Questions tagged [approximate-integration]
Use this tag for questions related to approximate integration, which constitutes a broad family of algorithms for calculating the numerical value of a definite integral.
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Most efficient quadrature formula
On the integral I am looking to obtain an approximation with machine precision, I've thought about applying a Gauss-Jacobi formula or Gauss-Legendre for:
$$ \int_{0}^{2}\sqrt{2-x}dx $$
I am not sure ...
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Numerical integration with modified Bessel function of second kind
I am working with the so-called screened Poisson PDE, whose solutions in two-dimensions are given in terms of the modified Bessel function of the second kind, $K_0$, for Dirichlet boundary conditions ...
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Integrating products of many oscillating functions
I'm working with the circular random matrix ensembles, in particular the circular unitary ensemble (CUE). For unitaries drawn from the CUE with dimension $N$, the distribution of its eigenvalues is ...
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Prove that $ \int_0^1 x(\sec x)^2 \,dx < 1.$
Prove that $ \int_0^1 x(\sec x)^2 \,dx < 1.$
I tried a few combinations but nothing seems to work, as the integral is quite close to 1 itself. And question strictly inhibits use of calculators to ...
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Stationary phase approximation, compact support needed
I am trying to understand the stationary phase approximation as described on Wikipedia https://en.wikipedia.org/wiki/Stationary_phase_approximation. As necessary condition, they mention a compact ...
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Applications of highly oscillatory integrals
I was reading a series of articles on numerical integration of highly oscillatory functions, e.g.,
S. Olver, Numerical approximation of highly oscillatory integrals
S. Xiang, H. Wang, Fast ...
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Asymptotic expansion of $\int_0^1 e^{x(tp-\frac{1}{2}t^2)}dt$ as $x\to\infty$ for different $p$
Let $p\in \mathbb{R}$, I would like to investigate the asymptotic behavior of the following integral:
$$\int_0^1 e^{x(tp-\frac{1}{2}t^2)}dt$$
as $x\to\infty$. In particular, I would like to know how ...
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Numerical method to integrate an exponentiated polynomial
Let $P_{n}$ be the set of polynomials of degree $n$. Then if $f \in P_{2n + 1}$ we can compute $$\int_{-1}^1 f(x) dx$$ precisely using $n$-point Gauss-Legendre quadrature.
I am interested in computing ...
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Huge error bounds for midpoint rule in calculating integral
[the value of K is 195]. 1I have this problem an integration approximation problem of: $$\int_0^{4\sqrt{\pi}}\sin(x^2)dx$$ with n = 4. The result is 5.01. But when I check the error bound using the ...
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Midpoint rule integration for a matrix-vector product
Consider a function $F=F(q)$ which is a symmetric, positive-definite matrix form of dimensions $n\times n$, where $q \in \mathbb{R}^n$ and $q=q(t)$.
When applying the midpoint integration rule on $F$, ...
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Is the Monte Carlo integration method(s) actually a viable way to accurately integrate functions?
Recently, I was playing around with some Monte Carlo simulations using Python to evaluate integrals of functions such as f(x) = x(1-x)sin²[200x(1-x)].
I am aware that Monte Carlo integration methods ...
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How to integrate arbitrary discrete-time linear and angular body fixed velocity to world space?
I have body fixed angular velocity values and linear acceleration values streaming in to my application. at some interval $\delta t$.
I need to get a world position from these, assuming the start ...
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Iterative formula for the integration of autonomous differential equations
Consider the following autonomous differential equation:
$$y'=f(y)$$
where $y=y(t)$ with $y_0 = y(0)$.
Let us suppose that $y^{[0]}(t)$ is a good approximation for $y(t)$. Using this function in the ...
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Do numerical integration methods ever utilize evaluations of the derivative of the function?
Suppose one is performing numerical integration of some function $f(x)$, but in addition to being able to evaluate its value at points $f(x_1), \dots, f(x_n)$, one also can additionally obtain its ...
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Proof that the Trapezoidal Method for solving IVP is $O(h^3)$
My question comes from Chapter 6 of the book Introduction to Numerical Analysis, 2nd edition, by Kendall Atkinson. In section 6.5, p.368-369, the author proves that the Trapezoidal method for solving ...
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