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
27 views

approximation of x=sin(x) error

The larger $x$ becomes, the worst the approximation of $x=\sin(x)$ becomes. How large can $x$ get before the error is no longer less than $1/(2\times10^{14})$? What I have tried: I know that the ...
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0answers
15 views

cos(x) approximation with taylor of second degree

there is an approximation to find cos(x) is 1 - (x^2)/2, until n = 2 degree of taylor, but I'm confuse how to find how good is its approximation, the one thing I know only I get its error is (sin(c) ...
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1answer
34 views

Uniform convergence and relative error?

I never took a class where uniform convergence was introduced, so I only know the Definition and not much more about it. I have an Approximation of some sequence of functions $f_n$ by some function ...
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0answers
7 views

approximation of a complex valued real rational function

all. I am now struggling with a approximation problem. Suppose we have a matrix-valued measure $\mathrm{d}\Lambda(\omega)$, with compact support $[a,b]$, then its Cauchy transform is a well-defined ...
3
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1answer
32 views

Approximation rules?

Let's say I need to approximate the expression $$\frac{1}{2}mv^2\left(\frac{M}{M+m}+1\right)$$ when $m<<M$. Here is what I would do: $$\frac{1}{2M}mv^2\left(\frac{M}{1+\frac{m}{M}}+M\right) ...
3
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2answers
101 views

Method of dominant balance and perturbation

Approximate the solutions of $$\epsilon x^4 + (x-1)^3=0$$ I can't perform a singular perturbation because if I let $\epsilon=0$ then I lose a root. My professor suggests The Method of Dominant ...
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0answers
24 views

Iterative algorithm for finding approximation functions for N-dimensional space

Say, I have billions of integral-valued vectors of the form $(0, 1, 3, 0, 0, 0, 3)$. My goal is to efficiently compute approximate distribution of values of each component of these vectors for each ...
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1answer
39 views

Find a polynomial using minimax approximation

Find a polynomial with the maximum 1. degree which best approximates the $f(x)=e^x$ function in terms of minimax approximation in $[0,1]$.
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0answers
186 views

How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing ...
3
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0answers
40 views

Approximating this definite integral

I ran into the following integral in my research that I believe has no closed-form solution: $$ I = \int_{s_0}^{s_1} \frac{(\alpha_x s + \beta_x)^{\lambda_x}}{(\alpha_y s + \beta_y)^{\lambda_y}} ds ...
2
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3answers
48 views

How was this approximation of transcendent equation solution found?

I have an equation for $\xi$: $$\xi\gamma=\cos\xi,$$ where $\gamma\gg1$. I've tried solving it assuming that $\xi\approx0$ and approximating $\cos$ by Taylor's second order formula: ...
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0answers
22 views

Multivariate Appoximation

I have a mathematical model for a complex system which I would like to approximate it. My idea is to run this complex model once and produce some outputs, and then fit a polynomial for these outputs. ...
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2answers
57 views

Upper approximation of $\mathrm{atanh}(x)$?

Is there are nice upper approximation of $\mathrm{atanh(x)}$? For example, $\ln(x)$ is nicely approximated by $x-1$ for $x$ around $1$.
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0answers
26 views

Calculation rules for quadratic approximations

In this link ...
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0answers
41 views

How good an approximation to the derivative is an arc-length based approximation?

Note - my original definition below was wrong. I hope this replacement is better. The usual approximation to $f'(x)$ with step size $h$ is $D_h(f, x) = \frac{f(x+h)-f(x)}{h} $. This has so many nice ...
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2answers
31 views

Quadratic approximation of $tan(x)$ at 0.

I have tried this: http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-2-applications-of-differentiation/part-a-approximation-and-curve-sketching/problem-set-3/ and ...
0
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1answer
86 views

Approximate the largest and the smallest values of the integral

How do I solve this:approximate the largest and the smallest values of the integral for
2
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0answers
66 views

How to find the approximate basic frequency or GCD of a list of numbers?

