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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System of Equations & Approximations

I am trying to derive Eq. (3.6) in the following thesis: http://drum.lib.umd.edu/bitstream/1903/14898/1/Khalil_umd_0117E_14726.pdf This is the equation I am trying to show: \begin{equation} ...
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1answer
17 views

Uniqueness of function approximation over three points?

Given a function $f(x)$, we want to approximate $f$ using $P(x)$, such that: $P(x_0) = f(x_0)$, $P(x_2) = f(x_2)$, $P'(x_1) = f'(x_1)$. Prove that such a $P$ is unique $\iff$ $x_1 \neq ...
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1answer
35 views

What can be said about $f''$ if the trapezoidal approximation is always an overestimate?

For any $a$ and $b$ the Trapezoidal approximation of the integral $\int_a^b f(x)\,dx$ is an overestimate. What can you conclude about the second derivative of $f$? I think it might mean that the ...
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1answer
18 views

Under which condition is $\hat\Sigma\approx\frac{1}{T-1}(X'X)$

Let $X$ be $T\times N$ random matrix. We are interested in the sample variance covariance matrix of $X$. It holds that \begin{align} ...
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2answers
30 views

Approximate summation of flooring function

I have this summation: $\sum_{k=0}^x \lfloor{\frac{k}{c}}\rfloor$ Do you have any ideas on any general expressions that can approximate this? P.S I know I can approximate it with a Fourier ...
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0answers
13 views

selecting points on a domain which represent the derivative of a function

I'm working on some algorithm part of which entails me to subdivide a domain based on the derivative of a function. Let's just consider the 1D case with a closed and bounded domain $[a,b]$ for a ...
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0answers
16 views

How do we derive efficiency from robustness in the virtual ant solution to the traveling salesman problem?

Using virtual ants/swarm intelligence to solve the Traveling Salesman Problem is an example of using a robust system to solve an efficiency problem. We normally think of robustness and efficiency as ...
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1answer
35 views

Is there an alternative to Taylor expansion of functions with more control over the error distribution?

As a physics student I see the Taylor series being (ab)used very often, mostly for the expansion of $\exp(x)$. The usual line of though is that the error (remainder) term of a high order is likely ...
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2answers
41 views

Asymmetric second difference quotient?

I need to find (approximate) the second derivative of a discrete function. Usually I would approximate the second derivative with $$f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}\tag{1}$$ In my case, ...
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31 views

What is your favorite approximation to the normal distribution? [closed]

I am asking this because my favorite is this one, which I independently discovered: $$N(x, 0, 1) =\frac1{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2} dt \approx \frac{1}{1+\exp(-ax)}, \text{ where } ...
4
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1answer
38 views

Order of Growth of a Sum

Let $n>k$ and $r$ be arbitrary positive integers. Define $q=k/n$. I want to show that $$ \sum_{i=0}^{rk} \binom{rn}{i}q^i(1-q)^{rn-i}(rk-i)=\Theta(\sqrt{r}) $$ as $r\rightarrow \infty$. I've ...
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3answers
91 views

Approximating $e^{x}/(e^{x} - 1)$

Is it correct to tell that we can approximate \begin{equation*} \frac{e^{x}}{(e^{x} - 1)} \end{equation*} by: \begin{equation*} \frac{1}{x} \end{equation*}
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15 views

How to approximate (with minimal error) a 16-dimensional linear equation system with inaccurate solutions?

I have a system of linear equations with 16 coefficients that I'm trying to solve. I have an effectively unlimited number of equations using those coefficients (i.e., ax1+bx2+...+p*x16=y). I know how ...
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3answers
32 views

Maclurin Series. (Approximation)

Given that $y=\ln \cos x$, show that the first non-zero terms of Maclurin's series for $y=-\frac{x^2}{2}-\frac{x^4}{12}$. Use this series to find the approximation in terms of $\pi$ for $\ln 2$. My ...
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0answers
37 views

Sofic groups alternative definition

I am trying to solve exercise two from the following document : http://mtm.ufsc.br/~daemi/soficworkshop/Course%20notes/Lupini%20Lecture%202.pdf I suspect there is an error in the exercise, but I'm ...
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34 views

Why do I get a big error when I compute this integral with Gauss-Legendre Quadrature?

