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:

$A\mapsto Q_{k\ell }^{T}AQ_{k\ell }=A'.\,\!$

${\begin{bmatrix}{*}&&&\cdots &&&*\\&\ddots &&&&&\\&&a_{kk}&\cdots &a_{k\ell }&&\\\vdots &&\vdots &\ddots &\vdots &&\vdots \\&&a_{\ell k}&\cdots &a_{\ell \ell }&&\\&&&&&\ddots &\\{*}&&&\cdots &&&*\end{bmatrix}}\to {\begin{bmatrix}{*}&&&\cdots &&&*\\&\ddots &&&&&\\&&a'_{kk}&\cdots &0&&\\\vdots &&\vdots &\ddots &\vdots &&\vdots \\&&0&\cdots &a'_{\ell \ell }&&\\&&&&&\ddots &\\{*}&&&\cdots &&&*\end{bmatrix}}.$

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

$Q_{k\ell }={\begin{bmatrix}1&&&&&&\\&\ddots &&&&0&\\&&c&\cdots &s&&\\&&\vdots &\ddots &\vdots &&\\&&-s&\cdots &c&&\\&0&&&&\ddots &\\&&&&&&1\end{bmatrix}}.$

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,

$a'_{hk}=a'_{kh}=ca_{hk}-sa_{h\ell }\,\!$

$a'_{h\ell }=a'_{\ell h}=ca_{h\ell }+sa_{hk}\,\!$

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

$a'_{kk}=c^{2}a_{kk}+s^{2}a_{\ell \ell }-2sca_{k\ell }\,\!$

$a'_{\ell \ell }=s^{2}a_{kk}+c^{2}a_{\ell \ell }+2sca_{k\ell }.\,\!$

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

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

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

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

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

$a'_{k\ell }=a'_{\ell k}=0\,\!$

$a'_{kk}=a_{kk}-ta_{k\ell }\,\!$

$a'_{\ell \ell }=a_{\ell \ell }+ta_{k\ell }\,\!$

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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