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Fast Givens rotations seem like a nice upgrade from standard Givens rotations. They take fewer multiplications to apply and avoid the calculation of a square root.

They are, however, given very little focus in the recent literature. They appear as an exercise in Golub and Van Loan. They get a short section in Gentle but, as far as I can tell, they are unmentioned entirely in Davis. Since Davis is very focused on efficient calculation this is surprising.

Is there some issue with fast Givens rotations? Do they not give performance advantages on modern computer architectures?

References:

Davis, T., 2006. Direct methods for sparse linear systems. Philadelphia: Society for Industrial and Applied Mathematics.

Gentle, J., 2007. Matrix Algebra. New York: Springer.

Golub, G. and Van Loan, C., 2013. Matrix computations. 4th ed. Baltimore: Johns Hopkins Univ Press.

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    $\begingroup$ When I experimented with fast Givens rotations before, I encountered a lack of numerical robustness. That may have been due to flaws in my implementation rather than a fundamental problem, but I did not pursue it further and instead focused on the efficient computation of rhypot(a,b) = $\frac{1}{\sqrt{a^2 + b^2}}$. From an efficiency standpoint, the performance of sparse systems processing is dominated by memory access, not the actual computation. $\endgroup$
    – njuffa
    Jan 2 at 20:20

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