# Tagged Questions

**7**

votes

**3**answers

258 views

### Multiprecision numerical evaluation of integral: Sage vs. PARI/GP vs. mpmath

I am trying to compute thousands of integrals of the below type, that comes up in a conformal mapping problem, to as many accurate digits as possible (preferably 50+):
$$
\int_{-1}^1\textrm{d}t ...

**6**

votes

**2**answers

146 views

### Numerical integration of legendre polynomials

I hope that numerical questions are also permitted here.
I want to expand a smooth functions $f \in C^{\infty}$in terms of Legendre polynomials. Thus I need to calculate integrals of the form ...

**1**

vote

**1**answer

166 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 ...

**1**

vote

**2**answers

107 views

### Numerical calculation of Fourier transform with a nice error bound

I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and ...

**2**

votes

**2**answers

144 views

### dense lattices in high dimensions

I want a collection of points $\{ x_1, \dots, x_m\}$ to sample a unit cube $[0,1]^n$ with $n >>1 $ in high dimensions so that summing over these points is approximate the integral over that ...

**4**

votes

**0**answers

126 views

### About arithmetic-geometric mean

It's well known that if we set $a_0=x \geq 0, \ g_0=y \geq 0$, and
$$ a_{n+1}=\dfrac{1}{2}(a_n+g_n), \ g_{n+1}=\sqrt{a_n g_n} ,$$
then both $\{a_n\}$ and $\{g_n\}$ will converge to $AGM(x,y)$. ...

**1**

vote

**0**answers

85 views

### Distribute Monte Carlo samples among dimensions

Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...

**4**

votes

**1**answer

227 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 ...

**3**

votes

**1**answer

191 views

### Practical error-estimates for (adaptive) Newton-Cotes Quadrature

I am looking for practical error estimates for Newton-Cotes Quadrature rules.
Most books on numerical methods I have found mainly deal with theoretical error bounds/estimates for the respective ...

**4**

votes

**1**answer

390 views

### Numerical multivariate definite integration

I need to compute a set of multivariate definite integrals with infinite integration domain
$$\displaystyle \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f(x_1,x_2, \ldots , x_n)\;\;dx_1 ...

**1**

vote

**0**answers

75 views

### Integrating B-Spline composed with log

If $f$ is a real B-Spline and $a, b$ are real numbers, then is there a numerically stable way to evaluate the following expression?
$\int_a^b f (\log x) \mathrm{d}x$

**1**

vote

**2**answers

326 views

### Gauss Legendre Method for Implicit Integration

Methods that are usually adopted for time integration in transport phenomena problems are either:
Euler (explicit, first-order accurate)
$\frac{dY}{dt}=f(t,Y)$
$Y^{n+1}=Y^n+\Delta t f(t,Y^n)$
...

**1**

vote

**3**answers

302 views

### do numerical integration with fixed abscissas

Can I do an integral, possibly using gaussian quadrature, when the abscissas are fixed (for reasons that I don't want to get into right now), i.e. is it possible to calculate the weights for fixed ...

**3**

votes

**2**answers

972 views

### Best Numerical Method for Evaluating a Hilbert transform

I have to evaluate a Hilbert transform for some $\mathcal{L}^p(\mathbb{R},\mathbb{C})$-function ($1\leq p<\infty$). I know there are a number of algorithms out there to do it, but I don't have a ...

**14**

votes

**1**answer

949 views

### A mass spring model for hair simulation

A strand of hair is represented by a set of particles connected by springs.
The velocity for a particular particle is calculated implicitly using the following formula:
...

**1**

vote

**2**answers

709 views

### integration of a laplacian

Hi,
I solved for a Poisson equation with finite elements, using piecewise linear basis functions on 2d triangles.
Now, I want to evaluate the following expressions:
$$ \int_\Omega \Delta u ~dx$$
and
...

**17**

votes

**6**answers

2k views

### Why not evaluate integrals using ODE-solvers?

Hello!
I have a question about the relationship between numerical integration and the solution of ordinary differential equations (ODE). Suppose I want to evaluate the integral
$I(x) = \int_{0}^{x} ...

**10**

votes

**6**answers

3k views

### Numerical integration over 2D disk

I have a real-valued function $f$ on the unit disk $D$ that is fairly well behaved (real-analytic everywhere) and would like to find the integral $\int_D f(x,y)dxdy$ numerically. After much searching, ...