest norm? When Rouche's theorem is used, it still seems that the polynomial's root with the largest norm still needs to be found if we ...
1
vote
0answers
85 views

Approximation or calculation of the probability of getting “clumps” when sampling from a uniform distribution

Suppose that there are $n$ independent samples $X_1,X_2,...,X_n$ sampled from the uniform distribution on $[0,1]$ with the pdf $f(x)=1$. Is there a good way to calculate or approximate the ...
1
vote
2answers
451 views

Smoothing of absolute value and sign functions for numerical integration

I'm doing Numerical integration of ODEs. for a special system that has an always positive coordinate s and a conjugated momentum ...
0
votes
0answers
290 views

Comparing norms of a vector

Let $a$ be a vector in $\mathbb R^m$, such that $\sum_{i=1}^{m}a_i=0.$ I would like to compare $\sqrt{2m(2m−1)}\|a\|_{\infty}$ and $\sqrt{2m}\|a\|_2$, in the case when the vector $a$ satisfies the ...
0
votes
1answer
132 views

inequality and equivalence for norms

Let $x \in R^m$. It is known that $\|x\|_{\infty}\leq \|x\|_2\leq \sqrt m\|x\|_{\infty}$. What the difference between above inequality and if we are saying that $\|x\|_2\sim \sqrt m\|x\|_{\infty}$? ...
1
vote
1answer
363 views

Spline Theory and Code

On P. Janert's book Data Analysis with Open Source Tools there is a discussion on splines, that they are "constructed from piecewise polynomial functions (typically cubic) that are joined together in ...
3
votes
0answers
82 views

Is that observation really a property of the log of coefficients of continued fractions (example: cf(log(3)/log(2))

I'm again looking at the problem of approximation of perfect powers of 2 to that of 3 (I assume $\small q_N = 2^S / 3^N \gt 1 $) and specifically using the continued fraction representation of $\small ...
2
votes
1answer
2k views

Degenerate critical points and higher order derivatives (Intuition)

The question I'd like to ask is about the relevance of higher order derivatives in determining the behaviour of a function around a critical point in the multivariate case. The question occurs towards ...
1
vote
1answer
1k views

How to approximate an integral using the Composite Trapezoid Rule

I'm trying to estimate the value of the following integral on the interval $[0,1]$ $$ I = \int_0^1 \frac{1}{1+x} dx $$ So, using the composite trapezoid rule (and with $n=4$, ie I'm only using the ...
1
vote
4answers
136 views

$\log_2$ approximation in $[1,2)$

this is realistically for a programming project, but is more math centric then CS centric. I am attempting to write a function that approximates a power function, but in order to complete I need to ...
1
vote
5answers
595 views

Find square root approximation function (tool)

first I have to apologize for any uncorrect naming or categorisation of my question, as I am an electrical engineer rather than a mathematican. I try to find a simple solution for my problem: I have ...
1
vote
1answer
276 views

Newton-Raphson's Method to find $\sqrt{2012}$

I am asked to find $\sqrt{2012}$ using Newton-Raphson's Method with the following recursive method $$x_{n+1} = \frac{1}{2} (x_n + \frac{a}{x_n}) $$ I notied that give same answers as using ...
4
votes
1answer
153 views

Numerical differentiation issues

I've been using this to compute the first order derivative's value of a function $f$ in a given point: $$f'(x) = \frac{f(x+\epsilon) - f(x-\epsilon)}{2\epsilon}$$ For some $\epsilon = 0.0001$ or ...
1
vote
1answer
700 views

Interpolation error

Working with a homework problem where I'm to derive an estimation of the interpolation error, and compare it with the actual error. This part is ok and I'm done with it. But while working with this in ...
2
votes
0answers
114 views

explicit error bounds for Multivariate interpolation

I want to interpolate a function of $d$ variables over a Cartesian grid, using multivariate interpolation, while characterizing interpolation error in terms of bounds on partial derivatives of the ...
3
votes
3answers
10k views

Relation between Simpson's Rule, Trapezoid Rule and Midpoint Rule

I am studying numerical approximation and verifying $S_{2n} = \frac{1}{3}\left(T_n +2 M_n\right)$. ($S_n$ refers to Simpson's Rule approximation, $T_n$ refers to Trapezoid Rule approximation and $M_n$ ...
7
votes
3answers
7k views

How to justify small angle approximation for cosine

Everyone knows the picture that explains instantly the small angle approximation to the sine function (as defined by the parametrisation of the unit circle): "what's the length of that arc?" "See how ...
1
vote
1answer
126 views

Estimation of $x$ if $x! = N^{\log N}$

If $x! = N^{\log N}\;,$ How can I estimate $x$ in terms of $N$?
1
vote
0answers
265 views

What “boundary conditions” can make a rectangle “look” like a circle?

