Jacobi rotation

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In numerical linear algebra, a Jacobi rotation is a rotation, Qk, of a 2-dimensional linear subspace of an n-dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries of an n×n real symmetric matrix, A, when applied as a similarity transformation:

It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable and well-suited to implementation on parallel processors[citation needed].

Only rows k and ℓ and columns k and ℓ of A will be affected, and that A′ will remain symmetric. Also, an explicit matrix for Qk is rarely computed; instead, auxiliary values are computed and A is updated in an efficient and numerically stable way. However, for reference, we may write the matrix as

That is, Qk is an identity matrix except for four entries, two on the diagonal (qkk and qℓℓ, both equal to c) and two symmetrically placed off the diagonal (qk and qk, equal to s and −s, respectively). Here c = cos ϑ and s = sin ϑ for some angle ϑ; but to apply the rotation, the angle itself is not required. Using Kronecker delta notation, the matrix entries can be written

Suppose h is an index other than k or ℓ (which must themselves be distinct). Then the similarity update produces, algebraically,

Numerically stable computation[edit]

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 of this quantity,

If ak 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 Qk by retaining only the three values k, ℓ, and t, with t set to zero for a null rotation.

See also[edit]

References[edit]