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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Streaming algorithm for polynomial fitting data?

The specific problem I'm trying to solve is: $$h_k(x, n) = \left(\frac{\alpha}{n} + 1 - \alpha\right) \sum_{i=0}^{k} c_ix^i.$$ Given $k$ and a stream of tuples $(x, n, h_k(x, n))$ (where the $x$'s ...
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Approximating a large number of data points using (cubic) splines in l1/l2 norm.

I have a pretty large dataset ($x,y$) consisting of a few million points. There is a lot of noise in the data. I want to find a smooth but simple approximation/representation for this dataset, so that ...
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Fittig N-D periodic functions

Besides fourier series, are there any stable way to fit N-D nonlinear functions that demonstrate some degree of periodicity, sigmoid neural networks works poorly in this domain.
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still trying to figure out details of Newton's method

I understand mostly what going on when using Newton's method to approximate some value but there are some details I am still hazy on (like how you come up with the correct $f(x)$ for the update rule). ...
180 views

Harlan J. Brothers's approximation to $e$ ad infinitum?

Consider the series generated by Harlan J. Brothers's method for the number $e$ http://en.wikipedia.org/wiki/List_of_representations_of_e Can they be improved again and again or is there a limit so ...
608 views

Lambert function approximation $W_0$ branch

I am looking for a simple, inexpensive and very accurate approximation of the Lambert function ($W_0$ branch) ($-1/e < x < 0$).
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the following inequality is true, but I can't prove it

The inequality $$\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)$$ holds for all integer $d\geq 1$. I use computer to verify it for $d\leq 50$, and find ...
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approximate $\int_{u=0}^b e^{-\frac{u^2 + ac}{au}}du.$

I'm trying to find an approximation (or exact solution if possible) for an integral of the form: $$\int_{u=0}^b e^{-\frac{u^2 + ac}{au}}du.$$ I was thinking of somehow applying a Gauss Hermite ...
258 views

Approximate Nash Equilibrium

I am sort of confused by the notion of approximate Nash equilibrium. I will try to express my confusion in the following exercise. Problem. Is it true that for every two player game where every ...
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determining sign of function containing logarithm.

I would like to know the sign of the following term in general. I tried approximately and it was negative. Is there any $m_0$ such that for all $n>m>m_0$, the following function is positive or ...
329 views

Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t)$

I try to solve this problem all day, but can't reach any progress in it. Can you give me some hints? Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t) = \frac{1}{\sqrt[4]t}$ ...
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Understanding weighted linear least square problem

I am having difficulty in understanding about weighted linear least squares. Could anybody explain me instead of minimizing the residual sum of squares why we need to minimize the weighted sum of ...
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Does this ODE have an exact or well-established approximate analytical solution?

The equation looks like this: $$\frac{\mathrm{d}y}{\mathrm{d}t} = A + B\sin\omega t - C y^n,$$ where $A$, $B$, $C$ are positive constants, and $n\ge1$ is an integer. Actually I am mainly concerned ...
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Function with $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$

I am looking for a function $f:\mathbb{R}\to \mathbb{R}$ and $\epsilon>0$ such that there is no $\delta>0$, for him any $x\in\mathbb{R}$: $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$ ...
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Complete normed vector space

I came across this question and was not sure how to proceed. Is the normed vector space of all polynomial functions on [0,1] complete with respect to the infinity norm? Thank you
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Simpson's Rule derived from Trapezoidal Rule

I was just wondering if I could have some assistance in regards to the Trapezoidal Rule and Simpson's Rule. I have a question where it asks to generalize the Trapezoidal Rule to the case of Simpson'...
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Series evaluated to $m$ terms, approximating the error

Given a series $\displaystyle\sum_{n=0}^\infty a_n$, how can we bound the error (which I shall denote with $R_n$) when we evaluate it to $m$ terms? $$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$ ...
Taylor Series for $e^x$ where $x = 1$, estimating the Error
I'm trying to calculate $e$ to a certain number of digits. The Maclaurin Series expansion of $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$. When $x = 1$ we can approximate the value of $e$ by ...