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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30
votes
1answer
722 views

A series problem by Knuth

I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem. Prove that $$\sum_{n=1}^\infty ...
0
votes
3answers
665 views

How to do this approximation?

Question: Of the following, which is the best approximation of $$\sqrt{1.5}(266)^{3/2}$$ $$(A)~1,000~~~~(B)~2,700~~~~(C)~3,200~~~~(D)~4,100~~~~(E)~5,300$$ I used $1.5\approx1.44=1.2^2$ and ...
5
votes
4answers
3k views

Simulate a double chance bet with two single bets

If you bet on the result of a soccer match, you usually have three possibilities. You bet 1 - if you think the home team will win X - if you think the match ends in a draw 2 - if you think the away ...
3
votes
4answers
712 views

Find approximation to $\sin(x)$

How to find the approximation to $\sin(1.58)$ ? By using the Newton's method $$x_{n+1} = x_{n} - \frac{f(x)}{f'(x)}$$ You always will get $0$. Using this method: $f(x+\Delta x) \approx f(x) + ...
12
votes
9answers
6k views

How to calculate $e^x$ with a standard calculator

Is there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...
1
vote
2answers
182 views

Approximation of a function with polynomials

I have a function of the form $$ f(k)=\frac{1}{a_1-a_{2}k^2e^{-(a_{3}-a_{4}k^2)}};\quad k=0...n$$ I approximated it with Taylor series expansion around $k=\frac{n}{2}$, but the results is not very ...
3
votes
1answer
79 views

Disc method of approximating volume

There are a couple things I'm unclear on regarding the disc method of approximating shape volume. Given $x=y^2$ and $x=4$, determine the approximate volume by revolving the shape around the line ...
1
vote
1answer
45 views

approximate the probability of fixed length string segments match

say, i have got 3x'a', 5x'b', 2x'c',4x'd', as char collection. 2 strings are formed, each consists of all the chars given. eg. string A = 'aaabbbbbccdddd' B='abcdabcdabbbdd' so both strings have ...
1
vote
2answers
2k views

Easy approximation of the incomplete beta function $\text{B}_x(a,b)$

I need to calculate $\text{B}_x(a,b)$ on the cheap, without too many coefficients and loops. For the complete $\text{B}(a,b)$, I can use $\Gamma(a)\Gamma(b)/\Gamma(a+b)$, and Stirling's approximation ...
4
votes
1answer
423 views

Approximating Lambert W for input below 0

As a small part of a much bigger project, I need to be able to approximate the numerical output of the Lambert W function. I have found decent approximations (good up to at least 4 decimal places), ...
7
votes
2answers
632 views

Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

To everyone: don't bother writing up another answer, i'm giving this bounty Antonio's answer. It just doesn't let me yet (24 hours delay). If you map the nth roots of unity $z$ with the function ...
6
votes
2answers
509 views

Neglecting higher order terms in expansion

Suppose we have a function $v$ of $x$ with a minimum at $x=0$. We have, for $x$ close to zero, $$v'(x) = v'(0) +xv''(0) +\frac{x^2}{2}v'''(0)+\cdots$$ Then as $v'(0)=0$ ...
1
vote
2answers
192 views

Decomposing a matrix into lower dimension sub-components

Given a matrix $M^{n \times n}$, I would like to decompose it into two smaller matrices $A^{n \times m}$ and $B^{m \times n}$, with $m < n$ so that the multiplication of both $AB = M'$ ...
3
votes
0answers
220 views

Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
2
votes
0answers
173 views

calculate the rate of change

I am trying to calculate the change frequency for a set of data. Each bit of data has the date-time it was created. I would like to say for a specific set of data the change frequency is hourly, ...
3
votes
2answers
541 views

Finding the real roots of a polynomial

Recent posts on polynomials have got me thinking. I want to find the real roots of a polynomial with real coefficients in one real variable $x$. I know I can use a Sturm Sequence to find the number ...
4
votes
1answer
205 views

Stuck on complex integral, approximate?

