Tagged Questions

64 views

Is there an example where the error of Gauss-Laguerre quadrature does not vanish?

The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum $$\sum_{i=1}^n f(x_i) w_i$$ where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
100 views

Seeking a class of functions for which sums approximate integrals well

Is there a "natural" class of integrable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ for which it is true (and, preferably, not too hard to prove!) that $\sup_{0 \leq a < h} |h S(a,h) - I|$ ...
85 views

Error of midpoint method for differentiable functions

Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$? ...
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Error of midpoint method for functions that are not twice-differentiable

All of the bounds I've seen for the error of the midpoint method of integration are expressed in terms of the second derivative of the function. What bounds are available when the function is not ...
Let $f_0,f_1,\ldots$ be a sequence of functions $f_n : [0,1] \rightarrow R$ defined as follows: $$f_0(x) =1+2x$$ $$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } ... 1answer 240 views The geometric-mean factorial Think of the factorial as f(n) = n \odot (n-1) \odot \cdots \odot 2 \odot 1, where \odot is the binary operator for multiplication, \cdot. This suggests exploring replacing \odot with other ... 1answer 190 views How to get an expression for this integral(Numerically/Analytically) I have the following problem: I need to evaluate the integral$$\int_{\cos(\alpha)}^{1} P_l(t)P_{l'}(t) dt $$for \alpha \in [0,\pi] and each combination of l and l', where P_l is the l-th ... 0answers 127 views Radius of convergence to be proved more precisely (differential equation) There is a differential equation in polar coordinates: r'^2+r^2=(kt)^2, r(t=0)=0, k- Const. It is possible to get a solution which is a power series (see below). However, I am looking for an ... 2answers 650 views Approximating erf by tanh It appears to be well-known that \tanh(x)\le \mathrm{erf}(x) on [0,\infty). It's off-handedly mentioned here, for example. Where can I find a formal proof? On the one hand, it's hard to imagine ... 2answers 620 views why do we need algorithms, and why is non-convex optimization difficult? [closed] A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global ... 1answer 149 views Estimating the volume of a semialgebraic set from above Suppose S is a subset of \mathbb{R}^n of finite volume defined by a system of finitely many polynomial inequalities with integer coefficients. Can anyone describe an algorithm that, given such a ... 1answer 694 views On an eigenvalue inequality Let \lambda_1 (\cdot) be the larger absolute value eigenvalue of a 2\times2 matrix and \lambda_2 (\cdot) the smaller absolute value eigenvalue of a 2\times2 matrix, i.e. |\lambda_1 (\cdot)| ... 1answer 458 views Acceleration via smoothing Is the following approach to accelerating the rate of convergence of (1+1/2+\dots+1/n)- \ln n (with n=1,2,3,\dots), and other sequences like it, in the literature? Let f(t)=(\sum_{1 \leq n \leq ... 1answer 346 views Numerically finding a Mercer expansion for a given covariance kernel Let c(r) be a nice, continuous function with compact support. For example, c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big) for r \in [0,1], and c(r) = 0 otherwise. On ... 0answers 304 views Evaluating Shintani cone zeta functions Hi everyone I am trying the evaluate sums of the form$$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)} ...
Suppose $f$ is continuous on $[a,b]$ with $I = \int_a^b f(x)\: dx$, and for every $\epsilon > 0$ let $\delta(\epsilon)$ be the largest $\delta > 0$ such that every Riemann sum arising from a ...