I'm using Gauss-Legendre Quadrature to solve the following integral: $\int_0^{1}x^xdx$ After I've compared the result with the MatLab vpa(int(...)) of the same integral I've noticed that the ...
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2answers
28 views

find which two points an arbitrary point is nearest to

I would like to solve for a point $P$ regarding its proximity to the line segment it resides within. I can make a guarantee that the point will be placed along a line. In the included example, we ...
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1answer
34 views

Alpha max plus beta min algorithm for three numbers

There exists fast algorithm to approximate length of 2D vector - Alpha max plus beta min algorithm. It says that $\alpha\cdot\max(x,y)+\beta\cdot\min(x,y)\approx\sqrt{x^2+y^2}$ for some constants ...
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2answers
36 views

$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$

I seek to prove the identity $$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$$ I was given the following hint: Split the integral into $\int_2^{f(x)}+\int_{f(x)}^x$ for a ...
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0answers
7 views

Approximating a grid-valued signed distance function with a continuous function

I want to solve a continous optimization problem using IPOPT. My optimization involves a signed distance function whose values are defined on a 2D grid. Since IPOPT can't handle piecewise functions, I ...
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1answer
25 views

Using SVD to approximate matrix-vector multiplication?

Given some matrix A, is it possible to use Singular Value Decomposition to approximate Ax for some vector x within some error bound? According to Efficient low rank matrix-vector multiplication, it ...
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1answer
14 views

Expectation of trigonometric functions involving random variables.

This is more a formulation question. I need help making a sales pitch (lol). I am working on an practical engineering problem where I encounter functions of the form: $\cos(\phi + d_\phi)$, $ ...
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how to approximate compute p(a|b, c) , using p(b|c), p(a|*, c), p(a), p(b), p(c)

how to approximate compute p(a|b, c) , using p(b|c), p(a|*, c), p(a), p(b), p(c) p(a|*, c), * means anythins. p(a|*, c) = $\sum_i p(a|i, c) $ error = | p(a|b, c) - f(p(b|c), p(a|*, c), p(a), p(b), ...
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1answer
25 views

Trapezoidal Rule in Stirling's Approximation

https://en.wikipedia.org/wiki/Stirling's_approximation Here $$\sum \limits_{i=1}^n log(i) $$ is approximated as $$\int^n_1log(x) dx\ +\ 1/2\ log(n)$$ but I would approximate it as ...
4
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3answers
88 views

Calculate fractional part of square root without taking square root

Let's say I have a number $x>0$ and I need to calculate the fractional part of its square root: $$f(x) = \sqrt x-\lfloor\sqrt x\rfloor$$ If I have $\lfloor\sqrt x\rfloor$ available, is there a ...
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0answers
7 views

Mapping Function from Generated Data

Assuming solution space data is 1:1 mapped to domain space, is there a good/well known approach to flushing out a mapping function/approximation having access to lots of mapped data? I imagine I am ...
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1answer
31 views

Finding the integer parts of irrationals

When working with continued fraction expansions, I sometimes have to calculate the integer part of irrationals quickly without a calculator, what would be an effective way to do this? For example, ...
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1answer
18 views

Approximation Error of Stirlings Formula

Stirlings Approximation : $n! \approx \sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}$. So $100!$ has an approximate percentage error of about $\frac{100}{12n} = \frac{1}{12}$. Using this information, how does ...
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2answers
52 views

what is the best approximation for sine?

can you tell me which is the best approximation for cosine/sine functions. It should also reduce the computational complexity. I've already tried the Bhaskara-1 approximation. Can you suggest me ...
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1answer
33 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| ...
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1answer
14 views

Linearization of an implicitly defined function

$f(x,y,z)=e^{xz}y^2+\sin(y)\cos(z)+x^2 z$ Find equation of tangent plane at $(0,\pi,0)$ and use it to approximate $f(0.1,\pi,0.1)$. Find equation of normal to tangent plane. My attempt: I found that ...
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1answer
157 views

Accelerating approximations for arccos

I have recently built a method to accelerate drastically the accuracy of the following approximation of $\arccos(x)$ : $f_n(x)=2^n\sqrt{2-2g^{n-1}(x)}$ where $g(x)=\frac{1}2\sqrt{2+2x}$ and ...
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3answers
35 views

Curve fitting as a linear least squares approximation problem?

So I have a problem from a textbook that will count for a bonus homework assignment, but I am having some trouble knowing where to start. Some more difficult curve fitting problems can be ...
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10 views

Is there a way to calculate expected entropy and confidence bounds in this unknown multinomial parameter problem?