I've been stuck on a particular integral I encountered. I don't need an exact solution, I doubt it even exists. $$f(x)=\frac{e^{-i (r+R-k) x} \left(i-2 e^{i (r+R) x} r x-R x+e^{2 i r x} (R ...
2
votes
2answers
168 views

Generate a Monte Carlo sample from a PDF defined by a Fourier Series

I have a probability distribution (PDF) defined by a Fourier series.. actually it's a purely cosine series over a known range. The PDF quite smooth, so most of the power is in the low 5 or so ...
2
votes
1answer
268 views

Solving an integral with Laplace method

I'm trying to approximate the sum $$\sum_{\alpha=1}^{\mu} \Big(1-\frac{(\alpha(2 \mu-\alpha))^2 \gamma_1 \gamma_2}{2n^2 \mu^4}\Big)^{\frac{\lambda}{2}}$$ with an integral ...
1
vote
0answers
95 views

Find a special type of subgraph of certain edges that minimizes the cost of the edges in that subgraph

Say we have a graph $G=(V,E)$. Each edge $e$ in $E$ has a cost $c > 0$. Now we want to find a subgraph $G'=(V',E')$ of $G$ such that there is at least $k$ edges, $k>0$, and ...
1
vote
3answers
1k views

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
1
vote
1answer
160 views

Find fast exact value for numbers in the form $\sum_{k=min}^{Max}\frac{1}{k}$

I know I could start multiplying by all denominators and try to get the exact value that way but is there some smarter way or shortcut? Let's take simple example: $\displaystyle ...
3
votes
2answers
149 views

Numerical Approximation Involving Trig

I have a graphics problem that reduces to this: (Computer equation) alpha = arctan(X / ((Y / (Z * cos(alpha) - k)) * Z * cos(alpha))) (LaTeX) $$\alpha = ...
4
votes
2answers
701 views

Approximation vs. Interpolation

Sorry if this is a silly question, i'm just getting back into math after a long time away. My question is regarding approximation and interpolation. In which cases is it appropriate for one technique ...
1
vote
0answers
177 views

Terminology: Galerkin approximation

Dear all, I'm preparing a paper in which I'm trying to prove that my numerical approximation (Galerkin) of some mechanical problem indeed provides an approximation of the solution. In order to do so, ...
1
vote
0answers
132 views

Efficient way to recompute weights when shifting range of Legendre polynomial bases

I am storing a 2D (Cartesian) density function as a 2D patch with known X/Y limits and a set of 11 coefficients of the third order 2D Legendre polynomial basis functions over that patch. This works ...
3
votes
1answer
842 views

Stirling's Approximation and Binomial Random Variable

I am trying to follow equation (1.13) in MacKay's Information Theory textbook (http://www.inference.phy.cam.ac.uk/itprnn/book.pdf). It is: $$ \ln \binom{N}{r} = \ln \frac{N!}{(N-r)! r!} \approx (N-r) ...
3
votes
2answers
328 views

Multidimensional Interpolation within a polygon

Apologies in advance if I get terminologies wrong (not sure if "multidimensional interpolation" is the right term), I'm not really that great at maths, but here goes: Suppose we have two 2D points, ...
2
votes
2answers
125 views

Calculate other tangents which are related

I am using a small microcontroller, which has limited processing resources. I need to calculate the three tangents: ...
5
votes
3answers
470 views

$\lim_{n\to\infty} f(2^n)$ for some very slowly increasing function $f(n)$

I should be able to answer this myself, but feel insecure anyway. I want to know, whether a function f(n) is bounded if n goes to infinity (and if it's bounded, the limit). Heuristically it appears ...
0
votes
1answer
234 views

How would you determine the measurement error in the following example?

I'm at a bit of a loss as to how to determine the error in measurement in a project I'm working on. The project involves taking a picture of an object, and then using the image to determine the width ...
19
votes
2answers
580 views

Maximum of Polynomials in the Unit Circle

Let $z_{1},z_{2},\ldots,z_{n}$ be i.i.d random points in the unit circle ($|z_i|=1$) with uniform distribution of their angles. Consider the random polynomial $P(z)$ given by $$ ...
1
vote
2answers
1k views

Approximation in $L^2$ by piecewise constant functions

Dear all, I'd like to know if there is any general result on the approximation of $L^2$ functions by piecewise constant functions. More specifically, I'd like to know if the following approximability ...
1
vote
2answers
497 views

best real approximation to complex numbers

I have a system of equations and its answers are complex, but I want real numbers. Is there any way to find the best real approximation to a complex number?
1
vote
1answer
192 views

bound of Erlang distribution

Is there any known polynomial bound of the Erlang distribution? I'd like to say that, given $k$ and $\lambda$ with probability p the r.v. is going to be less than some value x.
0
votes
2answers
6k views

With a few data points can a generate a close equation to meet them?

