Articles of approximation

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$

Weierstrass Approximation Theorem for continuous functions on open interval

I am studying for my introductory real analysis final exam, and here is a problem I am somewhat stuck on. It is Question 2, in page 3 of the following past exam (no answer key unfortunately!): Give an example of each of the following, together with a brief explanation of your example. If an […]

Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on simple manifolds (e.g. on a unit sphere) shows that computed eigenvalues of $D$ often have non-zero imaginary parts, even though they should all be real. (For simplicity […]

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 for a given value of $x$ I have a decent estimate of $y$. I expect the data to have […]

How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$

I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I decided to write a program for approximating $\tan(x)$. But I am having difficulty. I want to use the Taylor series of $\tan(x)$ to approximate […]

Are the Taylor polynomials of a function the results of a minimization problem?

Here is an example to better explain my question. Consider the function $f(x) = \cos(x)$. I want to approximate it in the set $[-\pi; \pi]$ using a polynomial $g(x) = a + bx + cx^2$ of order $2$. First approach – Taylor polynomials $$f(x) \simeq g(x) = 1 – \frac{x^2}{2} $$ Second approach – Minimization […]

Evaluating a limit involving binomial coefficients.

If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer: $$\lim_{n\rightarrow \infty} \binom{n}{k}\frac{\binom{\binom{n-k}{2}}{N_c}}{\binom{\binom{n}{2}}{N_c}}=\frac{e^{-2kc}}{k!}.$$ It feels like using Stirling’s approximation would help but I can’t quite figure out how… I ask this question because I am currently trying […]

How does rounding affect Fibonacci-ish sequences?

I’m curious how one might account for rounding in simple recurrence relations. $\textbf{Explanation}$ For a specific problem, suppose we have a sequence of positive integers $a_1, a_2, a_3,…$ with where each $a_i$ obeys the following rule $$a_n=a_{n-1}+\text{floor}\Big[\frac{a_{n-4}}{2}\Big]$$ Where “floor” here just means round down. For example if $a_1=5$ and $a_4=7$ then $$a_5=7+\text{floor}\Big[\frac{5}{2}\Big]=9$$ And if we […]

How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$

How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$. I am trying by calculus but don’t know how to use here in this problem. Any idea?

An integral and series to prove that $\log(5)>\frac{8}{5}$

A relationship between $\log(5) \approx 1.6094$ and $\dfrac{3}{2}=1.5$ can be justified by the harmonic approximation $$\log(5) \approx H_2=1+\frac{1}{2}=\frac{3}{2}$$ that can be obtained by regrouping Lehmer’s logarithm $$\log(5) = \sum_{k=0}^\infty \left(\frac{1}{5k+1}+\frac{1}{5k+2}+\frac{1}{5k+3}+\frac{1}{5k+4}-\frac{4}{5k+5}\right)$$ symmetrically around the negative term $$\log(5)=\frac{3}{2}+\sum_{k=1}^\infty \left( \frac{1}{5k-2}+\frac{1}{5k-1}-\frac{4}{5k}+\frac{1}{5k+1}+\frac{1}{5k+2} \right)$$ The corresponding integral is $$\log(5)-\frac{3}{2}=\int_0^1 \frac{x^2(1-x)(1+3x+x^2)}{1+x+x^2+x^3+x^4}\:dx$$ (answer by Olivier Oloa) This is a direct proof […]