The Kaczmarz method is an iterative algorithm used to solve systems of linear equations, particularly useful in tomography and image reconstruction. It's a row-action method, which means it iterates through the rows of the matrix, updating the solution vector at each step.
In that regard, the Archimedean spiral as an example, could be an optimized version as it is a 2D spiral, which is one of the simplest and most commonly studied spirals.
For example Let; $Ax = b$ be a system of linear equations, $m$ be the number of rows of A, and $a_i$ be the $i$th row of complex-valued matrix $A$, and let $x^0$ be arbitrary complex-valued initial approximation to the solution of $A x = b$. For $k = 0$ , $1$ , …
the method is computed as such:
$$x^{(k+1)} = x^{(k)} + \frac{b_i - \langle a_i, x^{(k)} \rangle}{\|a_i\|^2} a_i$$
Why the Archimedean Spiral?
The polar equation of an Archimedean spiral is given by: $$r=a+bθ$$
where:
r is the radius.
θ is the angle in radians.
a and b are constants.
To find the arc length S of an Archimedean spiral from $θ = θ_0$ to $θ = θ_1$, the integral is:
$$S=\int_{0= θ_0}^{0 = θ_1}\sqrt{r^2+r'^2}(drdθ)$$
For the Archimedean spiral, $$dr/dθ=b,$$
so:
$$S=\int_{0=θ_0}^{0=θ_1}\sqrt{b^2+(a+bθ)^2}(drdθ)$$
Here, the Spiral-Based Approach could potentially be used for spatial sampling problems, while not typically used for solving linear equations directly. It could be useful in a different context, such as path planning or spiral scanning in image processing.
Conceptual Outline:
1-Starting with an initial guess $x^{(0)}$
2- then the iterative update at each iteration $k$ by selecting the row index $i = (k \mod m) + 1$
3- then computing the standard Kaczmarz update $y$ $$ y = x^{(k)} + \frac{b_i - \langle a_i, x^{(k)} \rangle}{\|a_i\|^2} a_i$$
4- then Apply a rotational transformation to $y$ $$x^{(k+1)} = R_{\theta_k} (y - x^{(k)}) + x^{(k)}$$
where $R_{\theta_k}$ is the rotation matrix by angle $θk$ (which changes with $k$).
The rotation
For a 2D spiral, the rotation matrix $R_{\theta_k}$ is given by: $$ R_{\theta_k} = \begin{pmatrix} \cos(\theta_k) & -\sin(\theta_k) \\ \sin(\theta_k) & \cos(\theta_k) \end{pmatrix} $$
where $R_{\theta_k}$ can be a small, gradually increasing angle to induce the spiral effect.
Incorporating a spiral approach can be beneficial for certain problems where a simple linear approach may not converge efficiently or may get stuck in local minima. Adjusting the angle $θk$ and the frequency of rotation would need to be tuned based on the specific problem characteristics.
My question relates to the bigger picture though, do this spiral approach make sense? is it potentially useful?
