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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15 views

Fitted function - Which is better to use?

So I have some data for program running time, that follows a power law relation aN^b. I log-log plotted the data and saw that it became a straight line, so I calculated the slope of this line to get ...
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1answer
46 views

How to write approximations of a sequence $x_n = {1/3^n}$

Write three approximations of the sequence ${x_n} = {1/ 3^n}$, using the following scheme - $P_0= 1, P_1 = 0.33332$ and $P_n = (6/5)P_{n-1} - (1/5)P_{n-2}$ for $n = 2, 3,\dots$ Further, make a ...
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0answers
65 views

Does a greedy task selection algorithm find a c-approximate solution?

A scheduling problem can be stated as: Given a set $\{(s_i,f_i)\}_{1\le i\le n}\}$ of tasks identified by their start and end times, choose the maximum size subset of non-overlapping tasks. ...
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0answers
17 views

Finding when $(x^3 + …)/(x^4 + …)$ reaches $10^{-n}$

A problem I’m working on throws up equations like: $$ \frac{1}{4k + 9} + 3\frac{k+1}{(4k + 9)^2} + \frac{2k+1}{(4k + 9)^3} + \frac{k}{(4k + 9)^4} $$ I need to know the value of $k$ for which this ...
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1answer
20 views

Error evalution for Newton-Raphson method

I'm supposed to approximate a solution of an equation using the Newton-Raphson method, knowing one real solution to this , namely $x \approx 0.61803$. $$x^4 + 3x - 2 = 0 $$ Therefore I start by ...
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0answers
17 views

How to interpolate multidimensional functions?

I'm learning about interpolation and I wanted to ask if there's a "good" method to interpolate multidimensional functions (when the dimension can be even a few thousands)? Is there a theoretic limit ...
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34 views

Taking a stationary phase approximation of a multidimensional integral

I'm looking for a way to take a stationary phase approximation of an integral of the following form: $$ \int_{-\infty}^\infty d\vec{q} \exp\left(2 \pi i N \left(S(q_{n+1}, \vec{q}, q_1) - ...
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0answers
40 views

How can we evaluate the troublesome integral?

How can we evalutate the integral $$ I=\int_0^{\infty}\Big[\arctan\big(\frac{\pi}{\ln 5-5-x-\ln x}\big)-\pi\Big]\frac{dx}{x+5} $$ analytically? I tried to evaluate it by parametrical integral, i.e. $$ ...
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1answer
29 views

Question on probability and approximation

Okay I think you are all familiar to YouTube videos and some facts are: to comment, like and dislike on a video you need a Google account. when someone views the video the view count of the video ...
0
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1answer
24 views

Binomial cumulative probability

Here is the question I need help on : Let $X$ be a binomial random variable with p = 0.5 and n = 100. Give $P(X \geq 60)$ rounded to two decimal places without using a calculator (by using ...
2
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0answers
94 views

Lower bound on a polynomial far from its zeros

Let $p(x) = \sum_{i=0}^{d}c_{i}x^{i} \in \mathbb{R}[x]$ and assume that all its zeros are real and in $[-1,1]$. I am interested in lower bounding the value of $|p(a)|$ in case $a \in [-1,1]$ is far ...
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0answers
6 views

Root finding: Distance b/w 2 objects = 0. (and other examples of finding roots?)

Can someone explain general uses of finding roots? I understand you can find roots to help manually graph a function, but there's gotta be more. For example, in video games, I recall something about ...
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1answer
44 views

Approximating a power of a root of unity to within $\delta$

I have an estimate of $\omega$, a root of unity. I'm really wondering how small the error (in the estimate), which I give as $\epsilon$, has to be, so that when I take my estimate of omega to the ...
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0answers
41 views

Univariate Polynomial Approximation

I'm working on an algorithm in which I need to approximate the behavior of a polynomial by computing its roots to some $\epsilon$ precision. The problem can be defined as follows: Let $f(x) = x^n + ...
2
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1answer
50 views

How to approximate this nasty exponential function with an integral?

What is the best way to approximate a function of the following form, $$ \text{exp}\left(-\int_{y}^{+\infty} f(x)\ dx \right)$$ Any approximation to this, does taylor series work? The reason I am ...
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2answers
156 views

A Mathematical Coincidence, or more?

According to the paper "Ten Problems in Experimental Mathematics", $$\int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)dx \quad = \quad \frac{\pi}{8}\color{blue}{-7.407 \times ...
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1answer
28 views

Angle measurement

Assume I want to compute one of the angles of a right triangle doing $n$ measurements of the sides with a ruler. In order to increase the precision I make several measurements. After that I compute ...
2
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2answers
71 views

Methods for calculating $\pi$ that use the sphere?

