Yes, that's a great algorithm, I use it too sometimes. Michael Saunders was one of my dissertation readers. Wickedly smart man, and more importantly, his software *works*.
I agree with a previous poster who said it is unfair to compare this algorithm to Gaussian Elimination. Frankly, it seems to me that the poster has taken enough numerical analysis to know that GE is O(n^3), but is not familiar with the much wider body of research into efficient methods for solving structured linear systems.
Symmetric diagonally dominant linear systems are among the best conditioned systems you could possibly construct, which means that it is a rare case that you would ever use GE. Instead, you would use an iterative method, such as conjugate gradients or a stationary method like the Chebyshev method discussed in the paper. While it does seem that they are proposing a novel way to construct an efficient iterative method, the truth is that once you've entered the realm of iterative methods it's not likely there is going to be a single approach that works best in every instance. Even the paper's author agree that this is a "conceptually simple and possibly practical iterative solver." They rightly recognize that what works well on paper may not always prove useful in practice. So I think we should take the authors' own advice and wait for further research into this approach.
eldavojohn writes: "While nothing interesting was found by most scientific journals..."
As evidence he provides a reference to: the statement by a single journal. Surely that is not "most journals", is it? Where is the evidence that most journals have even commented on the story, much less rendered a verdict as to its seriousness?
To be fair, the statement might well be true, in the sense that "most scientific journals" have not issued any statement on the matter. And even if they did so, in the short period of time that has transpired, it could only represent the views of the editors, not the body of researchers that contributes to it.
So what we have here seems to be the gross magnification of one statement to reflect a broad consensus.
I don't believe they found the first trillion congruent numbers; rather, they tested the first trillion whole numbers for congruency.
The key point you've glossed over is the measurement "200CK". How much is 200CK? It means that the substance has undergone 200 100-to-1 dilutions. That means that the concentration has been reduced from full strength by a factor of 100^200. Yes, that's right---10^400. According to this article in Wikipedia, the number of observable atoms in the observable universe is approximately 10^80. Clearly, you will be the luckiest person alive, 10^40 or so times over, if even one atom of the active ingredient is left in your sugar pill.