I have 1x = -40, 2x = -41 , 3x = -54 and getting a few more. Can I generate a equation for a graph that gets close to this? I was trying to get Wolfram Alpha to ...
8
votes
4answers
4k views

Is there an analytic approximation to the minimum function?

I am looking for an analytic function that approximates the minimum function. i.e., $|f(x_1,x_2) - \min(x_1,x_2)| < \zeta$ for some $\zeta$ that may be related to $|x_1 - x_2|$. Or may be a series ...
5
votes
2answers
3k views

Approximations Involving Exponential Functions

I am reading a text and I am curious to know how certain approximations were reached. The first function approximations is: $$ 1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y}{x(1-p)}}) ...
1
vote
1answer
428 views

Having the eigenvalues how to find the eigenvectors?

This is a more practical question. In Pari/GP I have difficulties to use mateigen for some real matrices because of extreme long computation time and frequently "missing eigenspace" due to too low ...
1
vote
0answers
207 views

Gauss interpolation in 3D and friends

I was looking for approaches on how to adequately interpolate the values for a continuous 3D function for which I have the exact values in a 3D grid of equidistant points. I found that linear ...
6
votes
3answers
529 views

How is it that this shape can converge to what looks like a triangle but has a different perimeter?

I had this strange notion some time ago, and I recently wrote a blog post about it, as a mere curiosity. I don't really consider it a "serious" mathematical question; but out of interest, I wondered ...
3
votes
2answers
348 views

Asymptotic number of unlabeled graphs

A rather tight lower bound $c(n)$ of the number of unlabeled digraphs of order $n$ (loops allowed) seems to be $$c(n) = 2^{n^2}/n!$$ because there are $2^{n^2}$ labeled graphs, almost all of them ...
3
votes
1answer
633 views

Approximation for Lambert W function near zero

I am looking for a good approximation for the $W_0$ branch of the Lambert $W$ function. I am looking for values $0 < x < e$ only, so I expect something simpler than the general Taylor expansion. ...
0
votes
1answer
1k views

SVD for a complex matrix and approximation to a real matrix

Suppose M is a 20 by 3 complex matrix, and I'd like to SVD. For example, in Matlab, I can do easily with: [U, S, V] = svd(M); where U, S, and V are complex ...
2
votes
0answers
213 views

When is it valid to convert a function inside a probability integral to the indicator function?

I am faced with an approximation that replaces a probability density function with the indicator function and I am at a loss as to why this is valid. We want to model the lifetime $T$ of a website ...
9
votes
2answers
644 views

Complex Zeros of $z^2e^z-z$

Can anyone give me a hint on showing (in a relatively elegant way, as I know the answer from WolframAlpha), that the complex valued function $z^2e^z-z$ has at most 2 roots with norm less than 2? ...
67
votes
9answers
4k views

Find the average of $\sin^{100} (x)$ in 5 minutes?

I read this quote attributed to VI Arnold. "Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he ...
4
votes
3answers
535 views

How to find a Newton-like approximation for that function?

I want to find the complex fixpoint $t=b^t $ for real bases $b> \eta = \exp(\exp(-1))$. added remark: I'm aware that there is a solution using branches of the Lambert-W-function, but I've no ...
1
vote
0answers
465 views

Approximate linear density function for a normal distribution

I'm working on implementing Order Preserving Encryption for Numeric Data, and part of the algorithm includes approximating density of the distribution as a linear density function $f(p) = qp+r$ where ...
1
vote
2answers
1k views

How to find closest exponential approximation?

I have a bunch of data, and I'd like to find the closest exponential aproximation I can to fit the points. I'm guessing there's a (relatively) straightforward way to do this. For example, if I have ...