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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1answer
17 views

Second Order Approximation for a Polynomial

if I have an expression: $L=\frac{12a^3d^3-4wa^3d^2+16a^2d^2-4wa^2d+6ad+1}{12a^3d^3-4wa^3d^2-4a^2wd+16a^2d^2+7ad-aw+1}$ what is the second order approximation in $\frac{d}{w}$? I know that ...
0
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0answers
16 views

Approximating an integral with a change of integral

(I have previously found out $x_1 = -\frac{1}{\sqrt{3}}$ and $x_2 = \frac{1}{\sqrt{3}}$ ) Approximate an integral using the 2-point rule, with an appropriate change of integral, to approximate ...
0
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0answers
21 views

Trapezoidal Rule Error Bounding with the Absolute value of x

So, I am attempting to find a large enough n to allow for the Error from evauluating the Trapezoidal Rule to be less than 1/100. I know the equation is K(a-b)^3/12n^2 > 1/100 however I am running into ...
2
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1answer
24 views

Estimating accuracy of Taylor series approximations with 2 bounds

I have a question from a previous exam as such: Use Taylor's Inequality to estimate the accuracy of the approximation $f(x) \approx T_{3}(x)$ when $0.8 \leq x \leq 1.2$. I computed from an ...
3
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5answers
129 views

Approximation of $\sqrt{ x + y } - \sqrt{ x - y }$

I've been struggling to try and find a way to approximate the function: $\sqrt{ x + y } - \sqrt{ x - y }$ I should mention that $y$ is positive and a small number, so that $0<y<<1$. What ...
2
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2answers
46 views

function approximation using series

I was trying to approximate a function by another one using some kind of a series. Let $$ f(x) = m\cdot \left(1-\sqrt{\frac x2\biggm/\left(1+\frac x2\right)}\right) $$ I'm trying to approximate ...
0
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1answer
17 views

For small x, Taylor Series to determine constants n and C in the approximation

cos(x) - e^(-(x^2)/2) I have tried writing the taylor expansion for cosine and e but do not know how to combine them in one. In addition to this, i do not know how to approach the problem whilst ...
4
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4answers
178 views

How to compute a lot of digits of $\sqrt{2}$ manually and quickly?

After having read the answers to calculating $\pi$ manually, I realised that the two fast methods (Ramanujan and Gauss–Legendre) used $\sqrt{2}$. So, I wondered how to calculate $\sqrt{2}$ manually in ...
0
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1answer
36 views

Messing with the linear approximation… I don't get Taylor's formula??

Alright, so I was messing around with the linear approximation for a function; I wanted to see if I could get the formula for the 2nd degree approximation from it. I went about it like this: $$ f(x + ...
0
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0answers
21 views

Superstrong approximation

I am reading up on research in approximate groups and have noticed that one of the reasons for doing this research is because it has applications in superstrong approximation theory. I'm more or less ...
1
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0answers
22 views

Approximation using Stirling

In the article "Series evaluation of Tweedie exponential dispersion model densities" by Peter Dunn and Gordon Smyth it is stated that $$-\log\Gamma(1+j)-\log\Gamma(-\alpha j)\approx ...
0
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1answer
61 views

Approximation for $ e^{ - x^2 } $ , x>0.

what is the good approximate so that it works for a large range of values. My purpose is to calculate logarithm of likelihood ratios. $ \log \left( {\frac{{e^{ - x_1 ^2 } + e^{ - x_3 ^2 } }} {{e^{ - ...
0
votes
1answer
58 views

how to approximate this expression $\frac{1}{8}x^2(1-\frac{1}{12}x^2)/(1-\frac{1}{4}x^2)$

when x is small, for example <1, then the expression can be approximate by (from a book) $$ g(x)= \frac{-x^2}{8}{\frac { \left( 1-1/12\,{x}^{2} \right) }{1-1/4\,{x}^{2}} }= \frac{-x^2}{8} ...
0
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0answers
51 views

How to derive this integral?

I have relatively simple integral but I could not figure out how to solve. It is $$ \int_{0}^{1}dz \frac{(1-z)(1-z^2)A^2}{B^2z+(1-z)^2A^2} $$ $$B>>A$$ EDIT: You should use an approximation to ...
2
votes
2answers
152 views

Approximating $\ln(1+\exp(x)+\exp(y))$

Ok, so I know that $\ln(1+e^x)\approx x$ when $x$ is large. But what about $\ln(1+e^x+e^y)$ when both $x$ and $y$ are large? I can figure out cases when $x\gg y$ or $y\gg x$ since that simplifies ...
1
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0answers
14 views

Multi-Objective Approximation Algorithms

Can algorithm approximations be combined in some form for purposes of multi-objective optimization? The study of approximation algorithms is very new to me, but I have been having a lot of difficulty ...
0
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0answers
17 views

