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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11 views

interpolation preserving boundedness property

I'm trying to construct interpolation for a function $m$ such that \begin{equation*} 0\leq m(x)\leq 1,\quad\forall x\in\Omega\subset \mathbb{R}^1. \end{equation*} I tried to use ...
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0answers
9 views

How to compute norm bound error in robust approximation

I am reading convex optimization, and I am little confused about the following two prolems in norm bound error of robust approximation. How to compute $\{\|\bar{A}X-b+Ux\| | \|U\|\le a\}$ ? For the ...
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1answer
42 views

$N$ is approximately linear in $d$ for $N^d=\frac12 e^{N}$

let us look at the function $N^d e^{-N}$, for each $d\in \mathbb{N}$. The graphs of the function for various values of $d$ show a striking phenomenon: the graph look parallel, and with a near-constant ...
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1answer
28 views

Can a discrete function converge to a continous function?

Let $f\in C^{\infty}[a,b]$, let also $X \subset [a,b] = \left\{x_0,\ldots,x_k \right\}, Y = \left\{ f(x_0),\ldots, f(x_k) \right\}$. I guess that if I let $k\rightarrow \infty$ then some how I should ...
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1answer
20 views

Numerical Analysis: Approximations — Discrete Average Value Theorem

I am asked to compute approximations to $f'(1)$ using $h=\frac{1}{16}$ for $f(x)=\sqrt{x+1}$ with the following formulas ...
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0answers
21 views

Weight Function in gaussian quadrature

My question is pretty simple, although I know of the properties that the weight function must follow , such as being well defined,positive,continuos and integrable on the interval . I do not know how ...
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0answers
29 views

Determinant of Hessian approximation

I have a question regarding formula in SURF article by Bay et al. Theory Given a point $p=(x,y)$ in an image $I$, the Hessian matrix $\mathcal{H}$ in $x$ at scale $\sigma$ is defined as follows $$ ...
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0answers
29 views

Approximating the Heat Equation

Let us assume that we want to approximate the solution of $\partial_t a = \partial_{xx} a$ which is subject to the Dirichlet boundary condition $a(-1,t) = a(1,t) = 0$, with $t \geq 0$, by considering ...
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1answer
21 views

How to solve this question using approximation theory?

I am asked to find the first three terms in the taylor series of the function $$ f(x)=(x-1)\ln x $$ around $x_0=0$. Then to find the maximum error in my approximation in the interval $[0.5,1.5]$. ...
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2answers
70 views

Approximation of a quotient that involves the Lambert function.

I would like to find an asymptotic upper bound for $$\frac{-\ln n}{W(- \ln^{-c}n)}$$ where $c$ is positive and $W$ is the Lambert function. More precisely, I want something which dominates this ...
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0answers
12 views

Upper bound on the remainder of a polynomial (not taylor)

There are many ways of approximating a function with a polynomial, $\widehat{f}(x)\approx f(x)$. One way is the taylor polynomial. A nice property that goes along with the taylor polynomial is an ...
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1answer
82 views

Approximate roots of nonlinear equation (non-integer polynomial)

In case of pulsating bubble arising from underwater explosion, bubble radius satisfies the following equation. $x^3\dot{x}^{2} + x^3 + \frac{k}{x^{3(\gamma-1)}} = 1$ The minimum and maximum bubble ...
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2answers
20 views

What does it mean by the approximation $\int_a^bf(x)dx\approx\sum_{i=0}^nA_if(x_i)$ is exact for all polynomials of degree up to $2n+1$?

There is these notes about Gaussian Quadrature and I am trying to understand what does the sentence "is exact for all polynomials of degree up to $2n+1$" actually mean. Gaussian Quadrature - General ...
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2answers
19 views

A couple of inequality / similarity that don't make sense to me.

I was reading thru the proof for a combinatorics problem, but there were a couple places in there that gave me pause. In particular, one part of the proof had the following: $${n \choose ...
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0answers
15 views

Alpha Max Plus Beta Min Calculation

I read about the Alpha Max Plus Beta Min algorithm described here. Here is a screenshot from the wikipedia page: I think understand what the algorithm is supposed to do. It makes an approximation ...
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0answers
28 views

How one can approximate irrational raised to irrational power?

How one can evaluate irrational number raised to irrational power? Like is there an easy way to prove that $-0.685<\pi^e-e^\pi<-0.675$?
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2answers
42 views

How do I find arc length using the trapezoid rule?

The question asks, "Use the trapezoid rule (when $n=8$) to approximate the arc length of the graph of $y=2x^3-2x+1$ from $A (0,1)$ to $B(2,13)$" I first graphed this out and found the points to have ...
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4answers
52 views

What is the name of the approximation $ \left(1-\frac{1}{x}\right)^n \approx e^{-n/x} $?

