Tagged Questions

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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0answers
9 views

Angle measurment

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
53 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
10 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
33 views

Approximation of numbers [on hold]

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
29 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
10 views

Approximate the district function by an exponential function [on hold]

Define $f(N)= {[{(N-it)}/N]^{j+1}}$ where $\quad i,t,j \leq N$ .Approximate f by an exponential function.Also we know that N ,i,j belong to natural number.
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0answers
22 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) = ...
0
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1answer
31 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
39 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
16 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 $ ...
3
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0answers
25 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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0answers
41 views

A question about $f(x)\equiv C$

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 ...
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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
24 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
36 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
16 views

expectation approximation error

Let $X$ be a random variable with no mass taking values in $\mathbb{R}$, and $f:\mathbb{R}\mapsto\mathbb{R}$ be a "smooth" function. I want to approximate $\mathbb{E}[f(X)]$ with $\mathbb{E}[g(X)]$ ...
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0answers
12 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
34 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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0answers
27 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
20 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
33 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
26 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 ...
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2answers
44 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 $0\leq b,r \leq 1$. ...
2
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3answers
43 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
votes
2answers
49 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 ...
80
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3answers
5k 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
12 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
35 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 ...
1
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1answer
33 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
22 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
21 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 ...
1
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1answer
54 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
7 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
12 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
17 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 ...
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1answer
164 views

Function for which trapezoidal rule outperforms midpoint rule for every $n$

Is there a continuous elementary function $f:[0,1]\to [0,\infty)$ such that for every $n$ the trapezoidal approximation to $\int_{0}^{1}f(x)\,dx$ with $n$ trapezoids is strictly better than the ...
1
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1answer
38 views

Integration and Laplace-Stieltjes of a multiplied Weibull and Exponential distribution Function

I have a trouble for integrating a multiplied weibull and exponential distribution. The expression is as follows: $$ Y(t) = \int_0^t e^{-\lambda T}e^{-(T/\mu)^z}dT\,. $$ Then, I need to take ...
0
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1answer
39 views

The accuracy of approximating $ f(x) = x^{2/5}$ for $0.9 \le x \le 1.1$ using the cubic Taylor polynomial

For the equation $ f(x) = x^{2/5}$, $a=1$, $n=3$, $0.9 \le x \le 1.1$ I was able to approximate f by the following Taylor polynomial: $$ F_3(x) = 1 + \frac2 5 (x-1) - \frac3{25}(x-1)^2 + ...
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0answers
12 views

multivariate quadrature

Assume that $f:\mathbb{R}^n\ \to \mathbb{R} $. We want to approximate the integral, $\int_{I_d} f \, d\mu$. Let $U^{m_i}$ be a quadrature rule in $x_i$ in direction of $x = (x_1 , \dots , x_n)$, with ...
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1answer
59 views

Approximating a line segment with orthogonal components

I was thinking about this problem yesterday and was wondering if someone can provide some insight into it. Let's say we have two points in Euclidean space: $p_1$ at $\left(0, 0\right)$ and $p_2$ at ...
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0answers
22 views

How to do a linear approximation with several parts

if i have partial data set of $ \langle x,y\rangle\in \mathbb{R}$ for a given function $f$, and i want to approximate it by $n$ partial linear functions how would i calculate those linear functions, ...
0
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1answer
30 views

Trapezoidal rule over interpolation of higher dimensional vectors

According to a wikipedia and mathworld, the trapezoidal rule is: $$ \int_a^b f(x)\,dx \approx h\left[\frac{f(a) + f(b)}{2} \right], $$ where $h = (b-a)$. If you apply this rule to a function ...
1
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1answer
20 views

Rounding with significant digits

I'm dealing with significant digits right now, and recently I've been having a nagging question in my mind. When we have digits past the last significant digit in a quantity, do we round the last SD ...
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2answers
27 views

Approximation involving Gamma function: $\frac{\Gamma(j+d)}{\Gamma(j+1)\Gamma(d)}\approx(j-1)^{d-1}$

With $d\leq 1$ and $$ a_j=\frac{\Gamma(j+d)}{\Gamma(j+1)\Gamma(d)}=\frac{d(d+1)\ldots(d+j-1)}{j!},\quad j=0,1,2,\ldots $$ my professor wrote in class that $$ \sum_{j=N}^\infty ...
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3answers
33 views

Asymptotic behavior of $-gTt-gT^2e^{\frac{-t}{T}}$ for small $t$

I want to solve this using Taylor series expansion of $e^{f(x)}$ $$\begin{align}x=-gTt-gT^2e^{\frac{-t}{T}}+gT^2+x_0\end{align}$$ Show that for small values of t $(t\ll T)$, the equation for ...
1
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1answer
73 views

Subalgebra generated by selfadjoint operator $A_0\in\mathscr{L}(H,H)$

Let $\mathscr{L}(H,H)$ be the Banach algebra of bounded operators defined on a complex Hilbert space $H$ and let $B(A_0)$ be the subalgebra generated by the selfadjoint operator $A_0$, i.e. ...
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0answers
19 views

Approximating the max value of a function containing the half-sum of binomial series

I recently met a problem and I have been finding related materials about it. However it is still hard to handle. Precisely, I want to maximize a function related to $$S_n = \sum_{i=0}^{\lfloor ...
5
votes
2answers
135 views

Wikipedia wrong? Convergence of finite difference

Update: I have edited the Wikipedia page, so that the mistake no longer appears. On the Wikipedia article for "Finite difference" there is the claim Assuming that $f$ is continuously ...
0
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0answers
6 views

Has the degree to which a partial eigensystem of a large sparse matrix approximates the complete eigensystem been determined?

Does anyone know of any studies or results regarding the degree of approximation or the error in estimating the complete spectrum of a large sparse matrix by means of its first $n$ eigenvalues and ...