Approximations of the kind $x<<y$

I have an expression for a force due to charged particle given as $$F=\frac{kQq}{2L}\left(\frac{1}{\sqrt{R^2+(H+L)^2}}-\frac{1}{\sqrt{R^2+(H-L)^2}}\right)$$ where $R$, $L$ and $H$ are distance ...
0
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0answers
34 views

Approximating continuous functions by steps functions: Proof that the approximation error monotonically decreases as the number of intervals increase

Let $f$ be a continuous function defined on a compact set, $f: X \subset \mathbb{R} \rightarrow \mathbb{R}$. Let $\mathcal{P}_k = P_1,\ldots,P_k $ be partitions of $X$ such that $\mathcal{P}_k$ is an ...
2
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1answer
102 views

A Better Approximation of $e$

So, I'm trying to self-learn Analysis, and I don't have any solutions, so I hope you don't mind if I put my answer here for you guys to help me check it, as it seems I haven't solved it correctly. ...
1
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1answer
50 views

Linear Programming - deriving the Dual of the Primal

I've the following linear programming problem: This is the LP representation of the uncapacitated facility location problem. This is the dual representation of this problem: My question is how ...
0
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2answers
51 views

Approximation: What can I put under the $\cal O$s

First, let me specify that $\cal O (X)$ denotes an (infinitesimal) amount that is of the same order with $\cal X$, i.e., $\lim \frac{\cal O(X)}{\cal X}=\text{constant}\ne0$ as $\cal X\to 0$. For ...
3
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3answers
131 views

Am I cheating in this case to evaluate $\pi$?

Since $\lim_{x \to 0}$$\sin x \over x$$=1$,here let $x=$$\pi\over n$ , then we have $\lim_{{\pi\over n} \to 0}$$\sin {\pi\over n} \over {\pi\over n}$$=1$ , which implies $\pi=$$\lim_{n \to\infty}$$\ ...
2
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1answer
22 views

Approximate a positive multivariate function with a sum of squares of polynomials?

I am constructing approximation to a multivariate function which I know is positive. My question is the following: Let $f(x)$ be a multivariate positive and continuous function. Can we approximate ...
2
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0answers
22 views

Borel functions and continuous functions [duplicate]

Suppose we have a set $A\subset\mathbb {R}$ and let $f\in\mathcal{B}(A)$ and $g\in\mathcal{B}_b(A)$ (Borel function on $A$ and bounded Borel function on $A$, resp.) Is it possible to approximate $f$ ...
1
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0answers
39 views

Polynomial Approximation of Holomorphic Functions

Consider $\Omega \subseteq \mathbb{C}$ be open. Let $f:\Omega \rightarrow \mathbb{C}$ be holomorphic on $\Omega$. For any closed ball $B[a;r]$ in $\Omega$ does there exist a sequence of polynomials ...
0
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2answers
23 views

Approximation: is it logical to approximate to zero?

"What is the value of 0.02 cm rounded to the nearest centimeter?" Is it logical to approximate a real value (however small) to zero? I know that following a simple 'rounding' or approximation ...
0
votes
1answer
48 views

Why erf(a-b)+erf(a)+erf(a+b) is so close to 3erf(a)?

I am approximating an empirical distribution function with a sum of three gaussians, and noticed that for erf function $erf(a-b)+erf(a)+erf(a+b)$ is numerically very close to $3erf(a)$ for applicable ...
5
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0answers
56 views

Given a vector $x\in \mathbb R^n$, how can we find $z\in \mathbb Z^n$ which is closest to a scalar multiple of $x$?

I am looking for how to find integer approximations to scalar multiples of real valued vectors. This is close to the problem of finding a best rational approximation to a real number, but kind of ...
1
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2answers
35 views

Approximating the value of $\frac{1}{\sqrt{1.1}}$ using Linear approximation of $\frac{1}{\sqrt{1+x}}$.

How do I calculate approximately the value of $\frac{1}{\sqrt{1.1}}$ with Linear approximation of the function $\frac{1}{\sqrt{1+x}}$ around point $0$. And here is a follow-up question: Show that the ...
2
votes
1answer
44 views

Approximating a matrix so that 1) all rows sum to one and 2) all values have max 6 digits.

Let consider a big matrix with values ranging from 0 to 1 (included). Each row sums to values that are lower than 1, extremely ...
6
votes
2answers
135 views

Strange approximation of $\pi$?

