Questions tagged [numerical-integration]

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4 votes
1 answer
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Intractability of an integral by deterministic numerical methods

Suppose $X_1,\ldots,X_n$ is an i.i.d. sample from a probability distribution with continuous c.d.f. $F.$ Let $F_n$ be the empirical c.d.f. $$ F_n(x) = \frac 1 n \sum_{k=1}^n \mathbf 1_{X_n\le x} = \...
Michael Hardy's user avatar
2 votes
1 answer
161 views

On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$

Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...
user521337's user avatar
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2 votes
1 answer
700 views

Error in Gauss-Laguerre numerical quadrature scheme

The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0 ; \infty[$ by a finite sum, according to: $$ \int _0 ^{+ \infty} ...
MathTolliob's user avatar
1 vote
1 answer
103 views

Numerical solution to some functional equation

Let $z>0$ be fixed. Consider the function $p_a: \mathbb R^2_+\to\mathbb R_+$ given as $$ p_a(t,x):=\frac{1}{\sqrt{2\pi N_a(t)}}\left[\exp\left(-\frac{(x-z)^2}{2N_a(t)}\right)-\exp\left(-\frac{(x+z)^...
Fawen90's user avatar
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1 vote
0 answers
178 views

Numerically compute the Schwarz-Christoffel mapping to the square

I want to map the upper-half plane $$\mathbb H:=\{z\in\mathbb C:\Im(z)>0\}$$ to $[0,1)^2$ by a conformal map. If I got this right, then such a mapping is given by the Schwarz-Christoffel mapping to ...
0xbadf00d's user avatar
  • 161
1 vote
3 answers
833 views

Finding energy minimizing path

I'm trying to find an approximation for the optimal path for a material point, minimizing the integral associated with the total energy. I managed to write the exact formula for the energy along a ...
J. Doe's user avatar
  • 19
1 vote
0 answers
126 views

Numerical calculation of a double integral from the slowly-decaying oscillating function

Let us consider the following integral $$ I = \int\limits_{0}^{+\infty}dx\int\limits_{-\infty}^{+\infty}dy \left[f(x,y) + g(x,y) \right]. $$ We know several properties of these functions. There are ...
MightyPower's user avatar