Questions tagged [numerical-methods]
Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.
14,526 questions
4
votes
2
answers
85
views
How good is computing $\left\|f\right\|_p$ with large $p$ as a numerical approximation for $\left\|f\right\|_{\infty}$ for functions $f$?
Let's suppose that I wanted to compute $\left\|f\right\|_{\infty}=\sup_{t \in \mathcal{T}} \left|f(t)\right|$ for a $f$ that may not be easy to optimize. This is the infinity norm of a function, and ...
- 1,553
0
votes
1
answer
97
views
Numerical vs exact antiderivative using Fourier spectral method
I'm trying to compare my numerical vs exact solution to the anti-derivative of the piecewise function $$u(x)=\begin{cases}1,&x<\frac{\pi}{2}\\2,&\frac{\pi}{2}\leq x<\frac{3\pi}{2}\\1,&...
0
votes
1
answer
64
views
+200
Convergence of Chebyshev Pseudospectral Method
I am interested in how to show convergence of a numerical implementation with Chebyshev differentiation matrices and matrix-exponential time-stepping.
Consider a PDE in $\phi = \phi(t,x)$ of the form
...
- 584
0
votes
2
answers
53
views
Condition Number of $A$
Let $A = \begin{bmatrix}
1.0000 & 2.0000\\
1.0001 & 2.0000
\end{bmatrix}$. Suppose we wish to find $Ax = b$, where $b = (3.0000,3.0001)^T$. Instead of $x$, we obtain $x' = ...
- 1,171
0
votes
1
answer
68
views
Order of convergence of Newton Raphson Method (modified)
I understand the definition of the order of convergence: if $(x_n)$ is a sequence that converges to a root $r$, its order of convergence is the largest positive constant $\alpha$ such that
$$
\lim_{n\...
- 45
1
vote
0
answers
48
views
How to define the dimensionality and manifoldness of point data?
Consider the following scenario. A set of points is uniformly sampled from the surface of a sphere. Then a perturbation noise is applied to each point, displacing them in space.
As the magnitude of ...
- 3,825
5
votes
2
answers
194
views
$H^1$ functions and continuity
I’m sorry if this has already been answered; I couldn’t find a clear answer (maybe there is an indirect one and I failed to make the connection).
In the context of numerical analysis / finite elements ...
- 51
0
votes
1
answer
91
views
"Derivative" of a random process experiment of Itos Lemma
I though that the meaning of saying that the stochastic process given by $S_t = e^{0.5t + W_t}$ satisfies the SDE, $dS = dt+dW$ meant in particular that the best linear approximation to S around $t =0$...
- 211
0
votes
0
answers
30
views
How does one exactly implement exponential rosenbrock methods?
I will focus on the formulas from this paper
https://na.math.kit.edu/download/papers/rosei.pdf
Namely these sets of equations (eq 2.7a, 2.7b, the 3rd eq isn't given a number, eq 2.2b)
$$U_{ni}=u_{n}+...
- 11
0
votes
0
answers
29
views
Evaluating the error of adaptive step size methods for differential equation systems whose variables do not have the same units.
I'm trying to do a simulation run for a system of differential equations using an adaptive step size method such as Dormand-Prince or Runge-Kutta-Fehlberg.
When it comes to evaluating the error to ...
- 1
1
vote
0
answers
75
views
The simplest formula for stable, monotonic C2 interpolation
I want to interpolate $n+1$ data points $D=\{(x_0,y_0), \cdots, (x_n,y_n)\}$ by a smooth C2 function $f:\mathbb R\to\mathbb R$. Here the data is guaranteed to be strictly monotonic, that is, $x_i<...
- 381
0
votes
0
answers
62
views
Transforming a function in $\mathbb R^2$ to a signed distance function.
Assume you have an ordinary, smooth, real function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$.
These are also sometimes fall height fields as they can be represented via the notation $f(x,y) = h$ where $...
- 3,825
3
votes
2
answers
173
views
Solving a general 1st order ODE with a series expansion
I am interested in solving the general Cauchy problem:
$$\begin{cases}\frac{dx}{dt}=f(x, t) \\ x(t_0)=x_0\end{cases}$$
computationally. Of course, I know there are plenty of well-established methods ...
- 241
0
votes
1
answer
85
views
completing to 4th and 2nd power
With this formula:
$$B^{(j)}=(m+\frac{j-2}{2})_j$$
where $$(m)_j=\frac{m(m-1)...(m-j+1)}{j!}$$
see Zdeněk Kopal: Numerical Analysis, page 55,
how do I derive from $$B^{IV}-cB^{II}=\hat{f}(m),$$
this:
$...
- 3,888
0
votes
1
answer
37
views
condition number of the intersection point of two lines for infinitesimal perturbations
I am given a question that we consider two lines in $\mathbb{R}^2$ given by the equations $y=0$ and $ax+y=b$, with $a,b, \in \mathbb{R}$. We are to compute the intersection point $S$ of the two lines ...
- 1