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Results tagged with numerical-integration
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user 36721
1
vote
Accepted
Numerical solution to some functional equation
$\newcommand\erf{\operatorname{erf}}\newcommand\R{\mathbb R}$The functional equation in question is
\begin{equation*}
a=F(a) \tag{1}\label{1}
\end{equation*}
on $(0,\infty)$, where $a$ is in the c …
- 128k
5
votes
Numerical integration method that doesn't involve derivative in the error bound
$\newcommand\Z{\mathbb Z}\newcommand\R{\mathbb R}\newcommand\la{\lambda}$If $f$ is an arbitrary Lebesgue-integrable function, then, as is done in a definition of the Lebesgue integral, it makes sense …
- 128k
2
votes
Accepted
On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, w...
The key here is the simple change-of-interval/rescaling formula, found e.g. at the link in the OP, according to which
\begin{equation}
T_n(f)(x)=T_{n,[0,x]}(f)=x\sum_1^n w_i f(xx_i), \tag{*}
\end{eq …
- 128k
1
vote
Accepted
Numerical integration with integrable singularity
(i) Your purported error bound is of course incorrect: consider e.g. $f(t)=t$.
(ii) To get rid of the singularity, make the substitution $u=\sqrt t$, so that
$$\int_0^T dt\,f(t)=\int_0^T \frac{dt}{\sq …
- 128k