Acoustic wave

In numerical linear algebra, a Jacobi rotation is a rotation, Q_kℓ, 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:

$A\mapsto Q_{k\ell }^{T}A{Q}_{k\ell }=A^{\prime }.$

$\left[\begin{array}{ccccccc}*& & & \cdots & & & *\\ & \ddots & & & & & \\ & & a_{kk}& \cdots & a_{k\ell }& & \\ ⋮& & ⋮& \ddots & ⋮& & ⋮\\ & & a_{\ell k}& \cdots & a_{\ell \ell }& & \\ & & & & & \ddots & \\ *& & & \cdots & & & *\end{array}\right]\to \left[\begin{array}{ccccccc}*& & & \cdots & & & *\\ & \ddots & & & & & \\ & & a{\text{'}}_{kk}& \cdots & 0& & \\ ⋮& & ⋮& \ddots & ⋮& & ⋮\\ & & 0& \cdots & a{\text{'}}_{\ell \ell }& & \\ & & & & & \ddots & \\ *& & & \cdots & & & *\end{array}\right].$

It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable and well-suited to implementation on parallel processors Template:Fact.

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 Q_kℓ 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

$Q_{k\ell }=\left[\begin{array}{ccccccc}1& & & & & & \\ & \ddots & & & & 0& \\ & & c& \cdots & s& & \\ & & ⋮& \ddots & ⋮& & \\ & & -s& \cdots & c& & \\ & 0& & & & \ddots & \\ & & & & & & 1\end{array}\right].$

That is, Q_kℓ is an identity matrix except for four entries, two on the diagonal (q_kk and q_ℓℓ, both equal to c) and two symmetrically placed off the diagonal (q_kℓ and Q_ℓk, 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

$q_{ij}={\delta }_{ij}+({\delta }_{ik}{\delta }_{jk}+{\delta }_{i\ell }{\delta }_{j\ell })(c-1)+({\delta }_{ik}{\delta }_{j\ell }-{\delta }_{i\ell }{\delta }_{jk})s.$

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

$a{\text{'}}_{hk}=a{\text{'}}_{kh}=c{a}_{hk}-s{a}_{h\ell }$

$a{\text{'}}_{h\ell }=a{\text{'}}_{\ell h}=c{a}_{h\ell }+s{a}_{hk}$

$a{\text{'}}_{k\ell }=a{\text{'}}_{\ell k}=(c^2-s^2)a_{k\ell }+sc(a_{kk}-a_{\ell \ell })=0$

$a{\text{'}}_{kk}=c^2{a}_{kk}+s^2{a}_{\ell \ell }-2sc{a}_{k\ell }$

$a{\text{'}}_{\ell \ell }=s^2{a}_{kk}+c^2{a}_{\ell \ell }+2sc{a}_{k\ell }.$

Numerically stable computation

To determine the quantities needed for the update, we must solve the off-diagonal equation for zero Template:Harv. This implies that

$\frac{c^2-s^2}{sc}=\frac{a_{\ell \ell }-a_{kk}}{a_{k\ell }}.$

Set β to half of this quantity,

$\beta =\frac{a_{\ell \ell }-a_{kk}}{2a_{k\ell }}.$

If a_kℓ 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

$t^2+2\beta t-1=0.$

For stability we choose the solution

$t=\frac{\mathrm{sgn}(\beta )}{\left|\beta \right|+\sqrt{{\beta }^2+1}}.$

From this we may obtain c and s as

$c=\frac{1}{\sqrt{t^2+1}}$

$s=ct$

Although we now could use the algebraic update equations given previously, it may be preferable to rewrite them. Let

$\rho =\frac{s}{1+c},$

so that ρ = tan(ϑ/2). Then the revised update equations are

$a{\text{'}}_{hk}=a{\text{'}}_{kh}=a_{hk}-s(a_{h\ell }+\rho a_{hk})$

$a{\text{'}}_{h\ell }=a{\text{'}}_{\ell h}=a_{h\ell }+s(a_{hk}-\rho a_{h\ell })$

$a{\text{'}}_{k\ell }=a{\text{'}}_{\ell k}=0$

$a{\text{'}}_{kk}=a_{kk}-t{a}_{k\ell }$

$a{\text{'}}_{\ell \ell }=a_{\ell \ell }+t{a}_{k\ell }$

As previously remarked, we need never explicitly compute the rotation angle ϑ. In fact, we can reproduce the symmetric update determined by Q_kℓ by retaining only the three values k, ℓ, and t, with t set to zero for a null rotation.

See also

• Givens rotation

References

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