I posted the question below in Stackoverflow but then realized that it perhaps would find a better audience here. I am solving a fourth order non-linear partial ...
4
votes
1answer
2k views

Projection of Gaussian in Spherical Coordinates

Consider a point with spherical coordinates $\vec{r}_0=(r_0, \theta_0, 0)$. The spherical gaussian distribution centered at $\vec{r}_0$ is $f(\vec{r})=Ne^{|\vec{r}-\vec{r}_0|^2/A}$, where $N$ is the ...
2
votes
1answer
399 views

Sum of power series

Good morning, I have difficulties to find an approximation formula (or bound from the height) for the sum of the following power series $\sum \limits_{k=1}^\infty e^{-k^2}x^k$. Thanks
20
votes
8answers
4k views

Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
17
votes
3answers
1k views

Sine Approximation of Bhaskara

An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.) $$\sin x \approx ...
4
votes
3answers
507 views

A better approximation of $H_n $

I'm convinced that $$H_n \approx\log(n+\gamma) +\gamma$$ is a better approximation of the $n$-th harmonic number than the classical $$H_n \approx \log(n) +\gamma$$ Specially for small values of $n$. ...
2
votes
0answers
73 views

On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that $$ A_0 = A_1 = 0;\quad A_n = ...
2
votes
2answers
337 views

How does this square root approximation work?

I've come across an odd way of estimating the square root of a number, going like this: Given a number n, Subtract the odd numbers from n in a rising order (1, 3, 5...), until $n \leq 0$ Count how ...
1
vote
1answer
1k views

Generating control-point tangents for a Catmull-Rom spline

From Wikipedia, we have a few variations for calculating tangents when creating a spline based only on positions of control-points: Finite difference $$\mathbf m_k=\frac{\mathbf p_{k+1}-\mathbf ...
1
vote
2answers
903 views

Deriving the approximation formula

$f'(x) \thickapprox$ $\frac{1}{2h} [ 4f(x+h) - 3 f(x) + f(x + 2h)]$ I need to derive the approximation formula for the function above. And I need to show that it's error term is of the form ...
3
votes
1answer
56 views

Lower-Bounding angles in integer Lattices

Given an $n \times n$ integer grid I chose any two grid points $a,b$, draw a line $l$ through $a$ and $b$ and measure the angle between $l$ and a horizontal line. I can do this for any grid point pair ...
2
votes
1answer
312 views

A question about the coercivity of a lsc and convex function.

I was doing a proof and I need to show a result to conclude it: $X$ is a reflexive Banach space with a norm, $\|\cdot\|$, of class $\mathcal{C}^1$. $f:X\to\overline{\mathbb{R}}$ is lower ...
4
votes
2answers
195 views

solution to $\min \|A-BXC \|$

I have the following problem. Let $A$, $B$ $C$ be real-valued matrices of size $m \times q$, $m \times n$, $p\times q$ respectively. I would like to find matrix $X$ of size $n\times p$ and maximum ...
1
vote
0answers
106 views

Combining multiple posterior distributions

I am new to Bayesian statistics, and thus have problems to come up with a solution for the following problem: Using Approximate Bayesian Computation (ABC), I generate a posterior distribution from ...
5
votes
3answers
223 views

Approximate $\int_{0}^{\infty} \frac{\text{d} x}{1 + x^4}$

Now, I have been given this integral. And need to approximate it. My first idea was to use a Taylor series, but this series explodes, as x reaches infinity. Does anyone know how to approximate ...
0
votes
1answer
116 views

Approximation to $\mathbb{E}(X/Y)$

Let $X,Y$ are two random variables which are not necessarily independent. It is easy to get $\mathbb{E}(X)$ ann $\mathbb{E}(Y)$. I want to know: is there some approximation to ...
4
votes
1answer
508 views

Differentiability of Moreau-Yosida approximation.

I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is ...
21
votes
9answers
5k views

What is the purpose of Stirling's approximation to a factorial?

Stirling approximation to a factorial is $$ n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n. $$ I wonder what benefit can be got from it? From computational perspective (I admit I don't ...