Given low-count and fairly low dimensional count data (non-negative integers) $n_1,\ldots,n_d$ where say $d \leq 5$ and $n_i \leq 10$, the goal is to compute the posterior mean and posterior ...
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1answer
22 views

Assymptotically approximating a sum similar to binomial.

I am, through some combinatorial problems which I'm working on, trying to figure out what the following sum becomes as $n\rightarrow \infty$: \begin{equation*} \sum_{i=1}^{n-1} ...
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29 views

Boundary layer problem

This question is taken from Bender & Orszag "perturbation methods" $y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$ first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to ...
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1answer
47 views

How to approximate Heaviside function by polynomial

I have a Heaviside smooth function that defined as $$H_{\epsilon}=\frac {1}{2} [1+\frac {2}{\pi} \arctan(\frac {x}{\epsilon})]$$ I want to use polynominal to approximate the Heaviside function. ...
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1k views

Is this a valid attempt at the Riemann Hypothesis? [closed]

From Marcus Du Sautoy's book “The music of the primes”, there is a method of finding a very long list of N consecutive numbers which are not primes. e.g $101!+2, 101!+3,...,101!+101$ all of which are ...
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1answer
24 views

Origin of divergence in a divergent field (2D)

I have a field of measured vectors, see example of four vectors in image below. If there was no noise they would all point outward exactly from one "central point". i.e. there would be a circle whose ...
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23 views

Speed of the usual approximation of the exponential

Let's consider the usual approximation of the exponential function $f_n(x)=(1+\frac{x}n)^n$. What do we know about its speed of convergence to the exponential? That is to say, how can we characterize ...
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4answers
259 views

Explain this inequality, related to logarithms

I am trying to understand a proof of Stirling's formula. One part of the proof states that, 'Since the log function is increasing on the interval $(0,\infty)$, we get $$\int_{n-1}^{n} \log(x) dx ...
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1answer
30 views

approximation of $x^2$ in hilbert spaces

use the least squares to find the best linear approximation to $f(x)=x^2$ on [-1,1]. that is find the line $y=a_0+a_1x$ that minimizes $\int_{-1}^1|f(x)-y(x)|^2$ solution I used the theory of ...
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37 views

How to find the upper bound of a binomial coefficient by using binomial theorem?

I have a task to find a upper bound of the binomial coefficient for all $r \leq \frac{n}{2}$. I've already obtained by using such relation: $$\frac{\sum_{k=0}^{r}\binom{n}{k}}{\binom{n}{r}}$$ Which ...
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Two dimension Taylor approximation

Consider function $f:\mathbb{R}^2 \setminus \{(0,0)\}\rightarrow \mathbb{R}$ defined with $f(x,y)=\frac{y-x}{x+y}$. I'm trying to approximate this function on $(0, \epsilon)^{2}$ where $\epsilon ...
4
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3answers
66 views

How can a binomial coefficient can be approximated by using Stirling's formula?

I've met some difficulties with such question: How can we approximate a binomial coefficient by using a Stirling's factorial approximation. I've evaluate a little bit and got this How can I ...
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2answers
42 views

Can every continuous function on complex domain be approximated by polynomials pointwise?

Do you know any theorem that will help me with this question: Let $f$ be any continuous function on complex plane. Show that there is a sequence $(P_n)$ of polynomials such that $P_n$ converges ...
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1answer
53 views

Maclaurin Series Approximation of $\sin{x}$

Use first ten terms of the Maclaurin series for $\sin{x}$ to find an approximation to the values of both $\sin{\left(\frac{6\pi}{7}\right)}$ and $\sin{\left(\frac{20\pi}{7}\right)}$? One can say that ...
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1answer
19 views

Local Approximation of Real Valued Functions

I'm unsure where to begin. Any guidance would be greatly appretiated. Suppose that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ has continuous second order partial derivatives, and at the origin ...
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1answer
36 views

Show that S is a cubic spline (natural or clamped)

Please see question. I believe the answer should be: $S_0(2)=\frac12(x^3-3x+2)=2$ $S'_0(2)=\frac12(3x^2-3)=\frac{9}{2}$ $S''_0(2)=\frac12(6x)=6$ $S_1(2)=\frac12(x^3-12x^2+45x-46)=2$ ...
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22 views

expectation approximation

Note: You don't have to understand Approximation Algorithms to answer this Hello. I need to prove an algorithm approximation by using expectation. The algorithm takes $x_i \in {0,1,2}$ such that ...