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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145 views

Generating all positive integers from three operations

This question arose on sci.math, but since almost all the competent mathematicians there have migrated here, I thought I'd give the question a wider audience. Starting with an integer $t>2$, ...
2
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131 views

explicit error bounds for Multivariate interpolation

I want to interpolate a function of $d$ variables over a Cartesian grid, using multivariate interpolation, while characterizing interpolation error in terms of bounds on partial derivatives of the ...
2
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76 views

On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that $$ A_0 = A_1 = 0;\quad A_n = ...
2
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78 views

Algorithm to predict next 3D points

For example, having this data: year x/y/z 2007 10/20/70 2008 20/10/70 2009 30/10/60 2010 40/10/50 2011 40/15/45 We want to predict what will be the x/y/z in ...
2
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149 views

approximation of the sum

I have difficulties to find an approximation formula (or bound from the below) for the following sum: $$ \sum_{k=1}^n\left( \frac{1}{35}\right)^{k-1}(n-k)!\left(k-\frac 32\right)!. $$
2
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128 views

Approximating sums like $\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$

Can anyone tell me how to approximate the following functions? $f_3(n) = \displaystyle\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$ $f_4(n) = ...
2
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0answers
184 views

calculate the rate of change

I am trying to calculate the change frequency for a set of data. Each bit of data has the date-time it was created. I would like to say for a specific set of data the change frequency is hourly, ...
2
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224 views

When is it valid to convert a function inside a probability integral to the indicator function?

I am faced with an approximation that replaces a probability density function with the indicator function and I am at a loss as to why this is valid. We want to model the lifetime $T$ of a website ...
2
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307 views

Multivariate B-Spline Derivatives

To construct a multivariate B-spline, we simply take the Kronecker tensor product between the univariate basis functions constructed for each individual dimension. What I'd like to know is how do you ...
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32 views

Mixing Fuzzy Logic and Probabilistic interpretation of a dataset

A probabilistic data cloud is a set $M$ of data points $\{m_i\}_i$, where each data point $m_i$ is associated to an event $E_i$ expressing the set of the occurrences of $m_i$ in any possible ...
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47 views

Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ http://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...
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13 views

Cubic curve as approximation of Euler spirals?

I was reading the wiki article about Euler spirals and I reached this passage: Unaware of the solution of the geometry by Leonhard Euler, Rankine cited the cubic curve (a polynomial curve of ...
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63 views

Series for Stieltjes constants from $\gamma= \sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right)$

Euler's constant has the following representations (Euler-Mascheroni constant expression, further simplification, http://math.stackexchange.com/a/129808/134791, Question on Macys formula for ...
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55 views

Why is $\sum{\frac{1}{(j/2)!}(\xi)^j}\approx\sum{\frac{1}{j!}(\xi)^{2j}}$?

Why is $\sum{\frac{1}{(j/2)!}(\xi)^j}\approx\sum{\frac{1}{j!}(\xi)^{2j}}$? This equation is used for solving the Schrödinger equation of the harmonic oscillator at one point in Griffiths, ...
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25 views

Asymptotic behaviour of generalized binomial coefficient $\frac{an(an-1)…(an-n+1)}{n!}$

Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$? It looks like it might be possible to express this in terms of gamma functions and use ...
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28 views

Normalization in least-p'th minimax algorithm

In the book "Practical Optimization: Algorithms and Engineering Applications", the least-$p$th minimax algorithm is presented, for approximation of the minimax optimizer (Alg. 8.1): $Loss_x(k)$ = ...
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34 views

Approximating $|1-e^{i\delta}|$

Actually this question is derived from the Rudin's Real and Complex Analysis's lemma 15.17. In the lemma, given condition is below; Let $h(z) \in H(\Omega)$ such that Re $h(z) = \log |1-z|$, |Im ...
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34 views

Uniform error of approximating the Heaviside function by a partial sum of its Fourier series

Suppose $f(x)=H(x-.5)$ where H = Heaviside function on $0<x<1$ is approximated by the first five nonzero terms of its Fourier sine series. Compute the uniform error (i.e maximum error, max p(x) ...
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35 views

Asymptotic binomial ratios

I am in need of asymptotic version of $$\frac{ \displaystyle \binom{n^{1-s}}{n^s}}{\displaystyle \binom{n}{n^{s}}}$$ where $n\in\Bbb N$ and $s\in\big(0,\frac12\big)$ and $$\displaystyle \frac{ ...
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33 views

Benefit of Padé approximation

I am currently starting to read papers about Padé approximation of the matrix exponential $\exp(A)$, namely $\exp(A) \approx P_{n,m}(A)Q^{-1}_{n,m}(A)$ I am now seeking for a good motivation behind ...
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48 views

If $0.2349$ is approximated to $0.2299$, what is number of significant figures in such approximation?

If $0.2349$ is approximated to $0.2299$, what is number of significant figures in such approximation? Let $x_T=0.2349$, $x_A=0.2299$, then absolute error = $|x_T-x_A|=0.0050=\frac{1}{2}\times ...
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35 views

Least-squares fit of a nonlinear (polar) system

I want to determine the six unknown coefficients (uppercase letters) of the model $$x=X_c+(Au+B)\cos(Cv+D),\\y=Y_c+(Au+B)\sin(Cv+D)$$ given a set of data $(x_k,y_k,u_k,v_k)$, by least-squares. As ...
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23 views

How is inequality/approximation obtained?

