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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17 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 ...
2
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2answers
396 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
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0answers
16 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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1answer
138 views

Stone-Weierstrass theorem with $p(1/x)$

I am trying to prove that for a continuous $f\colon[1,\infty)\rightarrow\mathbb R$ and $f(x)\to a$ as $x\to\infty$ it could be approximated by $g(x)=p(1/x)$ where $p$ is a polynomial.
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0answers
31 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
24 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 ...
4
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1answer
880 views

Method for estimating the $n^{th}$ derivative?

When using numerical analysis, I often find that I am required to estimate a derivative (e.g. when using Newton Iteration for finding roots). To estimate the first derivative of a function $f(x)$ at ...
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2answers
41 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
41 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 ...
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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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3answers
284 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 ...
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2answers
48 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 ...
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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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0answers
14 views

What is the theory behind numerical integration such as adaptive quadrature and laplace approximation? [closed]

I am trying to understand the theory behind the numerical integration. How it is done and what it results? Thanks !!!
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2answers
28 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
37 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 ...
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1answer
495 views

How to calculate APR using Newton Raphson

I'm have a computer program to calculate apr using Newton Rhapson. I imagine most mathletes can code so i dont imagine the coding being an issue. The solution is based on this initial formula ...
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2answers
328 views

How to find upper bound on absolute error with composite trapezoid rule

Obtain an upper bound on the absolute error when we compute $\int_0^6 \sin x^2 \,\mathrm dx$ by means of the composite trapezoid rule using 101 equally spaced points. The formula I'm trying to use ...
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1answer
31 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 = ...
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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 $2$. ...
0
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1answer
16 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 ...
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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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4answers
4k views

How to simplify or calculate a formula with very big factorials

I'm facing a practical problem where I've calculated a formula that, with the help of some programming, can bring me to my final answer. However, the numbers involved are so big that it takes ages to ...
10
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1answer
162 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 ...
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0answers
6 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 ...
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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 ...
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1answer
1k views

Midpoint approximation over/under estimation

So left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and ...
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1answer
15 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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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 ...
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 + ...
1
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1answer
245 views

How to fit a 3-D parametric equation to datapoints

Consider that I have $3$ parametric equations as function of time and describe the motion of a body in space: $x = f(t)$ $y = g(t)$ $z = h(t)$ These curves are pretty simple and can be modeled ...
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0answers
11 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 ...
1
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1answer
58 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
20 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, ...
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1answer
72 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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3answers
254 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
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 ...
2
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2answers
47 views

Calculators using Taylor polynomials?

I've always heard that calculators (TI-84's and the like) use Taylor polynomials to approximate trigonometric/exponential/etc functions. Do any of you know this for a fact?
9
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1answer
97 views

Hardy's approximation for the cosine

I was reading about the Hardy's approximation for the cosine function (here and also in Mathworld): for 0<x<1 What I would like to know is, how was this approximation derived? What other uses ...
2
votes
2answers
218 views

How many significant figures are needed in base 2?

$x \in \mathbb{R}$ $2^{500}<x<2^{501} $ How many significant figures are needed in base 2, to know in high approximation whether $2^x$ is integer?
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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 ...
1
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2answers
370 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
1
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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 ...
1
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0answers
18 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 ...
23
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1answer
428 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
2
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2answers
12 views

Vertex Cover - Understanding the bounds

I was reading on wikipedia about the approximations of the Vertex Cover problem and saw that an approximation algorithm with an approximation factor of $\displaystyle 2 - \Theta \left( ...
5
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2answers
133 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 ...
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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 ...
3
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1answer
369 views

Differentiability of Moreau-Yosida approximation.

I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is ...