Numerically Stable Computation
To determine the quantities needed for the update, we must solve the off-diagonal equation for zero (Golub & Van Loan 1996, §8.4). This implies that
Set β to half this quantity,
If akℓ is zero we can stop without performing an update, thus we never divide by zero. Let t be tan ϑ. Then with a few trigonometric identities we reduce the equation to
For stability we choose the solution
From this we may obtain c and s as
Although we now could use the algebraic update equations given previously, it may be preferable to rewrite them. Let
so that ρ = tan(ϑ/2). Then the revised update equations are
As previously remarked, we need never explicitly compute the rotation angle ϑ. In fact, we can reproduce the symmetric update determined by Qkℓ by retaining only the three values k, ℓ, and t, with t set to zero for a null rotation.
Read more about this topic: Jacobi Rotation
Famous quotes containing the words numerically, stable and/or computation:
“I believe that the miseries consequent on the manufacture and sale of intoxicating liquors are so great as imperiously to command the attention of all dedicated lives; and that while the abolition of American slavery was numerically first, the abolition of the liquor traffic is not morally second.”
—Elizabeth Stuart Phelps (1844–1911)
“Man is not merely the sum of his masks. Behind the shifting face of personality is a hard nugget of self, a genetic gift.... The self is malleable but elastic, snapping back to its original shape like a rubber band. Mental illness is no myth, as some have claimed. It is a disturbance in our sense of possession of a stable inner self that survives its personae.”
—Camille Paglia (b. 1947)
“I suppose that Paderewski can play superbly, if not quite at his best, while his thoughts wander to the other end of the world, or possibly busy themselves with a computation of the receipts as he gazes out across the auditorium. I know a great actor, a master technician, can let his thoughts play truant from the scene ...”
—Minnie Maddern Fiske (1865–1932)