The area of the unit circle is $\pi$ and its circumference is $2\pi$. Consequently, many elementary methods for calculating and approximating $\pi$ use a geometric approach on the circle, such as ...
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0answers
18 views

MATLAB implementation Spline Fitting

Check the attached problem please. I am a beginner in spline fitting and have a few questions: 1) How to find the coefficients c[n]. Is it by DTFT? 2) I understand how to find the derivative but ...
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2answers
37 views

Approximation of numbers [closed]

How could we approximate an irrational number by rationals?? Could you give me some hints?? I don`t have any idea how we could approximate them by rationals...
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1answer
32 views

Finding a sequences so the image approximates integers

$x$ is fixed in $[0,1]$, $a_n,b_n,c_n$ are integers and not all of them are $0$ $y(a_n,b_n,c_n) = a_n x^3 + b_n x^2 + c_n x = Y_n$ Find an algorithm to go from $(a_n,b_n,c_n)$ to ...
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0answers
31 views

WKB approximation for multiple turning points

I'm working on a numerical program which approximates the eigenvalues of a Schrödinger equation by making use of the WKB approximation formulas. For example, if the Schrödinger equation is $$ y''(x) = ...
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1answer
46 views

Can we prove the Riemann-Lebesgue lemma by using the Weierstrass approximation theorem?

I'd like to prove the following version of the Riemann-Lebesgue lemma: Let $f: [0,1] \to \mathbb R$ be continuous. Then $$\int_0^1 f(x)\sin(nx) \, dx \xrightarrow{n \to \infty} 0$$ It's quite ...
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0answers
41 views

Are there ever cases where it's easy to get coefficients for the series representation for an integrand, but hard to approximate the integral?

WHY I'M ASKING THIS I'm working on a faster way to approximate integrals of series. So I'd like to know if this could be useful. THE QUESTION If we suppose that we can get a formula for the ...
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0answers
18 views

integration with pade approximant

given the function $$ \int _{0}^{\infty}\sqrt{x}exp(-x) $$ can we use Pade approximants to integrate this i mean let bhe te rational approxsiamtions of $ \sqrt{x}= \frac{A(x)}{B(x)} $ and $ ...
6
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2answers
146 views

Approximation of minimum among many binomials

We choose $k$ numbers independently from the binomial distribution $B(n,1/2)$, where we can think of $n$ as large. What is the expectation of the minimum of the $k$ numbers? Is there a good way to ...
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1answer
54 views

If a continuous function on $[0,\pi]$ integrates to zero against cosines, it is identically constant

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
0
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2answers
16 views

Use of small approximations to get the following formula

I cannot see how the following formula has been found $\displaystyle \cos(\theta + \frac{v\epsilon^2 \Omega}{L}+O(\epsilon^4))-\cos(\theta) = -\epsilon^2\frac{\Omega v}{L}\sin(\theta)+O(\epsilon^4)$ ...
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1answer
29 views

Approximating Pirrational Numbers

A while back I wrote this question on PPCG.SE about the numbers I termed Pirrational numbers. They are defined as follows: Let $P_i$ be the $i$th Pirrational number for some $i \in \mathbb{N}_0$ ...
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2answers
40 views

When $a\ll b$, how to approximate $f = \int_0^a \sqrt{b^2+x^2}/\sqrt{a^2-x^2} \, \, dx$?

Suppose $a\ll b$. How do I then approximate $$\int_0^a \frac{\sqrt{b^2+x^2}}{\sqrt{a^2-x^2}}dx$$ ? I think that maybe Taylor approximation may help, but I am not sure how to proceed. My physics ...
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0answers
14 views

How do they calculate players' chances of winning in 9-handed Hold 'em or Omaha poker with hidden information?

So Omaha poker is a card game where each player is dealt 4 private cards, and then 5 community cards are dealt in the middle, and each player makes the best possible 5 card poker hand by using 2 cards ...
0
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1answer
46 views

Linear Approximation with log

The problem: Find the linear approximation of the function $$ f(x,y) = \ln(e+x+y) $$ at point $(0, 0)$. Use it to approximate the value of the function at $(0.1, 0.2)$ What I have so far: I found ...
0
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1answer
64 views

Calculating originally arc approximated by cubic bezier curve

I have an cubic bezier curve, which is representing an arc by an approximation. The approximation was calculated with the kappa constant: $$ \\k = \frac43*(\sqrt{2}-1) $$ This means, that the ...
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0answers
23 views

Approximate Solution to Backwards Recurrence of Dynamic Game

Suppose we keep tossing a fair dice until we reach some cumulative sum greater than or equal to $N$. Then, let $S_k$ be the expected value of the final sum, given that the current sum is $k$. We have ...
2
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0answers
35 views

How do we know that the first few digits of an approximation for $\pi$ are correct?