Which approximation allows for the following? $$ \left(1-\frac{1}{x}\right)^n \approx e^{-n/x} $$ Here both $x$ and $n$ are variables.
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0answers
16 views

Summation of shifted sigmoids

I recently came across this statement in a paper, and I was not able to figure out how to prove it. Any ideas would be appreciated. $$\sum_{i>0} \phi(x-i+0.5) \approx \log(1+e^x) $$ where ...
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0answers
7 views

Approximation for minimum and multiple product

Is there an approximation for the following formula: $min\{1,w+1-\left((1-x )\left(\prod\limits_i(1-y_i)(1-z_i)\right)\right)\}$ where $0 \lt w,x, y_i, z_i \le 1$.
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2answers
54 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
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2answers
34 views

Error of Stirling’s approximation for Binomial with central limit theorem

So the question asks: Let $X_n$~Bin(2n,1/2),use Stirling’s approximation for $n!$ to show $P [X_n = n]$~ $1/√(πn)$ as $n→ ∞$, and show the error in the estimate for $P [X_n ≤ n]$, given by the central ...
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0answers
9 views

Expand or approximate entropy of a two-term Gaussian Mixture

Is it possible to create some expansion to approximate this $h(a)$ for $a>0$ near $a\rightarrow0$? $$N(x,v)\equiv\frac{1}{\sqrt{2\pi v}}e^{-\frac{x^{2}}{2v}}$$ $$ ...
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0answers
21 views

Approximation of function in a given interval

I need help with the following approximation problem. Given the orthogonal system of functions in $[-\pi,\pi]$. How do I approximate the function $f(x)=-1$, if$ -\pi\leq x\leq 0 $ and $f(x)=1$, if $ ...
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1answer
32 views

find monotonic, increasing function going exactly throught set of points

This questions stems from a problem that I encountered while writing a program. This problem was identical to this one. What I did on the spot was described as a "trick" with binary search over ...
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0answers
20 views

Effect of location of nodes for interpolation

I've been doing some numerical experiments to see how the location of the interpolating nodes affects the performance of the interpolator. I am just curious about this because it seems like the ...
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0answers
13 views

Partial Fourier Series Error

I calculated the coefficient of the complex fourier series of a trapezoidal signal to be: $$c_n = \frac{\tau}{T} \frac{sin(0.5n\omega_0\tau) \cdot sin(0.5n\omega_0\tau_r)}{(0.5n\omega_0\tau) \cdot ...
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0answers
43 views

How to improve Euler-Maruyama discretisation with analytical moments?

I'm trying to improve Euler-Maruyama discretisation by adding to it the analytical moments. To try it I made a very simple example on the stochastic process $X(t) = W(t)^2$, where $W(t)$ is a standard ...
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1answer
21 views

Choosing the knots for a linear interpolation

I want to approximate a function through piecewise linear interpolation and try to understand how I could set the associated interval points optimally. Take a continuous function $f: X \rightarrow ...
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20 views

What's the most accurate way to estimate a percentile from multiple partial percentiles?

There exists 3 sets of numbers. I have the 99th percentile (p99) of each set and the cardinality of the set, but not the values in the set themselves. p99: 540, cardinality: 215 p99: 288, ...
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2answers
68 views

Upper bound on ratio of incomplete Gamma function and Gamma function $\frac{ \Gamma \left( x; a\right)}{\Gamma(x)}$

I am trying to find a tight upper bound the following expression \begin{align} \frac{ \Gamma \left( x; a\right)}{\Gamma(x)} \end{align} where $\Gamma \left( x; a\right)$ is incomplet Gamma function ...
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1answer
40 views

Least positive integer such that $\cos^k \left(\frac{\pi}{2k}\right)\geq \frac{99}{100}$

I wrote a program to figure this out and found $k=123$. Writing $f(k)$ as the function, I showed that $f$ is strictly increasing on the positive integers, and Wolfram agrees that ...
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2answers
72 views

Is there integral or series for $\sqrt{10}-\frac{4^4}{3^4}$ (to prove the inequality)?

Both of these numbers are bad approximations for $\pi$, but they turn out to be much closer together: $$\sqrt{10}-\frac{4^4}{3^4}=0.00178$$ Since there is a lot of questions here about integrals and ...
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1answer
43 views

How to find an approximation of power series to correct to within $10^{-7}$ as faster?

