Constructing the field of values of decomposable and general matrices using the ZNN based path following method

This paper describes and develops a fast and accurate path following algorithm that computes the field of values boundary curve $\partial F(A)$ for every conceivable complex or real square matrix $A$. It relies on the matrix flow decomposition algorithm that finds a proper block-diagonal flow representation for the associated hermitean matrix flow ${ℱ}_A(t)=\cos (t)H+\sin (t)K$ under unitary similarity if that is possible. Here ${ℱ}_A(t)$ is the 1-parameter-varying linear combination of the real and skew part matrices $H=(A+A^{\ast })/2$ and $K=(A-A^{\ast })/(2i)$ of $A$. For indecomposable matrix flows, ${ℱ}_A(t)$ has just one block and the ZNN based field of values algorithm works with ${ℱ}_A(t)$ directly. For decomposing flows ${ℱ}_A(t)$, the algorithm decomposes the given matrix $A$ unitarily into block-diagonal form $U^{\ast }AU=\text{diag}(A_j)$ with $j>1$ diagonal blocks $A_j$ whose individual sizes add up to the size of $A$. It then computes the field of values boundaries separately for each diagonal block $A_j$ using the path following ZNN eigenvalue method. The convex hull of all sub-fields of values boundary points $\partial F(A_j)$ finally determines the field of values boundary curve correctly for decomposing matrices $A$. The algorithm removes standard restrictions for path following FoV methods that generally cannot deal with decomposing matrices $A$ due to possible eigencurve crossings of ${ℱ}_A(t)$. Tests and numerical comparisons are included. Our ZNN based method is coded for sequential and parallel computations and both versions run very accurately and fast when compared with Johnson's Francis QR eigenvalue and Bendixson rectangle based method and compute global eigenanalyses of ${ℱ}_A(t_k)$ for large discrete sets of angles $t_k\in 0,2\pi ]$ more slowly.