Re:Can We Solve Linear Algebra Problems at Extreme Scale and...
by mturmon on Friday September 17, 2021 @11:44AM
(#28566793)
Higham's writing is very readable and his book on numerical linear algebra is approachable and has information that's hard to find elsewhere.
His point about probabilistic error bounds for roundoff errors is excellent! The usual bounds are worst-case, and assume all roundoff errors complement each other. That doesn't happen in practice: the errors average out -- it's readily observable.
As Higham points out, this means intuitively that error magnitudes go from O(n) to O(sqrt(n)) which is huge for large n. I wasn't aware of his very recent joint work with Theo Mary that proves this under some conditions.
Re:Can We Solve Linear Algebra Problems at Extreme Scale and...
by techbio on Friday September 17, 2021 @11:54AM
(#28566949)
I've heard the phenomenon from your second paragraph called "the miracle of mitigating errors" in the context of old-school (pre-GPS) transoceanic navigation.
His point about probabilistic error bounds for roundoff errors is excellent! The usual bounds are worst-case, and assume all roundoff errors complement each other. That doesn't happen in practice: the errors average out -- it's readily observable.
As Higham points out, this means intuitively that error magnitudes go from O(n) to O(sqrt(n)) which is huge for large n. I wasn't aware of his very recent joint work with Theo Mary that proves this under some conditions.