Jacobi Methods - 2023.2 English

Vitis Libraries

Release Date
2023-12-20
Version
2023.2 English

Jacobi uses a sequence of plane rotations to reduce a symmetric matrix A to a diagonal matrix

\[A_{0} = A, \> A_{k+1} = J_{k}^{T}A_{k}J_{k}, \> A_{k} \rightarrow \Sigma \> as \> k \rightarrow \infty\]

Each plane rotation, \(J_{k} = J_{k}(i, j, \theta)\), now called a Jacobi or Givens rotation

\[\begin{split}\begin{equation*} J_{k}(i, j, \theta)=\begin{vmatrix} I &\, & & & \\ \, &\,c & &s & \\ \, &\, &I & & \\ \, &-s & &c & \\ & &\, & & &I \end{vmatrix} \end{equation*}\end{split}\]

where \(c=cos \theta\) and \(s=sin \theta\). The angle \(\theta\) is chosen to eliminate the pair \(a_{ij}\), \(a_{ji}\) by applying \(J(i,j, \theta )\) on the left and right of \(A\), which can be viewed as the 2x2 eigenvalue problem

\[\begin{split}\begin{equation*} \hat{J}_{(k)}^{T} \hat{A}_{(k)} \hat{J}_{(k)}= \begin{vmatrix} \, c &s \\ -s &c \end{vmatrix}^{T} \begin{vmatrix} a_{ii} &a_{ij} \\ a_{ji} &a_{jj} \end{vmatrix} \begin{vmatrix} d_{ii} &0 \\ 0 &d_{jj} \end{vmatrix}= \hat{A}_{(k+1)} \end{equation*}\end{split}\]

where \(\hat{A}\) is a 2X2 submatrix of matrix A. After the Givens rotations of the whole matrix A, the off-diagonal value of A will be reduced to zero-like value after 3-15 times iteration of the process.