If I approximate a $\displaystyle\int_{0}^{0.5} \frac{1}{1+x^7} dx$ correct to within $10^{-7}$. How to find it without using a calculator? Now I can't. I usually calculate every single term, and ...
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0answers
12 views

The Wronskian of parabolic cylinder function and the plane wave

Suppose equation $$ \tag 1 \ddot{y} + (t^2\theta (t - t_{i}) + p^2)y(t) = 0, \quad t \in (t_{0}, \infty) $$ (here $\theta (t - t_{i})$ is the step function) with initial condition $$ \tag 2 y(t \to ...
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0answers
5 views

Determine $(g \circ f)'(1, 1)$ and an approximate value of $(g \circ f)(1,01; 1,01)$. Approximation by differentials and chain rule.

First time posting. Excuse me for the formatting or grammar. Question Let $f: R^2 \to R^3$ be a differential function, such that $f(1,1) = (3, 1, 2)$ and $f'(1,1): R^2 \to R^3$ is given by the ...
4
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1answer
64 views

The 9 most significant digits in Fibonacci series (Project Euler 104)

With regards to project Euler, problem 104: https://projecteuler.net/problem=104 The essence of the question here is how to keep track of the 9 most significant digits of a Fibonacci series (Keeping ...
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1answer
31 views

approximation of binomial coefficient by exponentiation

Show that inequality holds for every n, k. $$\ { n \choose k} \leq \frac{n^k}{k!}. ( \, 1-\frac {k}{2n} ) \,^{k-1} $$ I used Stirling's formula but I stucked, which is $$\ { n \choose k} = ...
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2answers
32 views

How to arrive at this approximation? [closed]

I encountered an equation: $$\frac{1}{(ja + \delta{z_{n}} - \delta{z_{n-j}})^2} + \frac{1}{(-ja + \delta{z_{n}} - \delta{z_{n+j}})^2}$$ can someone tell me how it approximates to: ...
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1answer
28 views

Stuck in using Stirling's approximation to show and justify an approximation of the number of permutations with and without ordering

This is a problem from my applied mathematics class where we are currently working on using Stirling's approximation which is: $ n! \sim (\frac{n}{e})^n \sqrt{2 \pi n} $ and the context of this ...
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3answers
53 views

Rational approximation of square roots

I'm trying to find the best way to solve for rational approximations of the square root of a number, given some pretty serious constraints on the operations I can use to calculate it. My criteria for ...
2
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0answers
55 views

Does the continued fraction for $e^{3/n}$ have a pattern?

Wikipedia has patterns for the simple continued fractions $e^{1/n},e^{2/n}$, which made me wonder whether there is one known for $e^{3/n}?$ (by pattern, I mean that the partial quotients $a_n$ can ...
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1answer
22 views

propagation of error from product of Taylor Series

Say I have two functions $f(x)$ and $g(x)$, both of which I will be approximating with Taylor series $T_f(x)$ and $T_g(x)$ respectively. Lets say $f(x)$ is order $O(x^{n_1})$ and $T_f(x)$ has error of ...
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0answers
19 views

Approximating number of partitions of $n$, denoted $p(n)$, by $p(n)\ge e^{c\sqrt n}$

I was to show that $p(n)$, the number of partition of a positive integer $n$, satisfy: $p(n)\ge \max_{1\le k\le n}{{n-1\choose k-1}\over k!}$, which was obvious because every unordered collection of ...
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1answer
38 views

Linear Approximation for functions with first derivative as $0$

Linear approximation around a point through Taylor series requires the first order derivative to be non-zero unless you want to get only the value at that point. However this is only true when you are ...
2
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2answers
42 views

Approximating $f(1)$ given $f(0)=2$ and $\frac{1}{2}x^2≤f'(x)≤x^2$?

This is an exercise using the mean value theorem: Approximate $f(1)$ given $f(0)=2$ and $\frac{1}{2}x^2≤f'(x)≤x^2$ for $x≥0$. I've found (using MVT): $$\frac{1}{2}x^2+2≤f(1)≤x^2+2$$ and I can ...
3
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2answers
25 views

Minimal Subset that sums up to

Let $X \subsetneq \mathbb{N}$ be a finite set, and $c \in \mathbb{N}$ we are looking for a subset $$ Y \subseteq X $$ such that $\sum_{y \in Y} y \geq c$. Assuming a subset that satisfies the ...
2
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3answers
109 views

Approximating $\exp(\sin(x))$ with polynomials

I want to find explicit formula for the sequence $f_n$ of polynomials which uniformly convergent to $\exp(\sin(x))$ on $[0,2014]$. Taylor's expansion is terrible for this function so i think that ...
12
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1answer
224 views

Where am I violating the rules?

Being fascinated by the approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ proposed, more than 1400 years ago by Mahabhaskariya of Bhaskara I (a ...
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0answers
28 views

Help with basic arithmetic involving Big Oh

I'm trying to determine the resulting "Big Oh" when arithmetic operators are applied between two different functions, but I'm a bit unsure after looking at even the basic operators shown on wikipedia ...