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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11
votes
1answer
171 views

Is $e = \sum_n 1/n!$ the most efficient sequence of denominators for rational series for $e$?

The classical series $e = \lim_{n \to \infty} X_n$ where $X_n = \sum_{k=0}^n 1/k!$ is incredibly efficient. But is it known to be the most efficient series in terms of denominators for using fractions ...
2
votes
1answer
190 views

Curve fitting a power law to a linear fractional transformation

Good Morning ! I have a function $y = \frac{ax + b}{cx + d}$. I want to fit the curve $y_f = c_1 + c_2 x^{c_3}$, so that $||y_f - y||$ is minimized in some norm (say $L^2$), by varying $c_1, c_2$ ...
4
votes
4answers
1k views

Damped oscillation fit

We have some measurement data like this: The expected behavior of the data is a damped oscillation: $$y=a e^{d*t} cos(\omega t+\phi) + k$$ Where: $t$ Current time $y$ Current deflection ...
1
vote
1answer
39 views

Estimate a common formula

I know this should be easy, but I just can't find the proper search result. Thanks. $\left(1-\frac1n\right)^n$, what is the estimation value when $n$ is very large? Some follow-up, If $n = ...
0
votes
1answer
62 views

Approximation of logarithm of harmonic mean

Given a large $M\in\mathbb{N}$,if I could approximate $M\log\left(1+\frac{1}{\sum_{i=1}^M x_i^{-1}}\right)$ by $\sum_{i=1}^{M}\log(1+\frac{x_i}{M})$ in the case of $x_i\in\mathbb{R}$, $0<x\leq C ...
0
votes
1answer
576 views

Simpson's Error Bound Estimation

The problem: I need to use Error Bound to find n (least) to the $10^{-9}$ in approximating the integral of 5e^x^2 from 0 to 1 I'm using $$Error(Sn) \le \frac{k(b-a)^5}{180N^4}$$ I found the 4th ...
5
votes
1answer
131 views

Integral approximation with divergent integrand

Given a integral of a continuous function on a closed (small) interval, one can find an approximate value of the integral using Simpson's rule or other quadrature rules. What happens if even though ...
3
votes
2answers
212 views

Simple problem whose approximation ratio is still open.

I am preparing for a talk on "Approximation Algorithms", aimed at undergraduate students. In order to motivate the topic, I want to give them an example of a problem which is easy to describe and has ...
2
votes
4answers
141 views

When multiplying 2 positive integers, why when we round the larger number up and the smaller number down, the product will be lower and vice versa.

I experimented with 2 digits multiplication and found out that when we rounded the larger number up and the smaller number down, the product will be lower. When we rounded the larger number down and ...
1
vote
1answer
97 views

question about deriving the midpoint method

http://en.wikipedia.org/wiki/Midpoint_method $y'(t) \approx \frac{y(t+h) - y(t)}{h} \qquad\qquad (3)$ For the midpoint method, one replaces (3) with the more accurate $ y'\left(t+\frac{h}{2}\right) ...
1
vote
0answers
40 views

Approximation of series

Can anyone provide an approximation for the series: $$\sum_{i=0}^{\infty}\frac{\left(1-\Gamma\left(i+1,\frac{b^{2}}{N_{01}}\right)\right)v^{-i-1}}{\mu_{1}N_{01}^{i}}\\\times ...
0
votes
1answer
103 views

Landau-Lifshitz light aberration formula order of accuracy

I'm reading this part of Landau, Lifshitz, The Classical Theory Of Fields: I'm able to derive the unnumbered formula for $\sin\theta-\sin\theta^\prime$, finding expansion to Taylor series, but what ...
1
vote
0answers
71 views

Best simple exponential estimator

Given smooth enough positive $f(x)$. Find real $c$ such that $$\int_0^1 (e^{cx}-f(x))^2dx$$ is minimized. Note: it only looks innocent..
6
votes
1answer
199 views

Simple approximation to a sum involving Stirling numbers?

I have also posted this question at http://mathoverflow.net/questions/141552/simple-approximation-to-a-sum-involving-stirling-numbers#141552. I have an exact answer to a problem, which is the ...
0
votes
1answer
48 views

Fastest path to cover area

How do we determine the fastest path to cover an area using an object of some radius r? E.g. a machine that needs to spray a chemical onto a surface. I assume this is some kind of NP-hard problem.
1
vote
0answers
61 views

Kalman filtering with angle measurements

I'm designing an Extended Kalman Filter which will take several types of measurements and try to estimate a location. One type of measurement is the direction to the location. I've thought about this ...
4
votes
0answers
105 views

