910 reputation
310
bio website code.google.com/p/notebk
location
age
visits member for 2 years, 3 months
seen 14 hours ago

A mathematician, a programmer, etc. etc.


Dec
31
comment Evaluating the precision in the calculation of $\mathrm{e}$
@user21820 err, you seem to be forgetting the topic of discussion is computing $e$ to some desired precision. (I am not worried about $\exp(x)$ for general $x$, just $x=1$.) The 20th continuant of the continued fraction cited is good beyond 100 digits, requires 40 addition operations, 80 multiplication ops, and 1 division op. The Taylor series, OTOH, requires 38 terms, in Horner form that's 38 multiplication ops + 38 addition ops + the killer 38 division ops. That number of division operations makes it practically unacceptable...
Dec
30
comment Evaluating the precision in the calculation of $\mathrm{e}$
@user21820 A division operation is costlier than a multiplication operation. (About 5 times costlier, in fact, for x86 floating point...and rational arithmetic would be worse, computing the gcd and then performing 2 division operations...) The continued fraction expansion any sane person would have in mind could be googled in a second. Remember $e=\exp(1)$...
Dec
30
comment Evaluating the precision in the calculation of $\mathrm{e}$
@user21820 Well, look at (e.g.) Wikipedia's page for the general formulas for the numerator and denominator. You can compute them recursively, requiring (for each iteration) 2 additions and 4 multiplications. So $N$ iterations costs 1 division operation (the final division) + $2N$ addition + $4N$ multiplication, far better than the naive Taylor series.
Dec
30
comment Evaluating the precision in the calculation of $\mathrm{e}$
Unrelated but potentially useful to the OP: using the continued fraction expansion for $\exp(z)$ is more accurate, and if you compute the convergents $A_{n}$, $B_{n}$ recursively...it is faster since it uses fewer division operations...
Dec
29
comment Identifying a power series
This power series does have a real root at $x\approx-1.403761051217752$. Needless to say, all real roots are necessarily negative...but I think this is the only real root.
Dec
27
comment The difference between $\Delta x$, $\delta x$ and $dx$
Well, $\delta x$ means different things depending on the context. For example, it has a particular meaning in variational calculus, and a completely different one in functional calculus...
Dec
24
comment Learning math: still paper and pen
@AndreySokolov A pop review and a more thorough answer (with many references) at the personal productivity stackexchange. I think from there, you can find additional references to studies done...
Dec
19
comment Handbook of mathematical drawing?
Honestly, learning how to draw in general helped me with technical diagrams and mathematical figures. Perhaps it's the only way...
Nov
23
comment How to find this limit: $A=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}$
I'm interested: how did you derive your approximation? (I ask because I wouldn't have a clue where to begin with deriving it...)
Nov
6
comment Derive the equation of motion from Lagrangian of a particle moving in an electromagnetic field
Uh, perhaps plug the Lagrangian into the Euler-Lagrange equations...?
Oct
21
comment General Linear Group of a vector space
Well, to be technical, you only need to pick a basis to determine the components of the matrix representing the isomorphism.
Oct
20
comment Maclaurin series for $e^z /\cos z$.
Perhaps some words or explanations would help this jumble of equations make sense...
Oct
20
comment Converging or diverging series?
One quick trick for (a) is to note if $a_{n}=\sqrt{n}\cos^{2}(n)/(n^{2}-2)$, then $0\leq a_{n}\leq \sqrt{n}/(n^{2}-2)$, since $0\leq\cos^{2}(n)\leq1$.
Sep
28
comment Question on category theory
Well, you describe a "locally-small" category, where the $\hom(-,-)$ are sets. They can be "bigger" collections (classes), which would give us a "large" category (example of a large cat: Set). THEN the question is: are you talking about the category of small categories, or of large categories? (Or both?)
Aug
14
comment Summing the series $ \sum_{n=1}^{\infty} \int_0^{\infty} \frac{\mathrm dx}{n(1+x^3)^n}$
@Norbert, you are too clever for my feeble old mind! ;) Thanks for the explanation, I appreciate it greatly.
Aug
13
comment Summing the series $ \sum_{n=1}^{\infty} \int_0^{\infty} \frac{\mathrm dx}{n(1+x^3)^n}$
I'm old and forgetful, so I have a question: when you use your clever trick $\log(1+q)=\sum\dots$, setting $q=(1+x^{3})^{-1}$, don't you need to be careful with the limits of integration and take $\int^{\infty}_{\epsilon}\dots\,\mathrm{d}x$, then take the limit as $\epsilon\to0^{+}$? [I know this may be pedantic, but as I said -- I am old and forgetful, and I forgot whether you can skip it or if it's really necessary!]
Aug
9
comment Understanding mathematical set theory syntax
@user2666425: the one-to-one correspondence is between the powerset of $A$ and $2^{A}$. There is not necessarily a one-to-one correspondence between the cardinality of $A$ with $2$! There is equal cardinalities between $P(A)$ -- the powerset of $A$ -- and $2^{A}$, the set of characteristic functions for all subsets of $A$.
Aug
9
revised Understanding mathematical set theory syntax
added 461 characters in body
Aug
9
answered Understanding mathematical set theory syntax
Aug
9
revised Understanding mathematical set theory syntax
Improved TeX