I could't actually summarize the question in the title, so I'll explain my situation. I want to tell the integer numbers which act as the best approximate basic frequencies of a list of real numbers: ...
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0answers
22 views

Taylor's remainder in a compact

My impression is that each function enough regular ($C^\infty$ ) in a compact is equivalent to a polynomial. Is this true? Is there a way to prove it? The expression of the Taylor's remainder just ...
1
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1answer
11 views

Approximate solution to non-linear equation set

I have an equation set like this: $$ R_e(T_i)R_t(T_i) + R_e(T_i)R_p(T_i) = R_sR_t(T_i) + R_sR_p + R_pR_t(T_i) \\ R_e(T_f)R_t(T_f) + R_e(T_f)R_p(T_f) = R_sR_t(T_f) + R_sR_p + R_pR_t(T_f) $$ Where, ...
2
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4answers
93 views

Solution to a system of nonlinear equations

Do you know any method to solve the following system of nonlinear equations ? $\begin{equation} 141,3829=A+\frac{B}{323}+5,78C+F323^{E}\\ 69,07645=A+\frac{B}{333}+5,81C+F333^{E}\\ ...
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0answers
19 views

Is it correct approximation of Upper Incomplete gamma function?

I am trying to approximate the Upper incomplete regularized gamma function $P(s,t)$, at the constant value of $s=c$ by: $$Q(c,t) \approx 1-Q(t,c)$$ where: $c$ is some constant and $t$ is variable. ...
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1answer
23 views

Signifcant Figures — Why are rules for multiplying and adding true?

I found this other question that deals with this somewhat, but I am still unclear as to why the rules for adding/subtracting and multiplying/dividing significant figures are the way they are. In the ...
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0answers
5 views

Are there other names for multilayer perceptrons or multidimensional interpolants based on Kolmogorov's approximation work?

Are there other names for multilayer perceptrons that are used outside of the neural net community? At its core, multilayer perceptrons form a multidimensional interpolant of the form $$ ...
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0answers
20 views

Stirling like approximation for lower-incomplete gamma function?

May we have a similar approximation for lower incomplete gamma function $\gamma(s,x)$, as we have a Stirling's approximation for Gamma function $\Gamma(s)$.
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2answers
60 views

Continuous differentiable spline or function resembling floor

I'd need any (real-valued) function (whatever meets the following description at least approximately) continuous and thrice differentiable everywhere (or twice if 3 not possible), with the following ...
13
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2answers
365 views

Ramanujan's approximation for $\pi$

In 1910, Srinivasa Ramanujan found several rapidly converging infinite series of $\pi$, such as $$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}. ...
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0answers
14 views

Mapping from $\left(-\infty,\infty\right)$ to $[a,b]$ to reduce numerical error

Suppose $A = \left[\begin{array}{cc} \exp\left(x_1\right)&\exp\left(x_2\right)\\ \exp\left(x_3\right)&\exp\left(x_4\right) \end{array} \right]$, where each of $x_i\in\left(-1000,1000\right),$ ...
16
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3answers
275 views

Approximating $100!$

I participated in an Estimathon (a speed contest of Fermi problems) not long ago. It works as follows. Contestants are given questions and they must give a closed range $[a,b]$ which should contain ...
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0answers
19 views

Applications of low-rank matrix approximation

There was a similar question here Use of low rank approximation of a matrix that has unfortunately remained unanswered. Although being along the same lines, my question will be formulated in a little ...
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2answers
93 views

Constrained Newton-Raphson method

Peace be upon you, I want to solve a system of two equations in which the existence of $ln\left(\frac{\alpha}{\alpha+\beta}\right)$ function makes some limitations in iterations of the Newton-Raphson ...
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3answers
136 views

Approximate summation of the given equation

I have been trying from an hour to approximate the value of $M$ in the equation given below. $$ M = \sum\limits_{i=1}^n\left(\sum\limits_{j=1}^n\left(\sqrt{ i^2 + j^2 }\right)\right) $$ One thing I ...
0
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1answer
62 views

Maximum likelihood estimators

I have $X_1,X_2,\dots,X_n$ as random samples from a binomial distribution, with probability function: $$p_X(x) = Pr(X=x) = {m \choose{n}}\alpha^x(1-\alpha)^{m-x},x=0,1,2,\dots,m$$ where $m$ is given ...
0
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1answer
25 views

Approximate non-Lipschitz (but continuous) functions by Lipschitz functions

Is there any algorithm to approximate non-Lipschitz (but continuous) functions by Lipschitz functions ?
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2answers
149 views

Is there any reason why $4-\pi$ is quite close to $\frac{\sqrt{3}}{2}$?

