41 votes
Accepted
How does Mathematica do symbolic integration?
An overview by one of the developers of Mathematica, focusing on definite integrals, is at Symbolic definite integration: methods and open issues. Mathematica knows all of the entries in Gradshteyn-...
- 200k
26 votes
How does Mathematica do symbolic integration?
Maple uses the Risch algorithm; see Keith Geddes and George Lebahn, Symbolic and numeric integration in Maple
- 27.1k
18 votes
How does Mathematica do symbolic integration?
People usually mention the Risch algorithm first, as other answers have. Another approach, which is surprisingly successful, is to do what you or I would when solving integrals: look for patterns for ...
- 931
11 votes
How does Mathematica do symbolic integration?
One approach to symbolic integration is using representations as Holonomic functions (https://en.wikipedia.org/wiki/Holonomic_function): solutions to differential equations of the form: $$p_r f^{(r)} +...
- 489
9 votes
Accepted
Numerical integration method that doesn't involve derivative in the error bound
If $f$ is of bounded variation $V(f)$, then Koksma's inequality (Theorem 5.1 in Kuipers & Niederreiter, Uniform Distribution of Sequences) says $$ \left|{1\over N}\sum_{n=1}^Nf(x_n)-\int_0^1f(t)\,...
- 40.7k
7 votes
A numerical calculation for an integral
Nemo's representation of $F(\eta)$ in terms of a hypergeometric function can be evaluated without difficulty for large $\eta$: $$F(\eta)=\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; ...
- 200k
6 votes
Applications of Fourier Transforms in Number Theory
Do you know the Riemann zeta function ? $\displaystyle\frac{\log \zeta(\sigma+2i\pi \xi)}{\sigma+2i\pi \xi}$ is the Fourier transform of the prime counting function $\displaystyle J(e^u) e^{-\sigma u}...
- 3,434
5 votes
Numerical integration method that doesn't involve derivative in the error bound
$\newcommand\Z{\mathbb Z}\newcommand\R{\mathbb R}\newcommand\la{\lambda}$If $f$ is an arbitrary Lebesgue-integrable function, then, as is done in a definition of the Lebesgue integral, it makes sense ...
- 142k
4 votes
Intractability of an integral by deterministic numerical methods
For small $n$ Monte Carlo integration is not needed. For $n$ up to 100 see Kolmogorov-Smirnov Tests when Parameters are Estimated with Applications to Tests of Exponentiality and Tests on Spacings (...
- 200k
4 votes
Accepted
Any convergence rule for ${\mathbf X}_k={\mathbf A}{\mathbf X}_{k-1}{\mathbf B}$?
For every square matrix $C$, let $r(C)$ denote its spectral value. We say that a complex number $\lambda$ is a dominant eigenvalue of $C$ if $\lambda$ is the only eigenvalue of $C$ with modulus $r(C)$....
- 13k
4 votes
Expected number of lines meeting four given lines or "what is 1.72..."
If anyone is still interested, Antonio Lerario and I recently published a paper: Probabilistic Schubert Calculus: Asymptotics, in which we give a more convenient formula to compute this number. What ...
- 41
4 votes
Fourier series of $e^{(\cos(\pi x) - m)^2}$
For large $s$ a single sum in terms of a hypergeometric function may be useful, $$c_{p}=\frac{(-1)^p}{2^p p!}\sum_{n=0}^\infty\frac{(n)_p}{(2s)^{n}n!}(m+1)^{n-p} {}_2F_1\bigl(p+1/2,p-n,2p+1,2/(m+1)\...
- 200k
3 votes
Integration algorithm and analytic property
The article gives an existence proof of an algorithm, but does not say anything about effectively using that algorithm. In practice, given an arbitrary real analytic function $f$ one cannot determine ...
- 41.7k
3 votes
Why exactly is Simpson's rule better than the Trapezoidal rule?
This is a textbook question. The Simpson rule is exact (no error at all) when $f$ is a polynomial of degree $\le3$, while the trapezoidal rule is exact only for degree $\le1$ (affine functions). The ...
- 53.1k
3 votes
Error in Gauss-Laguerre numerical quadrature scheme
The function $f(x)=e^{-x+\sqrt{x}}$ belongs to the space $C_{0}^{3}[0,\infty)$ defined, for $q\geq p\geq0$, by $$C _ { p } ^ { q } [ 0,\infty ) : = \{ f \in C ^ { p } [ 0,\infty ) \cap C ^ { q } ( 0,\...
