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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2
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1answer
243 views

Polynomial approximation of $\chi^2$ distribution pdf

The $\chi^2$ distribution PDF is $$f_{\chi^2}(x;k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2 - 1} \mathrm{e}^{-x/2} \mathbf{1}_{x \geq 0}$$ I am trying to find a polynomial approximation to this density ...
3
votes
0answers
233 views

Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
1
vote
1answer
47 views

Approximation of closest k-coloured points?

I'm a working software engineer faced with the following problem: I have a set of points on a 2d plane. Each point can have one of $k$ different colours. I wish to select one point of each colour that ...
4
votes
1answer
107 views

How can we approximate $\sum_{j=0}^n{\sum_{k=0}^j{c^j k^{1/2}}}$ by integrals?

"Difference Equations" by Walter G. Kelley and Allan C. Peterson, 2nd Edition, gives an example on how to approximate $\sum_{k=1}^n{k^{1/2}}$ using integrals and Bernoulli numbers. I'm interested in ...
0
votes
1answer
214 views

Solve $\displaystyle\int_{0}^a \left(3^{\frac{1}{3} \left( x^3 - 3x \right) }-1\right)\, dx = 0$ using elementary methods

A friend of mine came upon the following problem. Solve for $a$ the equation $\displaystyle\int_{0}^a \left(3^{\frac{1}{3} \left( x^3 - 3x \right)}- 1\right)\, dx = 0$. By typing the problem ...
2
votes
2answers
1k views

Help finding the absolute error with $n$th degree Taylor polynomials

I am trying to estimate the absolute error in approximating $\ln 1.09$ with the $3$rd-order Taylor polynomial centered at $0$. It's been a while since I've taken the Calculus and I'm afraid I need ...
1
vote
1answer
51 views

Euler's approximation of $m' = -\frac{m}{V}v$

Continuing my previous question from today - I should code a program that finds approximate value of the aforementioned differential equation. The full text of the assignment is: Water purifier with ...
2
votes
2answers
311 views

Triple Recursion Relation Coefficients

I am reading Atkinson's "An Introduction to Numerical Analysis" and I am trying to understand how a certain equation was reached. It is called the "Triple Recursion Relation" for an orthogonal family ...
1
vote
2answers
1k views

Approximating a sum of exponential distribution with a normal distribution

Here is the actual question: $A$ is random variable representing the lifespan of a component. It is an exponential law with an average of 10. Considering a system with $n$ components $A$, what is the ...
2
votes
1answer
278 views

Euler's method approximation

I'm supposed to write a program for approximating the value of function $y = y(x)$, which is given as: $$y' = \frac{1+y}{1 + x^2}$$ I also know that $y(0) = 0$. I should approximate the value for ...
3
votes
2answers
566 views

Limit Question with Bernstein Polynomial approximating $f(x) = x^2$

I am reading Atkinson's "An Introduction to Numerical Analysis". I am trying to verify a limit on page 198 involving the Bernstein polynomial approximating $f(x) = x^2$. The statement in the book is ...
1
vote
1answer
745 views

Approximate the integral $\int_0^1 \sin(x^2) dx$

I'd like to ask if someone can please give me a little push with this assignment: Approximate the value of the integral $\int_0^1 \sin(x^2) dx$ using only $\mathbb{N}$ numbers and basic operations ...
3
votes
1answer
290 views

Approximation with complex polynomials on $S^1$ - can it be done?

Can one uniformly approximate a function 'similar' to identity on $S^1$ with complex polynomials? I mean a function like: $f(z)=z \cdot (1+h \cdot \sin(m\cdot Arg(z)))$, for $|h| < 1,\ m \in ...
3
votes
2answers
407 views

Approximate a positive Sobolev function by positive smooth functions

Here is a problem that I have encountered in PDE book several times. But I have never seen a proof of it. I will be very grateful if someone could give me a proof. Question: Let $B$ be the unit ball ...
3
votes
1answer
827 views

With Euler's method for differential equations, is it possible to take the limit as $h \to 0$ and get an exact approximation?

I was recently watching a tutorial on Euler's method for approximating differential equations, and the whole time I was thinking "why can't you just take the limit of the step size $h$ as it goes to ...
1
vote
1answer
196 views

Midpoint Rule, Trapezoidal Rule, etc.: When the number of intervals increases by a factor of $q$, the approximation error decreases by $r(q) =\;$?

I'd like to look at this problem in terms of the definite integral $I = \int_0^5 e^{\sin\sqrt x}dx$, and in terms of the Midpoint Rule. (Then, hopefully, I'll be able to figure out the left-point ...
1
vote
0answers
143 views

Spline function: Parametric representation of a curve

I have this problem: I have a curve/figure on a sheet of graphpaper. Then I have to select points, read the x,y-values and label them $t_{0} = 1.0$ and $ t_{1}=2.0$ ect. I now have to obtain a table ...
1
vote
2answers
96 views

Low degree approximation of the polynomial extension of the logical-or function

Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$). Consider ...
17
votes
5answers
3k views

Approximation of $e^{-x}$

Is there a method to mentally evaluate $e^{-x}$ for $x>0$? Just to have an idea when computing probabilities or anything that is an exponential function of some parameters.
1
vote
1answer
385 views

How to select a sensible tolerance when making approximations in MATLAB

I'm approximating $\pi$ using a series in MATLAB. I can approximate to within a relative error of $3\times 10^{-10}$ of MATLAB's built-in value. How would I choose a sensible tolerance for my ...
3
votes
2answers
568 views

How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$

I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I ...
8
votes
4answers
213 views

How could we manually approximate $\sum_{i=1}^{50} i! = 3.1035 \times 10^{64}$?

How could we manually approximate $$\sum_{i=1}^{50} i!$$ to the value $ 3.1035 \times 10^{64}$? I faced this question in my aptitude test,there were four option given,I couldn't solve it during ...
10
votes
2answers
4k views

Approximating the logarithm of the binomial coefficient

We know that by using Stirling approximation: $\log n! \approx n \log n$ So how to approximate $\log {m \choose n}$?
9
votes
1answer
669 views

Tensor products of functions generate dense subspace?

Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
5
votes
2answers
2k views

How can I calculate non-integer exponents?

I can calculate the result of $x^y$ provided that $y \in\mathbb{N}, x \neq 0$ using a simple recursive function: $$ f(x,y) = \begin {cases} 1 & y = 0 \\ (x)f(x, y-1) & y > 0 \end ...
29
votes
1answer
696 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
647 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
675 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
5k 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
178 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
77 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 ...
0
votes
2answers
1k 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
406 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
592 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
463 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
180 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
208 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
169 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
527 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
164 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
267 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
91 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
159 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
146 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
605 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
160 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, ...