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
11 views

How can I express this in terms of Gauss-Hermite Quadrature?

I am having the following expression. This is the PDF of Nakagami-Lognormal Distribution. I want to express in terms of Gauss-Hermite abscissas and weights. How can I do it? ...
2
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0answers
26 views

Uniform approximation. Two boundary layers?

Find uniform approximation up to order $O(\epsilon)$: $$ \begin{cases} \epsilon y''+\epsilon y' - y^2=-1-x^2 \\ y(0)=2 \\ y(1)=2 \end{cases} $$ At $\epsilon=0$ solutions $\pm \sqrt{1+x^2}$ don't ...
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0answers
23 views

Approximate $e^{0.01} \sin(0.02)$ by using its linearization

$f(x,y) = e^x \sin(y)$ $f_x = e^x \sin(y)$ $f_y = e^x \cos(y)$ $L(x,y) = f(0,0) + f_x(0,0)x +f_y(0,0)y$ Solving: $f(0,0) = e^0 \sin(0) = 0$ $f_x(0,0)x = e^0 \sin(0) \cdot x = 0$ $f_y(0,0)y = ...
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0answers
64 views

Kalman filtering with angle measurements

I'm designing an Extended Kalman Filter which will take several types of measurements and try to estimate a location. One type of measurement is the direction to the location. I've thought about this ...
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1answer
21 views

For $f(x)=x^4$, find its projection $f(x)^*\in P^2(-1,1)$ onto $W$

Consider the vector space $V=C[-1,1]$ and $W=P^2[-1,1]$. $V$ is an inner product space withe inner product $\langle f, g\rangle=\int_{-1}^1f(x)g(x)dx$. Consider a function $f(x)=x^4$ whcih is in ...
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0answers
5 views

Accuracy Rebonato Swaption Approximation Formula among Different Strikes

Can somebody explain me if the Rebonato swaption volatility approximation formula is accurate for only ATM strikes, and if yes why? Can it also be used for ITM and OTM strikes? My foundings: Let $0 ...
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0answers
7 views

Error bound for function limit with arbitrary $\Delta h$

Say we have some function $f\in C^1$. I would like to somehow bound the error that is made when the limit is approximated using some arbirary $h$, that is to find $E(x,h)$ such that: ...
20
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3answers
668 views

Maximum of Polynomials in the Unit Circle

Let $z_{1},z_{2},\ldots,z_{n}$ be i.i.d random points in the unit circle ($|z_i|=1$) with uniform distribution of their angles. Consider the random polynomial $P(z)$ given by $$ ...
2
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1answer
28 views

Approximating Fresnel integrals with standard functions

I would like to approximate the Fresnel S and Fresnel C with standard functions. I've started with the $ S(x) $ function: $$ approxS(x) = sgn(x) * \left ( sgn(x)* \left ( \frac{ \sin( \frac{x^2}{2} ...
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0answers
21 views

Given the 3x3 matrix D2 is a second order differentiation matrix, for a function u defined and twice differentiable on the interval [a,b].. [on hold]

Given that the 3x3 matrix D2 is a second order differentiation matrix, for a function u defined and twice differentiable on the interval [a,b].. u"(a) ≈ D2 * u(a) u"((a+b)/2) ≈ D2 * ...
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0answers
35 views

Quadratic Approximation Using Chebyshev Economization

For the quadratic approximation of given function, we use following: $\ Q(f) = f(a) + f'(a)*x + f''(a)*(x^2)/2$ In the question, it wants me to find quadratic approximation using chebyshev ...
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0answers
20 views

Approximation, Truncation argument, Sobolev space

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
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0answers
16 views

Big 'O' Notation - Taylor Series

Q) Use the Taylor Series Expansion to show the first derivative f '(x) can be approximated by $$-(3f(x) -2f(x+h) - f(x-h) / h ) $$ What is the precision? Now I found after using the Taylor ...
5
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0answers
72 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
2
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2answers
36 views

Approximate value of k

How do you solve $k$ in $\frac{(k-1)^{k-1}}{k^{k-2}}=n$ at least with a good approximation? Is there tight approximation?
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4answers
2k views

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
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0answers
10 views

Approximating integral of Erf with certain available functions.

I am developing certain software that deals with symmetric 2D Gaussian densities. One of the most common operations in that software is integrating those Gaussians over various 2D shapes. These ...
0
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0answers
37 views

A fast converging limit for $\ln x$ (or why $\ln 2 \approx \sqrt{\sqrt{42}-6})$

I've read a lot about approximating logarithms recently, and apparently it's not easy. It can be done by Taylor series (slow convergence), by continued fractions (also slow) and also by some limits. ...
1
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1answer
41 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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0answers
8 views

Approximating the integral of a large product

I would like to approximate the following integral of a product: $$ I = \int dz\, f(z)\prod_{i=1}^n\left(1 - \rho_i(z)\right) $$ The functions $f$ and $\rho_i$ are differentiable for all $i$, ...
0
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0answers
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 ...
1
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1answer
1k views

Smooth approximation of maximum using softmax?

Look at the Wiki page for Softmax function (section "Smooth approximation of maximum"): https://en.wikipedia.org/wiki/Softmax_function It is saying that the following is a smooth approximation to the ...
0
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1answer
431 views

Need some help understanding notation for composite gauss quadrature formula

Reading through some notes on 2-point gauss quadrature, I came across the following general formula. I'm currently doing an assignment with 3-point quadrature, and have gotten to a similar formula, ...
0
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0answers
8 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 ...
1
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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
18 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 ...
0
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0answers
18 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 ...
0
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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 $$ ...
0
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0answers
28 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 ...
0
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2answers
68 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 ...
11
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3answers
6k views

Approximating the error function erf by analytical functions

The Error function $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$ shows up in many contexts, but can't be represented using elementary functions. I compared it with another function ...
0
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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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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 ...
0
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0answers
48 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 ...
1
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2answers
19 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 ...
11
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9answers
608 views

What are better approximations to $\pi$ as algebraic though irrational number?

I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to ...
1
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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 ...
1
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0answers
13 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 ...
0
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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 $-0.685<\pi^e-e^\pi<-0.675$?
1
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2answers
41 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 ...
20
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6answers
6k views

Approximation of $e^{-x}$

Is there a method to mentally evaluate $e^{-x}$ for $x>0$? Just to have an idea when computing probabilities or anything that is an exponential function of some parameters.
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2answers
321 views

$\pi$ polynomials whose real zeros approximate $\pi$

Let's have the following polynomials $$x^4+105x^2-1134=0,$$ $$x^6+126x^4+10395x^2-115830=0,$$ $$3x^8+550x^6+45045x^4+3378375x^2-38288250=0$$ The positive real zeros of these equations are good ...
0
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4answers
51 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.
0
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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 ...
1
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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 ...
0
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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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0answers
28 views

linear approximation of surface

Suppose a surface determined by 4 points in the 3-D space like the following: Plane described by four points I would like to make a linear approximation in order to determine the z-dimension of point ...
0
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2answers
52 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 ...
0
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2answers
32 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 ...
0
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1answer
107 views

Archimedes' Approximation of Square Roots

Supposing a square root $\sqrt{X}$, let $x$ be the approximation of $\sqrt{X}$, then we get these 2 formulas to estimate $\sqrt{X}$: $x_{n+1}=\frac{x_n+\frac{X}{x_n}}{2}$ and ...