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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6
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10answers
1k views

How can I find the value of $\ln( |x|)$ without using the calculator?

I want to know if there is a way to find for example $\ln(2)$, without using the calculator ? Thanks
0
votes
1answer
90 views

Can we get an analytical solution to this equation involving the Lambert W function?

Can we get an analytical solution to the variable $t$: $$H\left(1+W\left(A\exp\left(Bt\right)\right)\right)=1+W\left(X\exp\left(Yt+Z\right)\right)$$ $W(x)$ is the Lambert W function.$A$ $B$ $X$ $Y$ ...
1
vote
2answers
133 views

How to approximate this series?

How to approximate this series, non-numerically? $ S_n = \sum_{n=1}^{50} \sqrt{n}$
0
votes
1answer
163 views

Which form of Euler-Maclaurin formula to use?

This question may be rather elementary, but I am sort of confused about various forms of the Euler-Maclaurin summation formula and their use. For instance, let us suppose that we want to approximate ...
11
votes
2answers
15k views

Explanation and Proof of the fourth order Runge-Kutta method

Runge-Kutte 4th order method is a numerical technique used to solve ordinary differential equation of the form $dy/dx=f(x,y), y(0)=y_0$ It gives $y_{i+1}$ in the form $y_{i+1} = ...
0
votes
0answers
50 views

Coefficient $a_k$ of generating function

Given a generating function $F(z)$, am I right to say that the coefficient $a_k$ of $[z^n]$ is computed by $\frac{F^{(k)}(0)}{k!}$ $(1)$. Since we have the definition of $F(z)$ is: $F(z) = \sum_{i ...
1
vote
3answers
67 views

Estimating $\sum_{k=1}^N a_kb_k$ given the means $\bar a_k,\bar b_k$ and determining the error

I need to calculate the following expression: $$\sum_{k=1}^N a_k b_k$$ I know the average values of $a_k$ , defined as $\overline {a_k} = {\sum_{k=1}^N a_k \over N } $ and $b_k$ , defined as ...
2
votes
1answer
35 views

Approximate sector between two lines?

I need to approximate a red figure. I know coordinates of three points (little transparent circles). I also know a count of segments I need to divide this figure. The angle may be from 0 to Pi and ...
0
votes
1answer
194 views

Cubic splines on a grid

I trying to work out how to interpolate on a grid with cubic splines. Let the point at which I'm trying to interpolate be at {xp,yp}. At the moment I am fitting splines across the rows and then ...
0
votes
0answers
190 views

Approximations other than taylor series and pade approximation

I have a function which has the following form: $$ f(x)=K_1 \coth (Q_1 Q_2 \sqrt{x})^2 + \frac{1}{x}\left[K_2 + K_3 \coth(Q_1 Q_3\sqrt{x})\sqrt{x}\right]$$ and I want to find $x$ for $f(x)=1$. I'm ...
3
votes
1answer
412 views

Upperbound approximation to the sum of Euler's totient function

I am currently working on a solution to a problem related to the density of finite coprime sets. I believe that I have found a solution to this problem - though it can only be expressed in terms of ...
0
votes
1answer
258 views

Proximal functions

I am a little bit new to proximal functions and I am currently stuck with the following problems How would I derive the prox function for the regularizer (h(x) function) : $\alpha\sum_{k+} $ and for ...
1
vote
1answer
322 views

Proximal operator

What is a proximal operator and how would one derive it in general for a function? In particular, if I had a function: $ f(x) = x^TQx + b^Tx + c $ How would I get the proximal operator for this if Q ...
0
votes
1answer
1k views

Normal approximation to the log-normal distribution

Intuitively, it seems that a lognormal distribution with a tiny $\sigma/\mu$ ratio might look quite a bit like a normal distribution. Can this be formalized in any way (e.g., by stating upper bounds ...
5
votes
4answers
1k views

Can I approximate sine and cosine without derivatives?

Assuming I don't know derivatives (and Taylor series) can I manage to approximate sine and cosine of a generic given (rational) angle in radians?
3
votes
0answers
98 views

Representation for a function that, when added/multiplied/composed with another function of the same form, yields a new function of the same form

I apologize for the possibly unclear wording of the title. I'm not well versed in math terminology. I'm after a concrete representation of a function, eg $y(x) = Ax^p$ (where $A$ and $p$ are ...
0
votes
1answer
175 views

How to calculate error size from division of two ratios

How one would calculate an error from division of two rations? I am given 1/y and x/y as their decimal representation (numbers ...
4
votes
0answers
161 views

Approximate the integral $\int_0^\pi \sin(x^3)\mathrm{d}x$ with a standard pocket calculator

I came over the following integral $$ \int_0^\pi \sin(x^3) \mathrm{d}x $$ when a friend of mine tried to approximate it. The most obvious way is to use taylors formula, and then turn the integral ...
1
vote
0answers
23 views

How $(P_{r+1}-P_r)$ is maximized?