In this question obviously the error of our "approximation" is $4-\pi=0.858...$ . I tried to reconstruct the false argument with $\tau=2\pi$, and the error in that case would be $8-\tau=1.716...$, ...
0
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1answer
34 views

How can missing data be organised or classified (Interpolation vs Approximation)?

I'm looking for a way to distinguish between the various types of missing data techniques? Can someone help to clarify or organize these categories in sub-sections or indicate similarities or ...
0
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0answers
34 views

Series representation for $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 \pi ^2 A^2+W^2}+6 \pi ^2 A^2+3 W^2}}{\sqrt{2}}$

My question is, is there a series representation or other function of $L$ and $A$ I can use when I solve the following equation for $W$? $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 ...
1
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1answer
28 views

Curve approximation by some known points on the curve

I want to approximate a curve by some known points on the curve. I can choose these point. My curve is shown as below: I have to use such a equation: ...
0
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0answers
48 views

Determining when the approximation fails

Context Suppose we have a grid-based game where a unit has a range parameter that serves as the upperbound of the sum of the costs of his movements in a single turn. Moving orthogonally to an adjacent ...
2
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1answer
39 views

Closed-form term for this expression

I have a normal Distribution $X \sim N(\mu, \sigma)$. Is there an easy way to give an asymptotic estimate with small error (I would prefer with relative error $\rightarrow 0$) for $P[X \geq k]$? We ...
1
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1answer
26 views

Bound on the derivative of a cut-off function

Let $\rho$ be a smooth function in $\mathbb R^n$ such that $0 \leq \rho \leq 1$ and $\rho$ is supported in the unit disk and let $\rho_\epsilon(x) = \epsilon^{-n}\rho(\epsilon^{-1}\|x\|)$. If $f$ is ...
1
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1answer
51 views

Approximating a Continued Fraction

From a paper I was reading, If: $$w=\frac {1}{3}\left\{ \frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac ...
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2answers
44 views

When to Taylor expand in a differential equation

I've come across a differential equation: $\dfrac{d\theta}{dx} = f(\theta)$, whose analytic solution is very complicated, but in this situation it is valid to Taylor expand functions around $\theta ...
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0answers
19 views

Represent this differential equation as a set of n+1 equations w n+1 unknowns

Given the following differential equation: $$s''w'' + 2s'w''' + sw'''' = q$$ We use these approximations: $$w''''(x_i) \approx \frac { { w }_{ i+2 }-4{ w }_{ i+1 }+6{ w }_{ i }-4{ w }_{ i-1 }+{ w ...
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0answers
9 views

Vertex Cover Approximation

Is there any Vertex Cover approximation algorithm that gives the optimal solution for some graphs but otherwise near-optimal solutions for other graphs? Would an algorithm like that be useful? From ...
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( ...
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0answers
29 views

extrapolate boundary data of a 2d function

I have a function defined over a 2d domain. I want to keep all the internal data, but the information near the boundaries are less accurate and I want to extrapolate them from the inner points. Is ...
2
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1answer
43 views

How do you derive the secant method formula from the equation below?

The Secant Method forumula is; $$ x_{i+1}=x_i - \frac{f(x_i)(x_i-x_{i-1})}{f(x_i)-f(x_{i-1})}.$$ Derive the formula from the equation below; ...
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0answers
40 views

Archimedes' Apprxomation 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 ...
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2answers
54 views

I do not quite understand what destructive cancellation is, can someone explain it please?

I have been given this example 2.0013−2.0005=0.0008. The destructive cancellation is the large common value, here the 2, that disappears.