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
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