This is from DeGroot's "PROBABILITY and STATISTICS"(Second edition)(Cf. pages 87 to 92).I am rewriting the relevant stuff. Let $r$ be a positive integer and $r\leq n$. ...
0
votes
1answer
75 views

Minimum moves to transform a list to another?

Given two list of n positive elements. We are allowed to perform only one transformation which is to increment each element of the list except one. What are the minimum number of transformation ...
4
votes
2answers
973 views

approximating a maximum function by a differentiable function

Is it possible to approximate the max{x,y} by a differentiable function? f(x,y)=max{x,y} x,y>0
3
votes
3answers
190 views

How is $\frac{\big(\frac{3}{2}\big)^{99}-1}{\big(\frac{3}{2}\big)^{100}-1}\approx\frac{1}{\big(\frac{3}{2}\big)}$

I read somewhere that $$\frac{\big(\frac{3}{2}\big)^{99}-1}{\big(\frac{3}{2}\big)^{100}-1}\approx\frac{1}{\big(\frac{3}{2}\big)}$$I don't know how to have it. Please let me know how this is ...
2
votes
1answer
132 views

About continued fractions as best rational approximations

I'm reading this notes about continued fractions: http://www.math.jacobs-university.de/timorin/PM/continued_fractions.pdf I had no problems understanding everything there, except one thing that has ...
0
votes
1answer
86 views

Solving a set of linear equations for variables with non-constant values

Consider the following set of equations: $$ax+by=g$$ $$cx+dy=h$$ $$ex+fy=i$$ which could be represented as: $$ \begin{bmatrix} a & b\\ c & d\\ e & f\\ \end{bmatrix} \begin{bmatrix} x\\ y\\ ...
30
votes
2answers
769 views

On Shanks' quartic approximation $\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)$

In Mathworld's "Pi Approximations", (line 58), Weisstein mentions one by the mathematician Daniel Shanks that differs by a mere $10^{-82}$, $$\pi \approx ...
2
votes
1answer
138 views

Brownian bridge distribution: $\sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}}, 0 < a < b <1 $

If $W^0$ is a tied-down Wiener process (Brownian bridge) on the range $(0, 1)$, what is the distribution of \begin{equation} \sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}} \end{equation} ...
2
votes
1answer
183 views

Projectile Trajectory with Air Drag

Given the following equations and values, Find an initial theta value to maximize horizontal range with air drag. $f(x)=\tan(\theta)*x-16+(x/(200*\cos(\theta))^2$: height with no air drag ...
1
vote
0answers
123 views

How many solutions to $x^3+y^3 = z^3\pm 1$ for $z$ less than a bound?

Assume $a,b,c, N$ as positive integers, let primitive be $\gcd(a,b,c) = 1$ and, $$a^2+b^2 = c^2\tag{1}$$ Supposing you want to know how many solutions there are with $c$ less than a bound $N$. ...
3
votes
1answer
79 views

Why does this pattern fail (sometimes) for the continued fraction convergents of $\sqrt{2}$?

This is connected to my post on the continued fraction convergents of pi. Motivated by Calvin Lin's comment whether a similar pattern exists for other constants, I checked $\sqrt{2}$. Its convergents ...
11
votes
1answer
768 views

A strange “pattern” in the continued fraction convergents of pi?

From the simple continued fraction of $\pi$, one gets the convergents, $$p_n = \frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102}, \frac{104348}{33215}, ...
0
votes
0answers
65 views

Asymptotic for Taxicab number.

The taxicab numbers are sums of 2 cubes in more than 1 way. First few are - 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, ...
0
votes
2answers
269 views

Approximation using Euler's method.

Consider the initial value problem $$\dfrac{dy}{dx} = y,y(0) =1$$ Approximate $y(1)$ using Euler's method with a step size of $\dfrac{1}{n}$, where $n$ is an arbitrary natural number. Use this ...
0
votes
1answer
68 views

How to solve this using using stirling approx?

I have a relation $log(n!)=\Theta(n\log n)$ . And i really don't know how to reduce this using using stirling approx. ? But do know and tried with few logrithmic property like " Even if its ...
0
votes
1answer
71 views

Order of magnitude of a variable.

What will be the order of magnitude of a variable whose value varies between 0 and 1? And why?
11
votes
1answer
172 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
194 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
586 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
214 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
41 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
106 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
72 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
50 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
65 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 ...