The Energy of Bidiagonal Matrices – Nick Higham

Maybe probably the most fascinating result’s that if we’ve a factorization of a matrix right into a product of bidiagonal matrices then we are able to compute the situation quantity in flops and to excessive accuracy in floating-point arithmetic. Each matrix has such a factorization and in some instances the entries of the elements are often called specific formulation. For instance, we are able to compute the situation variety of the Hilbert matrix to excessive accuracy in flops.

		Relative error for quick algorithm
4	2.84e4	1.28e-16
8	3.39e10	2.25e-16
16	5.06e22	3.67e-17
32	1.36e47	1.75e-15
64	1.10e96	1.77e-15

Any try to compute by explicitly forming is doomed to failure by the rounding errors incurred within the formation, it doesn’t matter what algorithm is used for the computation.

All this evaluation, and way more, is contained in

• Nicholas J. Higham, The Energy of Bidiagonal Matrices, ArXiv:2311.06609, November 2023.

which relies on the Hans Schneider Prize speak that I gave at The twenty fifth Convention of the Worldwide Linear Algebra Society, Madrid, Spain, June 12-16, 2023.