- 4,294
3 votes
Gauss quadrature for products of multilinear functions on a simplex
It looks to me like you are searching for what are called monomial cubature rules on the simplex in the literature on numerical integration. As an example, a 4th-degree monomial in 2-D $(x,y)$ would ...
- 131
3 votes
Accepted
Gaps between roots of consecutive Hermite polynomials
The answers are yes and yes. Consider the Hermite Gauss functions : $\psi_n(x)=e^{-x^2/2}H_n(x) $, we have two properties: $$\psi_n''(x)+(2n+1-x^2)\psi_n(x)=0 $$ and $$\psi_{n+1}=\psi_n'(x) +x\psi_n(...
- 4,491
2 votes
Accepted
Numerical methods for IDE
I must premise that I am not a specialist in numerical analysis, therefore I may be not right when talking about more popular methods in this field pertaining the solution of IDEs. Said that, however, ...
- 6,830
2 votes
Accepted
On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$
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{...
- 142k
2 votes
Accepted
How to integrate the $L^2$ function $1/|x|$ numerically
Following Michael Renardy's suggestion, write $g=\phi\psi$, and write $$ g(x)=g(0)+x\cdot G(x) , $$ where $G$ is a smooth function. Then the integral becomes $$ \int_\Omega fg = g(0)\int_\Omega f + \...
- 3,372
2 votes
Accepted
Quadrature methods for high-dimensional Gaussian integration
You may want to use a stochastic algorithm. Entry points to the literature (which is large) could be A stochastic algorithm for high-dimensional integrals over unbounded regions with Gaussian weight (...
- 200k
2 votes
Accepted
Integrating a B-Spline basis function with respect to the standard normal PDF
Since a B-spline is a piecewise polynomial function, the question is whether there exists an exact equality formula for the integral $\int_{-a}^{b}u^pe^{-u^2/2}du$. This integral equals an elementary ...
- 200k
2 votes
Why exactly is Simpson's rule better than the Trapezoidal rule?
This question is likely to be closed as not research level. In any case, the point is that one typically is not interested in numerically integrating functions about which nothing whatsoever is known (...
- 3,751
1 vote
Understanding quadrature rule of a function multiplied by another $C^{\infty}$ function
If I understand your question correctly, we could have $g(x_j) = 0$ for all $j = 1, \ldots, n$. Thus, $Q_n[f g] = 0$. Further, if $f$ is a low degree polynomial, the right-hand side of (1) vanishes. ...
- 1,819
1 vote
Accepted
Numerical integration with integrable singularity
(i) Your purported error bound is of course incorrect: consider e.g. $f(t)=t$. (ii) To get rid of the singularity, make the substitution $u=\sqrt t$, so that $$\int_0^T dt\,f(t)=\int_0^T \frac{dt}{\...
- 142k
1 vote
Numerical method with rational nodes and weights to compute exact value of definite integral?
It can be done with $n$ abscissae for odd $n$, though I'm not sure it can always be done with positive weights. Take $a=-1$, $b=1$ without loss of generality. I'll do $n=5$. Consider $$I=w_0(f(-x_0)+...
- 38.3k
1 vote
Approximation for a Bessel function integral
Q: The OP seeks a "reasonable approximation" for large or small $\rho_0$ of the function $$P(x)= \int_0^{x} e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0)\,d\rho,\;\;x\geq 0.$$ Consider the ...
- 200k
1 vote
Accepted
Numerical solution to some functional equation
$\newcommand\erf{\operatorname{erf}}\newcommand\R{\mathbb R}$The functional equation in question is \begin{equation*} a=F(a) \tag{1}\label{1} \end{equation*} on $(0,\infty)$, where $a$ is in the ...
- 142k
1 vote
Accepted
What is the minimum number of stages $s$ required for a Runge-Kutta type numerical method of given order $p$?
For implicit methods, you can achieve order $2s$ with $s$ stages. Note that this result is the same if one considers the simpler problem of numerical integration (quadrature). Update as of 2024: a 16-...
- 1,253
1 vote
Accepted
Proving that $\lim_{j \to i} Z_{ij} = [\ln(\frac{\Delta s_i}{2})-1]\Delta s_i$
You want the relation between $\phi(x)$ evaluated at a point $x$ on the boundary and the derivative $\partial\phi/\partial n$ evaulated at the same point $x$, which is given by $$\phi(x)=\frac{\...
- 200k
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