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Hot answers tagged numerical-integration

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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 the entries in Gradshteyn-...
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How does Mathematica do symbolic integration?

Maple uses the Risch algorithm; see Keith Geddes and George Lebahn, Symbolic and numeric integration in Maple
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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 ...
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Expected number of lines meeting four given lines or "what is 1.72..."

The integrand is periodic modulo $\pi$ in each variable, so it suffices to integrate each variable over $[0, \pi]$ and replace the constant factor by $2^{-7}$. If we were to apply a change of ...
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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 (...
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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 $r(C)$...
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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 ...
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Real world example of use of Monte Carlo method for high dimensional integrals

Here are a few papers that discuss high-dimensional Monte Carlo integrals, together with quotes from Math Reviews. MR2719643 (2011i:65038) Griebel, Michael; Holtz, Markus; Dimension-wise ...
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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, ...
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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 T_n(f)(x)=T_{n,[0,x]}(f)=x\sum_1^n w_i f(xx_i), \tag{*} \end{...
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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 ...
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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 ...
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Finding numerical solution for nonlinear Poisson-like equation using finite difference method

Since it is a non-linear differential equation you cannot expect to obtain a linear system at the end. Think about using a non-linear solver like a Newton solver instead.
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Gaussian quadrature, with no exact result over polynomial, but on inverse functions

The book Stroud A. H., Secrest D., "Gaussian Quadrature Formulas". Prentice-Hall, Englewood Cliffs, N.J., 1966 gives some answer to my question. See Section 3.2.2 "Finite to semi-...
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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-...
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