Eigenvectors

[jte]

(12-27-2018 04:32 PM)parisse Wrote:  ⋮
approx algorithms should not be the same as exact algorithms.
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numeric algorithms do not follow the same path as exact algorithms for eigenvalues/eigenvectors computation. This is also true for many other algorithms.

Very true!

There’s no end to the fun one can have designing and implementing algorithms. As well as considering the (ill- / well-)conditioning of a problem, one can also consider the arithmetics used in implementation (e.g., how round-off or range limits in an employed floating-point arithmetic will interplay with the numerical computations carried out by the algorithm).

When I was adding the Sketch feature to the Function Plot view, I had concerns over the use of the already-present floating-point arithmetics when solving 3x3 and 4x4 matrix equations. Since I had a full plate of things to work on, I added higher-precision signed-magnitude arithmetics (1 sign bit + 128 binary digits and 1 sign bit + 160 binary digits, for just a few arithmetic operations) to rule out catastrophic cancellation in sums. (The extended precision arithmetics allowed for absolutely no round-off to be introduced until very near the end of calculation chains, where bounding it was entirely straightforward.)