Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.
18
votes
0answers
282 views
Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field
If $K$ is a number field, whose Galois closure over the rationals has degree 24 or so, and whose discriminant is around $163^4$, then what is a numerically efficient way of computing the first few ...
7
votes
0answers
168 views
Can I trust this numerical triple integral from Matlab?
Math people:
I am a novice at numerical methods and Matlab. I am attempting to evaluate the following sum of two triple integrals (it can obviously be written more simply, but you still cannot ...
5
votes
0answers
124 views
Trying to solve $\lambda^3 - 3.250\lambda^2 + \lambda - 0.063 = 0$ using Newton-Raphson method
This is what I've atempted so far in solving $\lambda^3 - 3.250\lambda^2 + \lambda - 0.063 = 0$. The following are the steps:
step 1: $f(\lambda) = \lambda^3 - 3.250\lambda^2 + \lambda - 0.063 $
...
5
votes
0answers
623 views
Restricted Three-Body Problem
The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
4
votes
0answers
145 views
How can I solve the Poisson PDE efficiently and fast in cylindrical coordinates?
I am trying to numerically solve the Possion PDE in cylindrical coordinate system.
$$\Delta f = {1 \over \rho} {\partial \over \partial \rho} \left(\rho {\partial f \over \partial \rho} \right) + {1 ...
4
votes
0answers
71 views
Inexact Newton method.
Let's a nonlinear function
$
f:[-1,+1]^N\subset\mathbb{R}^N\to\mathbb{R}^N,\; N\in\mathbb{N},
$
such that the the sequence generated by the method of Newton-Raphson
$$
...
4
votes
0answers
198 views
Simulating from a Multivariate Gaussian without Cholesky
I'd like to draw a sample from a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ is the mean vector (can assume it to be $\boldsymbol{0}$), and $\Sigma$ is a sparse positive ...
4
votes
0answers
241 views
Find roots of sum of sinusoids
Given this function and an initial point, find the next root:
$$
\begin{align}
f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - ...
4
votes
0answers
269 views
An optimization problem involving orthogonal matrices
Let $X\in\mathbb{R}^{3\times 3}$ be an orthogonal matrix. Then $\mathrm{vec}X\in\mathbb{R}^9$ is a 9 by 1 vector formed by stacking the columns of the matrix $X$ on top of one another. Given a matrix ...
4
votes
0answers
219 views
Find the error approximation for the function $f(x) = \dfrac{1}{1-x}$
Problem
Let $f(x) = \dfrac{1}{1-x}$, find the Taylor polynomial $P_n(x)$ about $x_0 = 0$. Find a value of $n$ such that the approximation is within $10^{-6}$ on $[0, 0.5]$.
To find $P_n(x)$ is ...
4
votes
0answers
189 views
Computing complex principal value integral - sgn-function?
I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$)
$$ PV ...
4
votes
0answers
560 views
Convergence of Gauss-Newton method for piecewise linear functions
Notation for Gauss-Newton method
Non-linear least squares problems are often solved by the Levenberg-Marquardt algorithm, which can be viewed as a Gauss–Newton method using a trust region approach.
...
4
votes
0answers
205 views
Evaluating matrix-continued fractions?
I have a matrix-valued continued fraction defined in the following way: $\alpha_n$ and $\beta_n$ are matrices, and I am interested in the quantity
$A_1$, where all the $A_n$, $n = 1, 2, \dots$ are ...
4
votes
0answers
103 views
Shintani cone zeta function
Is there a procedure/algorithm for calculating sums of the form
$$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$
where
$$ L_i(n_1,\ldots, n_r) ...
4
votes
0answers
180 views
Block Toeplitz Matrix
What is the name of the following matrix:
$$
\begin{pmatrix} a & b & 0 \\
c & d & 0 \\
0 & a & b\\
0& c& d& \\
b & 0 & a \\
d & 0 & c\end{pmatrix}\
$$
It looks like a Block Toeplitz matrix, but ...