Using formal power series to solve nasty equations

Consider a function $f:[0,\infty)\times \mathbb R\to\mathbb R$, and suppose that given some $a>0$, I would like to solve for $x\in\mathbb R$ satisfying \begin{align} f(\delta, x) = a. \end{align} ...
0
votes
1answer
46 views

a question on decreasing sequence of subspaces II

This is related to this question see here Let $V= \mathbb{Q}^{\mathbb{Z}}$, considered as an infinite dimensional linear space over $\mathbb{Q}$. And assume $W=\mathbb{Q}^F$ is a finite dimensional ...
3
votes
1answer
87 views

Does projection onto a finite dimensional subspace commute with intersection of a decreasing sequence of subspaces: $\cap_i P_W(V_i)=P_W(\cap_i V_i)$?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
2
votes
1answer
249 views

A counter example of best approximation

Construct a point $f\in C[0,1]$ and a closed subspace $V\subset C[0,1]$ such that $f$ does not have a best approximation in $V$. Definition: $C[0,1]$ is the set of countinous function with the norm ...
3
votes
2answers
157 views

How find $\Gamma{\left(\frac{8}{9}\right)}=\frac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$

show that $$\Gamma{\left(\dfrac{8}{9}\right)}=\dfrac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$$ where Gamma function:http://en.wikipedia.org/wiki/Gamma_function I found this problem is ...
2
votes
0answers
73 views

Extending $W^{2,n}(\Omega)\cap C^2(\Omega)$-functions to $W^{2,n}(\Omega)$

I have the following theorem: Let $u\in C^2(\Omega)\cap W^{2,n}(\Omega)$ satisfy $Lu\geq f$ for $f\in L^n(\Omega)$. Then for any ball $B_{2R} = B_{2R}(y)\subset\Omega$ and $p>0$ where $u$ is ...
1
vote
1answer
173 views

Absolute Error Confusion

I don't understand the following concept. It was given by the textbook. The dimensions of a cylinder are measured to the nearest millimeter using a measuring tape. The circumference is measured to be ...
2
votes
1answer
50 views

What are the fields for Low-rank approximation and Principal component analysis

I would like to start with practical applications of: Principal component analysis Low-rank approximation matrices the problem that I found is that with some internet researches I can't really ...
4
votes
1answer
341 views

Any simplification or approximation of sqrt

As you know, we could write $(a+b+c)^2$ as $a^2+b^2+c^2+2ab+2ac+2bc$. what about $(a+b+c+\cdots)^{1/2}$? is there any expansion for $(a+b+c+\cdots)^{1/2}$? Any simplification or approximation of ...
1
vote
1answer
250 views

Thinking through a Taylor error bound for arcsine

In lecture, we went through solving a Taylor error bound for arcsine. I followed most of it except for where it talks about the odds divided by the evens divided by $2n+1$ gaining in accuracy by a ...
0
votes
3answers
108 views

Sum of sequence precision

I came up with this answer in stackoverflow. It states a question: ...
1
vote
1answer
56 views

Algorithm to approximate the closest nonlinear formula(funciton) for an arbitrary set of points?

I have a table which concists of XY points (so I have a set of points hehe), where X represents the Velocity and Y the Real World Speed. Those points are not linear. With two points it's easy to ...
0
votes
0answers
26 views

Dependence on Parameters of the Solution of a Non-linear Equation

I have the following equation for the delay in a queue\begin{align} d(f)=\frac{c(1-f)^2}{2(1-a)}+\frac{\lambda b}{2f(f-a)}\end{align} where $0\le f\le 1;\quad c,a=\lambda\tau, \ b=\tau^2$ or ...
1
vote
1answer
58 views

Show that the method is of order 3 if a =-5 and of order 2 if a is not equal to -5

I am studying ODEs and came across this exam question: I have the solution here also: I have been working on this exam question all day and have been stuck for hours. What I don't understand is ...
13
votes
2answers
2k views

Approximation to the Lambert W function

If: $$x = y + \log(y) -a$$ Then the solution for $y$ using the Lambert W function is: $$y(x) = W(e^{a+x})$$ In a paper I'm reading, I saw an approximation to this solution, due to "Borsch and ...
0
votes
1answer
305 views

truncation error - help

I am trying to understand the concept of local truncation error and came accross this in my lecture notes: what I don't understand here is where the term 'O' comes from and what it stands for in ...
1
vote
1answer
169 views

How to approximate unknown two-dimensional function?

I have a surface defined by values on a two-dimensional grid. I would like to find an approximate function which would give me a value for any arbitrary point within certain range of xs and ys. My ...
2
votes
1answer
61 views

What information is needed to solve or approximate this simple equation?