For Gregory–Leibniz series, wikipedia has - "after 500,000 terms, it produces only five correct decimal digits of π.". But how do you know that those five decimal values are correct when you reach ...
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0answers
29 views

Bounds on constant for Stirling approximation

Stirling's approximation says that $$n!\sim\sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n.$$ What is known about constants $c_1$ and $c_2$ such that $$c_1\sqrt{n}\left(\dfrac{n}{e}\right)^n\le n!\le ...
2
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2answers
50 views

Estimating the behavior for large $n$

I want to find how these coefficients increase/decrease as $n$ increases: $$ C_n = \frac{1}{n!} \left[(n+\alpha)^{n-\alpha-\frac{1}{2}}\right]$$ with $\alpha=\frac{1}{br-1}$ and $0\leq b,r \leq 1$. ...
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3answers
53 views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
3
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2answers
57 views

Correcting Error in the Leibniz $\pi$ formula… why does it work?

You are probably familiar with the Leibniz $\pi$ formula: $$ 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \frac{\pi}{4} $$ For a CS homework assignment I had to write a ...
83
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3answers
6k views

A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
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0answers
18 views

Minmax approximation

Let $f(x)=a_nx^n+....+a_1x+a_0, a_n\neq0.$Find the minmax approximation to $f(x)$ on $[-1,1] $by a polynomial of degree$\leq n-1 ,$and also find the error $\rho_{n-1}(f).$ This problem is from one of ...
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2answers
40 views

Evaluate the following function using as many significant figures as required to get a final result of 4 digits accuracy

I need to evaluate $$ f_5(0.2) = 5! \left[ e^{0.2} - \left( 1 + (0.2) +\frac{(0.2)^2}{2!}+\frac{(0.2)^3}{3!} + \frac{(0.2)^4}{4!} +\frac{(0.2)^5}{5!} \right) \right] $$ using as many as required ...
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1answer
44 views

Linear approximation by rational number to square root of 15

Use linear approximation of the function $f$ given by $f(x) =\sqrt{16-x}$ at the point x = 0 to find an approximation of $\sqrt{15}$ by a rational number (i.e. fraction). What I have so far: $$L = ...
3
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1answer
24 views

Are the Taylor polynomials of a function the results of a minimization problem?

Here is an example to better explain my question. Consider the function $f(x) = \cos(x)$. I want to approximate it in the set $[-\pi; \pi]$ using a polynomial $g(x) = a + bx + cx^2$ of order $2$. ...
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1answer
26 views

Approximation and Monte Carlo simulation.

I am a bit up over my head here, I will present an argument and then I hope you guys will say if my reasoning is correct or what should be changed, ultimately I am hoping to say something qualified ...
2
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1answer
58 views

Approximating a Gaussian integral

I have been struggling with an approximation to the following integral \begin{equation} \text{p.v.}\int_{-\infty}^{\infty} {e^{-s^2/2v} \over (e^{-2s}- q a)^2} {ds \over \sqrt{2 \pi v}} \end{equation} ...
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0answers
17 views

How to find a spirograph that approximates another curve?

A spirograph is a curve of the form $$\vec r(t)= \sum_n (a_n \cos b_n t\ i + a_n \sin b_n t\ j)$$. This class of curves includes things like epi/hypocycloids and trochoids, but also a number of other ...
1
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1answer
30 views

solve for a constant value?

Can we solve for $g$ when $\varepsilon$ is small? $\newcommand{\sinc}{\operatorname{sinc}}$ $$3\sinc\left(-1+ \frac\varepsilon T \right)-3\sinc\left(1+\frac\varepsilon ...
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0answers
13 views

Decomposing a matrix into a finite set of elements

I'm looking to approximate real, symmetric matrix $\mathbf{A}$ of size $N$ with a unique restriction. First, it is known that the matrix has zero mean $\left < A_{ij} \right>=0$. Next, it is ...
0
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1answer
18 views

Approximation of $T10$ for integral $\int_0^1\sin(x^2) dx$ Trapezoid Approximation

I got through most of the work with finding the approximation of $T10$ which comes out to be $=.3111708111$, I also found the error of $Et10$ when I plugged into the formula of $K(b-a)^3/12(n)^2$ . My ...