All Questions
Tagged with integration numerical-methods
1,043
questions
-1
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answers
39
views
How to compute Contour Integral Numerically
So I am trying to compute the integral from $0$ to $a$ on the real axis, shown in the picture, for a function that is completely analytic on the upper half plane except at $E + i\epsilon$ where $\...
- 9
0
votes
0
answers
50
views
Approximation of a Riemann sum.
Given a twice continuously differentiable function $f\in C^2([0,1])$, is there a theorem/result/algorithm on how to place $0<x_1<\ldots<x_{n-1}<1$ so that adding $x_0=0$ and $x_n=1$,
$$
\...
- 208
0
votes
0
answers
32
views
Proof of Non-Exactness for Polynomials of Degree p in Quadrature Formulas
Given a Quadrature Rule
$$ \int_{a}^{b} f(x)dx \approx (b-a) \sum_{k=1}^s b_k f(a+c_k(b-a))$$
of order $p$
$$\frac{1}{q} = \sum_{k=1}^s b_k^{q-1} for\:all\: q=1,...,p\:, but\:not\:q=p+1$$
Show that ...
- 117
4
votes
2
answers
261
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Could we approximate $\int_0^1\frac{1}{x^4}dx$ using a Riemann sum?
We know that in one dimension, the integral $\int_{0}^{1}\left(1/x^{4}\right){\rm d}x$ is not finite. But could we approximate this integral using a Riemann Sum ?.
...
- 329
0
votes
0
answers
51
views
How to accelerate calculation of a nested integral
Setup
Let $A(t)$ and $B(t)$ be positive functions defined on $t \in [t_1, t_2]$. They are sampled uniformly within this interval, with a timestep of $\Delta t$.
Question
I want to calculate the ...
- 2,310
2
votes
0
answers
30
views
Expected value of mean reverting spot price - triple numerical integral
I have been reading this paper: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=7d38b03cfc62a15bdfd755c793d4e70a821725cc and having trouble trying to implement the expected ...
- 21
0
votes
0
answers
38
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Line integral of an exponential
Let $V_1, V_2\in \mathbb{R}^2$ be the vertices of a segment $s$, that is
\begin{equation}
s\triangleq\{z(\alpha) = (V_2-V_1)\,\alpha + V_1 \}_{\alpha\in[0,1]}
\end{equation}
Now let $y\in\mathbb{R}^2$ ...
- 729
0
votes
0
answers
48
views
Consistency of Runge-Kutta methods
Consider the Runge-Kutta method given by
\begin{equation*}
y_{n+1} = y_n + \Delta t \phi(t_n,y_n,\Delta t),
\end{equation*}
with
\begin{equation*}
\phi(t_n, y_n, \Delta t) = \sum_{i=1}^s b_i ...
0
votes
0
answers
56
views
How to do this integral numerically in $O(N)$?
Consider a region $V$ with finite volume, and bounded smooth function $g: \mathbb R^3\to\mathbb R$ and a kernel $K$, and define $f: \mathbb R^3\to\mathbb R$ by:
$$
f(\mathbf x) = \int_V K(\mathbf x, \...
- 103
0
votes
1
answer
122
views
Numerical integration with variation of parameters
I'm having a lot of trouble with a numerical integration problem for an astrodynamics course. We're starting with the function for a perturbed oscillator
$$ \ddot{x} + 3 x + x^3 = 0 $$
whose ...
- 1
1
vote
1
answer
53
views
Lagrange interpolation and orthogonal polynomials
Suppose that $\{p_i(x)\}_{i=0}^{n}$ are pairwise orthogonal polynomials on the interval $[a,b]$, It means,
$$
\int_{a}^{b} p_i(x)p_j(x)dx = 0\ , \;{i\neq j}
$$
wherein $p_i(x)$ for all $i$ is a ...
- 188
0
votes
1
answer
47
views
Two equivalent statements yielding different values when evaluated?
Assume $v_i$ and $v_j$ are tent functions.
We have the expression:
$$ \int^b_a v_iv_j'' dx $$
Integrating by parts gives:
$$ \int^b_a v_iv_j'' dx = -\int^b_a v'_j v_i' dx + [v_iv_j']^b_a $$
Now let'...
- 3,439
0
votes
0
answers
25
views
Figuring out simple quadrature formula from composite formula
Introduction
I'm given this quadrature formula:
$$\int_a^b f(x)dx = h \sum_{i=0}^{n-1}\left[ f(x_i) + \alpha h f'(x_i) + \beta h^2 f''(x_i) \right] = Q_n(a,b;f)$$
where
$$h = \frac{b-a}{n}, \ x_i = a +...
- 21
2
votes
1
answer
69
views
Approximation of $\int_0^{\infty} e^{-bx^2}sin(ax^2)dx$ when a>>b
This is from an exercise in Migdal's "Qualitative Methods in Quantum Theory".
For the case for where b>>a, we can arrive at an estimate by re-writing the integral in the following way:
...
- 73
1
vote
0
answers
55
views
Numerical Integration of Bessel K_0 and Hankel function.
I am trying to perform a numerical integration of a large integrand which includes the K_0 or the Hankel function as a part. I began with the study of behavior of K_0 and Hankel function alone and ...
- 275