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).

learn more… | top users | synonyms

1
vote
1answer
33 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). $$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$ ...
2
votes
0answers
18 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 $\widetilde{T}_{n}=\frac{{T}_{n}}{2^{n-1}}$, ...
1
vote
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,...
1
vote
2answers
43 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{...
0
votes
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 ...
2
votes
3answers
49 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 ...
1
vote
1answer
32 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) = - ...
0
votes
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 ...
3
votes
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 ...
1
vote
0answers
21 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), but cannot catch the idea of some ...
1
vote
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 $\sqrt{2}.$(...
0
votes
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 ...
0
votes
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? ...
33
votes
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
votes
1answer
71 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
votes
1answer
30 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_{(...
0
votes
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, $...
2
votes
0answers
54 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 ...
1
vote
1answer
39 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
votes
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
votes
1answer
55 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 ...
5
votes
3answers
115 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 \...
0
votes
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 ...
0
votes
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 ...
1
vote
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( \...
2
votes
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 ...
1
vote
0answers
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_{...
0
votes
0answers
22 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 ...
5
votes
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 ...
0
votes
2answers
76 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 ...
1
vote
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}\...
1
vote
0answers
29 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
votes
0answers
44 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 ...
0
votes
0answers
7 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 &...
0
votes
0answers
10 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
votes
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 $V$ ...
2
votes
1answer
34 views

Approximating Fresnel integrals with standard functions

I would like to approximate the Fresnel S and Fresnel C with standard functions. I've started with the $ S(x) $ function: $$ approxS(x) = sgn(x) * \left ( sgn(x)* \left ( \frac{ \sin( \frac{x^2}{2} ...
0
votes
0answers
39 views

Quadratic Approximation Using Chebyshev Economization

For the quadratic approximation of given function, we use following: $\ Q(f) = f(a) + f'(a)*x + f''(a)*(x^2)/2$ In the question, it wants me to find quadratic approximation using chebyshev ...
0
votes
0answers
21 views

Approximation, Truncation argument, Sobolev space

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), j=1,...
0
votes
0answers
20 views

Big 'O' Notation - Taylor Series

Q) Use the Taylor Series Expansion to show the first derivative f '(x) can be approximated by $$-(3f(x) -2f(x+h) - f(x-h) / h ) $$ What is the precision? Now I found after using the Taylor ...
2
votes
2answers
38 views

Approximate value of k

How do you solve $k$ in $\frac{(k-1)^{k-1}}{k^{k-2}}=n$ at least with a good approximation? Is there tight approximation?
0
votes
0answers
12 views

Approximating integral of Erf with certain available functions.

I am developing certain software that deals with symmetric 2D Gaussian densities. One of the most common operations in that software is integrating those Gaussians over various 2D shapes. These ...
0
votes
0answers
46 views

A fast converging limit for $\ln x$ (or why $\ln 2 \approx \sqrt{\sqrt{42}-6})$

I've read a lot about approximating logarithms recently, and apparently it's not easy. It can be done by Taylor series (slow convergence), by continued fractions (also slow) and also by some limits. ...
0
votes
0answers
9 views

Approximating the integral of a large product

I would like to approximate the following integral of a product: $$ I = \int dz\, f(z)\prod_{i=1}^n\left(1 - \rho_i(z)\right) $$ The functions $f$ and $\rho_i$ are differentiable for all $i$, $\...
0
votes
0answers
11 views

interpolation preserving boundedness property

I'm trying to construct interpolation for a function $m$ such that \begin{equation*} 0\leq m(x)\leq 1,\quad\forall x\in\Omega\subset \mathbb{R}^1. \end{equation*} I tried to use Lagrange-...
0
votes
0answers
9 views

How to compute norm bound error in robust approximation

I am reading convex optimization, and I am little confused about the following two prolems in norm bound error of robust approximation. How to compute $\{\|\bar{A}X-b+Ux\| | \|U\|\le a\}$ ? For the ...
1
vote
1answer
42 views

$N$ is approximately linear in $d$ for $N^d=\frac12 e^{N}$

let us look at the function $N^d e^{-N}$, for each $d\in \mathbb{N}$. The graphs of the function for various values of $d$ show a striking phenomenon: the graph look parallel, and with a near-constant ...
1
vote
1answer
29 views

Can a discrete function converge to a continous function?

Let $f\in C^{\infty}[a,b]$, let also $X \subset [a,b] = \left\{x_0,\ldots,x_k \right\}, Y = \left\{ f(x_0),\ldots, f(x_k) \right\}$. I guess that if I let $k\rightarrow \infty$ then some how I should ...
-1
votes
1answer
20 views

Numerical Analysis: Approximations — Discrete Average Value Theorem

I am asked to compute approximations to $f'(1)$ using $h=\frac{1}{16}$ for $f(x)=\sqrt{x+1}$ with the following formulas $$f'(x)-\frac{f(x+h)-f(x)}{h}=-\frac{1}{2}hf''(\xi_{x,h})=\mathcal{O}(h)....(1)...
0
votes
0answers
21 views

Weight Function in gaussian quadrature

My question is pretty simple, although I know of the properties that the weight function must follow , such as being well defined,positive,continuos and integrable on the interval . I do not know how ...