engleski

Jacobi method for symmetric 4x4 matrices converges for every cyclic pivot strategy

The paper studies the global convergence of the Jacobi method for symmetric matrices of size $4$. We prove global convergence for all $720$ cyclic pivot strategies. Precisely, we show that inequality $S(A^[t+3])\leq\gamma S(A^[t])$, $t\geq1$, holds with the constant $\gamma<1$ that depends neither on the matrix $A$ nor on the pivot strategy. Here $A^[t]$ stands for the matrix obtained from $A$ after $t$ full cycles of the Jacobi method and $S(A)$ is the off-diagonal norm of $A$. We show why three consecutive cycles have to be considered. The result has a direct application on the $J$-Jacobi method.

Eigenvalues ; Jacobi method ; global convergence

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