I was playing with my calculator (Casio fx-991MS) the other day. I input $$\arcsin(\sin(2))$$ The result came out as $$1.141592653\ldots$$ I immediately noticed that the digits seem to resemble $\pi$. ...
0
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1answer
41 views

Gamma function and Stirling's approximation

I am interested in strong upper and lower bounds on $\frac{\Gamma(n+\alpha)}{\Gamma(n)},$ where $n$ is a large non-integral number and $\alpha$ is a small constant like $3.5.$ I know the answer is ...
0
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2answers
28 views

Quick binomial test for high number of trials

I just wanted to perform a quick binomial test for an experiment (Bernoulli trial) with 185 successes out of 459 trials and a (hypothesized) success probability of 0.2. I do not have any mathematical ...
3
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0answers
38 views

How good is my approximation of this complicated sinc function? (plot included)

Part 1: The following function (for $N=256$) has the plot shown nexr $$ G(x) = \frac{1}{N}\text{exp} \bigg( j \frac{\pi}{2} \, x(N-1)\bigg) \frac{\sin (\frac{\pi N}{2} x)}{\sin (\frac{\pi}{2} x ...
1
vote
1answer
42 views

How exacting must we be for these powers of roots-of-unity?

If, given a root-of-unity approximation $\omega \approx e^{2 \pi i/n}$, and we want to take this approximation to the power $m$, how exacting must we be with the approximation so that: $$\left|\left( ...
1
vote
1answer
36 views

Closest rational approximation of $\sqrt x$ with denominator having prime powers $\lt n$

I am representing denominators in rational numbers with powers of their prime factors for easy multiplication and division in lowest terms (by adding and subtracting the prime powers). I would like ...
0
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1answer
31 views

approximation of rational functions

Given a multivariate rational function $p(\vec{x})= \frac{f(\vec{x})}{g(\vec{x})}$ over $[0,1]^n$ with $p(\vec{x})\in [0,1]$, how can we come up with a polynomial approximation of $p$, say ...
0
votes
1answer
48 views

Natural Cubic Spline Unequal Spacing

I'm currently trying to create a natural cubic spline by setting up linear equations to solve for the coefficients of the spline. Where the spline follows as : ...
3
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0answers
80 views

Solution for an ODE given only at discrete points

The problem I have: For each $n \in \mathbb N$ I have $$\begin{align} x_0^n & \in \mathbb R \\ h_n & \in \mathbb R \\ x_k^n & = x_0^n + k \cdot h_n \text{ for } k \in \{0,1,\ldots n\} \\ ...
0
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0answers
21 views

What is the meaning of sub-constant error?

The error is defined as $E \geq \frac{1}{2ab(1+ab)}$, where $a$ and $b$ are both positive . The claim is: if we fix the value $a$, then to get a sub-constant error $E$, we must ensure that ...
1
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0answers
18 views

Approximating ArcCos(x) without Radicals

Take $$f(x)=2x\arccos\left(\frac{x^2+d^2-1}{2xd}\right)$$ and try and find $$ I(x)=\int_{d-1}^{3}dx f(x) \sqrt{\left(\frac{x-1}{x}\right)}\left(3-x\right)^3 $$ You'll find the result is messy (see ...
4
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3answers
302 views

How to calculate $\exp(-x)$ using Taylor series

We know that the Taylor series expansion of $e^x$ is \begin{equation} e^x = \sum\limits_{i=1}^{\infty}\frac{x^{i-1}}{(i-1)!}. \end{equation} If I have to use this formula to evaluate $e^{-20}$, how ...
0
votes
1answer
35 views

Can a comparison network obtain all the n! permutations of a vector?

I want to permute a vector using comparison networks. This is the only method I have at my disposal. My original idea is to use a sorting network like Batcher or Bitonic. Basically I place my vector ...
1
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3answers
64 views

Numerical Approximation for 2D Curvature

I have a list of points (x, y) that are taken from an unknown 2D parametric curve $\vec{f}(t)$. These points are monotonically increasing in t (ie: they're a "connect the dots" version of ...
1
vote
1answer
33 views

Taylor theorem for f(x+h)

I am following a proof that applies Taylor's theorem on this document (http://www.gautampendse.com/software/lasso/webpage/pendseLassoShooting.pdf) I am not understanding one of the terms explained on ...
1
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1answer
54 views

Using $\ln (\cos x)=\frac{-x^2}{2}-\frac{x^4}{12}+…$, approximate $\ln 2$ in terms of $\pi$

Using $f(x)=\ln (\cos x)=\dfrac{-x^2}{2}-\dfrac{x^4}{12}+\dots $, approximate $\ln 2$ in terms of $\pi$. I know $\cos(x)$ will never be two - so what can I actually substitute in to get something ...
8
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0answers
236 views

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
0
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0answers
11 views

Ratio of fpras approximations

If I need to compute the ratio $\frac{A}{B}$ and if there exists an FPRAS that approximates the numerator and the denominator separately, that is, $\exists A_{fpras},B_{fpras}$: $Pr(A(1-\epsilon)\le ...
1
vote
1answer
43 views

polynomial approximation - basic chebyshev question

I was asked to find the best linear approximation to $f(x)=x^2$ in $x \in [0,1]$ using chebyshev polynomials, meaning, using the known property that $2^{1-n}T_n(x)$ is the best approximation to $0$ at ...
3
votes
0answers
39 views

Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...