Suppose I have some vector $x = a + b = (a_0 + b_0, \dots, a_{n-1} + b_{n-1})$ of length $N$. Now, if only $x$ is known, what is the minimum amount of information that is needed to either solve or ...
4
votes
2answers
132 views

Approximation of $ \frac{n!}{(n-2x)!}(n-1)^{-2x} $

I would like to find an approximation when $ n \rightarrow\infty$ of $ \frac{n!}{(n-2x)!}(n-1)^{-2x} $. Using Stirling formula, I obtain $$e^{\frac{-4x^2+x}{n}}. $$ The result doesn't seem right! ...
3
votes
2answers
81 views

Prove approximation given by the physicist Max Born

In an old book about optics, I have found a nice approximation, that for large l one has: $$P_l(\cos(\theta)) \sim \sqrt{\frac{2}{l \pi \sin(\theta)}} \sin \left((l+\frac{1}{2}) \theta + ...
3
votes
1answer
679 views

approximating diagonal of inverse sum of low rank and diagonal matrices

I was wondering if there is any theorem or algorithm to approximate the diagonal elements of the inverse of sum of low rank symmetric positive semi-definite and non-negative diagonal matrix. Let me ...
0
votes
1answer
137 views

Deconvolution of a convolution product with $Ax\ /\ (x^2+l^2)^{3/2}$

This is not a homework, and I have no idea whether it could be one. It is only a request for help, as I do not have any experience using Fourier transform. The origin of the problem is from physics. ...
6
votes
1answer
210 views

How to calculate this complementary Bessel function?

I am trying to calculate this complementary Bessel function $$\Psi(a,b,\gamma)=\int_0^\infty\Phi({a\over \sqrt{u}}+b\sqrt{u}){u^{\gamma-1}e^{-u}\over \Gamma(\gamma)}du$$ where $\Phi$ is the standard ...
75
votes
14answers
10k views

Can the golden ratio accurately be expressed in terms of e and $\pi$

I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$ And so, I took it upon myself to try to find a way to express the golden ratio in terms of the ...
3
votes
2answers
2k views

Approximate inverse matrix

Does a method exist to calculate the approximate inverse matrix for a matrix whose elements are small? To be clearer, let's suppose the elements of the matrix $A$ to be function of a small parameter, ...
4
votes
1answer
105 views

$f$ is approximated uniformly on $R$ by $p_n(x)$, then $f$ is a polynomial

Suppose $p_n(x)$ is a sequence of polynomials which converge to a function $f$, uniformly on $\mathbb{R}$. Show that $f$ is a polynomial. If there were a uniform bound $M$ on the degree of $p_n(x)$, ...
1
vote
0answers
335 views

Approximation of an exponential sum

Consider the flowing exponential sum with $x,\,y \ge 0$ and $c_i,\, d_i$ is a real number for $i=1,..,N$ $$ E=\sum\limits_{i = 1}^N \exp \left(- \left( x - c_i \right)^2 - \left( y - d_i \right)^2 ...
1
vote
0answers
100 views

Turn ugly series into a nice approximation

I am currently struggeling with a series(actually two). The problem is, that I can do nothing with them, since this expression is so ugly. I would love to hear about any kind of approximations that ...
0
votes
1answer
91 views

piecewise linear approximation of $x^2$

Take the function $f(x) = x^2$. I read that it can be written as $$f(x) \approx k_1\cdotΔx_1 + k_2\cdotΔx_2 + \dots +k_n\cdot \Delta x_n,$$ where $k_n$ is the slope of the $n$-th piecewise linear ...
0
votes
1answer
51 views

Approximation in $L^2$ of functions with values in a convex set

Here is my problem : Let $K$ be a convex set of $\mathbb{R}^m$ ($m\in \mathbb{N}^*$), such that $0$ belongs to the interior of K, I want to approximate (in $L^2(\mathbb{R}^m,\mathbb{R}^m)$) a function ...
5
votes
1answer
81 views

Approximations of fixed points of tangent.

This question comes from an exam, years ago. Show that $f(x)=\tan x-x$, for every positive integer $n$, has exactly one root $x_n$ in the interval $(n\pi,n\pi+\pi/2)$. And show that ...
2
votes
1answer
439 views

Uniqueness of approximations like the Taylor polynomial

Given a function $f: \mathbb {R} ^n \to \mathbb {R} $, I am curious about the uniqueness of a $k$th-order approximation at $c \in \mathbb {R}^n $, i.e. a function $\phi(x)$ such that $$ \frac {f(c ...
1
vote
1answer
1k views

How to fit non-linear matlab data?

I'm working on a problem in scientific computing namely fitting data to this equation $c(z) = 4800 + p_1 + p_2 \cdot z/1000 + p_3 \cdot e^{ -p4 \cdot z/1000}$ The data is in a background question ...
30
votes
7answers
1k views

Pi Estimation using Integers

I ran across this problem in a high school math competition: "You must use the integers $1$ to $9$ and only addition, subtraction, multiplication, division, and exponentiation to approximate the ...