On the Convergence of Cyclic Jacobi Methods

Jacobi’s method is an iterative algorithm for diagonalizing a symmetric matrix A = A(1), in which at iteration k a plane rotation J _k =J _k (i _k ,j _k ,θ _k ) is chosen so as to annihilate a pair of off diagonal elements in the matrix A(k)$$ \left( {\begin{array}{*{20}{c}} c&{ - s} \\ s&c \end{array}} \right)\left( {\begin{array}{*{20}{c}} {a_{ii}^{(k)}}&{a_{ij}^{(k)}} \\ {a_{ji}^{(k)}}&{a_{jj}^{(k)}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} c&s \\ { - s}&c \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {a_{ii}^{(k + 1)}}&0 \\ 0&{a_{jj}^{(k + 1)}} \end{array}} \right) $$ where c = cos θ _k and s=sin θ _k .