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
21 views

Boundary of the limit when $a<<1$

Let's say I have the expression $-b+\frac{3}{2}b\cdot a^2$. Can I say that by taking the limit when $a<<1$, that expression is $\approx -b$ ? can a constant number, like $\frac{3}{2}$ can "ruin" ...
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1answer
26 views

Numerical Approximation of Differential Equations with Midpoint Method

I want to proof that the local truncation error of the Midpoint Method is $d_{k+1}=O\left(h^{3}\right)$ Approach The local truncation error is defined as: ...
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3answers
27 views

Use a linear approximation to estimate the given number $64.07^{2/3}$

That's all they give you. I tried putting it into the linear approximation equation of: $$ f(a)+f'(a)(x-a) $$ but I get almost the same value as $64.07^{2/3}$, which is around $16.0116$. Just not ...
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2answers
45 views

Use a linear approximation to estimate the number $64.07^{2/3}$

That's all they give you. I tried putting it into the linear approximation equation of: $$ f(a)+f'(a)(x-a) $$ but I get almost the same value as $64.07^{2/3}$, which is around $16.0116$. Just not ...
1
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0answers
13 views

How to analytically find these rounding issues

Let's say we have a fixed yearly amount that we have to divide equally among an amount of days. For instance for $1,600 we may have: ...
1
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1answer
17 views

Approximate using differentials when partial derivatives are given?

I have ran into this problem on my online math assignment, this week we are covering partial derivatives and higher order partial derivatives, but I don't think I have learnt anything that can help me ...
2
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1answer
29 views

least squres vs. lagrange interpolation

can some one tell me the differences between these two approximation techniques, what are the strengths and weaknesses of each, and which better one to use. Thanks
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1answer
22 views

How does $\left[\nabla f\left(\mathbf{x}\right)\right]^\mathrm{T} \nabla f\left(\mathbf{x}\right)$ approximate $\mathbf{H}$?

Page 3 of a guide to Levenberg-Marquardt optimization says that $\left[\nabla f\left(\mathbf{x}\right)\right]^\mathrm{T} \nabla f\left(\mathbf{x}\right)$ approximates the Hessian matrix of $f$. I do ...
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0answers
20 views

Estimating special values of the Riemann zeta function on the critical line

If $p,q$ are primes, is it necessarily true that $$\left|\zeta\left(\frac{1}{2} + i\frac{p}{q}\right)\right| > (p+q)^{–(p+q)} ?$$ (Here $\zeta$ is the Riemann zeta function.)
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0answers
22 views

Approximation to the negative-binomial and negative-hypergeometric distributions

It is known that a binomial distribution can be approximated by a normal variable with the same mean and variance, for sufficiently large $n$. What approximations are known for the negative binomial ...
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1answer
29 views

Approximate an integral using Monte Carlo method

I have a question on an assignment Calculate the value of the integral I = $\int_0^\pi sin^2(x)dx$ using the Monte Carlo Method (by generating $ 10^4 $ uniform random numbers within domain [0, ...
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3answers
28 views

Errors and Taylor Polynomials

For $g(x)=x^{1/3}$, $a=1$, degree $3$ I found the Taylor polynomial: $$p_3(x) = 1 + (x-1)/3 - ((x-1)^2)/9 + (5(x-1)^3)/81$$ How do I use the error formula for the Taylor polynomial of degree 3 to ...
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1answer
43 views

if $x\ll 1$ is it safe to assume that $x\ll \frac{1}{2}$

I know that: if $x\ll 1$ then we can write $\frac{x}{x+1}\rightarrow x$ but is it safe to write $\frac{2x+1}{x+1}\rightarrow 1$?
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0answers
50 views

Historical calculations of $tan^{-1}x $ and $e^x$

SineBhaskara_I One reads that $tan^{-1}(x) $ series expansion existed in early (Indian) history. But like the Sine trigonometric function, did any similar approximation exist as well? The query ...
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4answers
107 views

Parabolic sine approximation

Problem Find a parabola ($f(x)=ax^2+bx+c$) that approximate the function sine the best on interval [0,$\pi$]. The distance between two solutions is calculated this way (in relation to scalar ...
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1answer
27 views

Uniform Approximation by Finite Wavelet Sum

Suppose $\psi:\mathbb{R}\rightarrow\mathbb{C}$ is a rapidly decreasing, bounded function with zero integral. For $j,k\in\mathbb{Z}$ define a function $\psi_{jk}(x):=\psi(2^{j}x-k)$. Suppose is the ...
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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 ...
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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 ...
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0answers
19 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
20 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 ...
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5answers
127 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
44 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
13 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 ...
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4answers
173 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 ...
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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 + ...
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0answers
20 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 ...
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0answers
21 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
60 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
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1answer
56 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} ...
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0answers
49 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
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2answers
150 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 ...
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0answers
11 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 ...
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0answers
16 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 ...
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0answers
30 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
101 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. ...
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1answer
30 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 ...
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2answers
46 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 ...
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3answers
130 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
20 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 $f$ ...
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0answers
34 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 ...
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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 ...
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1answer
47 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 ...
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2answers
33 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
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1answer
40 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
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2answers
130 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 $\pi$. ...
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1answer
38 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 ...
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2answers
24 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 ...