I am reading on combinatorics - probabilistic methods. In one particular problem I came across the inequality $$\binom{n}{k}(1-2^{-k})^{n-k} < n^k e^{-(n-k)2^{-k}}$$ I understand that ...
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21 views

Piecewise linear approximation of a (low order) trigonometric polynomial, with quantization

The Problem Given a trigonometric polynomial of order $K$: $$y(t)=\sum_{k=-K}^K c_k \ e ^ {j k \bar \omega t} \ , \qquad c_{-k}=\bar c_k$$ we want to find the best approximation to it using a ...
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35 views

Solving of numerical equation with integrals

Let's have equation $$ \cosh(2 \pi x) = \cos\left[\text{Re}\int \limits_{0}^{2 \pi}\sqrt{A - 2q\cos(2z)}dz \right]\times \cosh\left[ \text{Im}\int \limits_{0}^{2 \pi}\sqrt{A - 2q\cos(2z)}dz\right], $$ ...
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13 views

Least Square solution for approximating a sequence

Suppose I have a sequence of length N $a_1,...,a_N$ I want to approximate this sequence by $k^1,...,k^N$ where $k$ is my variable. What is the least square solution of this? is there a closed ...
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24 views

Trapezoidal Rule yielding the exact value of the integral

It is clear that if a function $f(x)$ is linear over the domain $a \leq x \leq b$, then one application of the trapezoidal rule, over the same domain, will yield the exact value of ...
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31 views

Please check this perturbation solution of polynomial root and truncation order.

I have a quintic polynomial where the coefficients depends on a parameter $c$, i.e. $$ a_0(c)+a_1(c)x+a_2(c)x^2+a_3(c)x^3+a_4(c)x^4+x^5 $$ I know that the roots of the polynomial are real and ...
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26 views

How to compare experimental data with teorethical prediction

I would like to know, what is the method to approximate experimental data to teorethical one. I have heard about polynomial regression. After calculating particular matrices and solving set of ...
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0answers
57 views

Is there an analytic approximation to integral of this form?

Started working on trying to find an analytical approximation to this integral and not getting very far. Any assistance or direction is greatly appreciated! Thanks Vince $$\int_{0}^{t} ...
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100 views

Derivation of approximation of Error function

In Abramowitz and Stegun there are some formulas for approximation of Error funtion. I am intrested in the formulas $7.1.25$ to $7.1.28$, here is one of them ($7.1.26$). ...
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73 views

Iterative algorithms to approximate unknown multivariate convex real-valued functions by a set of linear upper/lower bounds

I am looking for iterative algorithms that can approximate a multivariate convex real-valued function $f(\vec{x})=y,\, \Bbb{R}^n\rightarrow \Bbb{R}$. The function is not known beforehand, but it is ...
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85 views

Interesting approximation of distribution of numbers in a Farey sequence

I was investigating the distribution of the numbers in a Farey sequence and found some pattern. It is known that the number of elements in Farey sequence can be found using Euler totient function. So ...
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0answers
19 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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56 views

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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0answers
29 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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51 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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0answers
21 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: ...
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0answers
36 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 ...
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25 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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64 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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20 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 ...
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63 views

Chebyshev polynomials approximation - Is there a way to generalize this

In an exam I was given this question: let $f(x)=x^3$. We want to find the best linear approximation (best in the sense that the maximal error is minimized) of $f$ in the interval $[-1,1]$ using ...
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47 views

Is the Taylor Expansion a good approximation

Say I use a computer to sum the first 26 terms of $e^{-5}$ (degree 25), will this taylor expansion provide a good approximation? It summed to $.0067$ To me this seems like a good approximation, but I ...
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91 views

How to show$\frac{\pi}{4} = \frac{2\cdot4\cdot4\cdot6\cdot6\cdot8 \dotsm}{3\cdot3\cdot5\cdot5\cdot7\cdot7 \dotsm}$?

I am doing the exercises of Structure and Interpretation of Computer Programs. In exercise 1.31 the following equation is casually shown as an approximation of $\pi$: $$\frac{\pi}{4} = ...
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43 views

$L^2$ inequality for derivatives of polynomials on triangles

I'm reading a paper which states the following inequality, but the (presumably) elementary proof is cited to be in a document, which is too old to get access to. Let $p: \mathbb{R}^2 \to \mathbb{R}$ ...
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42 views

Fitted function - Which is better to use?

So I have some data for program running time, that follows a power law relation aN^b. I log-log plotted the data and saw that it became a straight line, so I calculated the slope of this line to get ...
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42 views

Univariate Polynomial Approximation

I'm working on an algorithm in which I need to approximate the behavior of a polynomial by computing its roots to some $\epsilon$ precision. The problem can be defined as follows: Let $f(x) = x^n + ...
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82 views

MATLAB implementation Spline Fitting

Check the attached problem please. I am a beginner in spline fitting and have a few questions: 1) How to find the coefficients c[n]. Is it by DTFT? 2) I understand how to find the derivative but ...
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37 views

integration with pade approximant

given the function $$ \int _{0}^{\infty}\sqrt{x}exp(-x) $$ can we use Pade approximants to integrate this i mean let bhe te rational approxsiamtions of $ \sqrt{x}= \frac{A(x)}{B(x)} $ and $ ...