# Tagged Questions

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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### numerical approximation to logarithm

we know that $$\ln(x)=\frac{1}{x} \int_{0}^{1}dt \frac{1}{1+xt}$$ then given a cuadrature formula inside $(0,1)$ is that true $$\ln(x)= \frac{1}{x}\sum_{i}\frac{w_{i}}{1+xt_{i}}$$ wht other ...
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### Approximate function using non-orthogonal basis

I'm currently trying to wrap my head around somebody's (very concise) description of Finite Element Analysis (FEA): ...
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### Polynomial Interpolation and Error Bound

Problem: Use the Lagrange interpolating polynomial of degree three or less and four digit chopping arithmetic to approximate cos(.750) using the following values. Find an error bound for the ...
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### 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 $t>2$, ...
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### Estimate the drift and diffusion function numerically

I have a 1D problem as following $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x} \Big[ \frac{1}{2} \frac{\partial (g(x) f)}{\partial x} -\mu(x)f \Big]$$ I have a time-series of ...
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### A geometric proof for the “small angle approximation” for the sine, cosine and tangent

How can be deduced the so called "small angle approximations" for sine, cosine and tangent, namely $\sin \theta \approx \theta$ $\tan\theta \approx \theta$ $\cos\theta \approx 1-\frac{\theta^2}{2}$ ...
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### Reduced Chebyshev approximation?

Few days ago my teacher mentioned a method of approximation for a function. I think it was called "reduced chebyshev approximation", where you find the taylor series of degree $N$ then subtract from ...
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### How can I prove two empirically derived graphs are topologically equivalent?

I have two graphs that I've derived from an empirical data set and I suspect that they're topologically equivalent. It seems much easier to show that these graphs are not equivalent than to show that ...
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I am currently trying to place an order of precision on the approximation $$\prod_{r=s}^t \left(1-\frac{b}{r}\right) \approx \left(\frac{s}{t}\right)^b$$ This follows because $$\prod_{r=s}^t \left(1-\... 0answers 40 views ### Mixing Fuzzy Logic and Probabilistic interpretation of a dataset A probabilistic data cloud is a set M of data points \{m_i\}_i, where each data point m_i is associated to an event E_i expressing the set of the occurrences of m_i in any possible non-... 0answers 57 views ### Three almost-integers of the form ce^{H_a+H_b}\approx 2^k\pm1 The approximation$$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 ... 0answers 32 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 ... 0answers 79 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 Euler-... 0answers 56 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, ... 0answers 36 views ### Softmax for Continuous Functions? The softmax \log \sum_{i=1}^n \exp(f_i) of vector f is a smooth upper bound on \max_i f_i. However, the same cannot be said of \log \int_{X} \exp(f(x))dx in relation to \max_{x \in X} f(x) (... 0answers 30 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 ... 0answers 35 views ### Normalization in least-p'th minimax algorithm In the book "Practical Optimization: Algorithms and Engineering Applications", the least-pth minimax algorithm is presented, for approximation of the minimax optimizer (Alg. 8.1): Loss_x(k) = E(... 0answers 33 views ### Approximation of integrable function by polynomials Assume f\in \mathscr{R}(\alpha) on [a,b], and prove that there are polynomials P_n such that$$\lim \limits_{n\to \infty}\int_{a}^{b}|f-P_n|^2d\alpha=0.$$Proof: Let \varepsilon>0 be given ... 0answers 36 views ### Approximating |1-e^{i\delta}| Actually this question is derived from the Rudin's Real and Complex Analysis's lemma 15.17. In the lemma, given condition is below; Let h(z) \in H(\Omega) such that Re h(z) = \log |1-z|, |Im h(... 0answers 42 views ### Taylor polynomial approximation with absolute error less than 3 decimal places Assume that f is a function with |f^{(n)}(x)| \le 11, for all n and all real x. Let T_n(x) denote the Taylor polynomial of degree n for f(x) about the point x=0. What is the least integer n ... 0answers 46 views ### Uniform error of approximating the Heaviside function by a partial sum of its Fourier series Suppose f(x)=H(x-.5) where H = Heaviside function on 0<x<1 is approximated by the first five nonzero terms of its Fourier sine series. Compute the uniform error (i.e maximum error, max p(x) ... 0answers 39 views ### Asymptotic binomial ratios I am in need of asymptotic version of$$\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{ \...
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 ...