This paper describes and develops a fast and accurate algorithm that computes the field of values boundary curve for every conceivable complex square matrix A, i.e., for matrices with or without repeated eigenvalues and with or without Jordan block structures  . It relies on a matrix flow decomposition method that finds the coarsest block diagonal flow representation for the hermitean matrix flow H\!K  (t) = \cos(t) H + \sin(t) K  . Here H\!K  (t) is a 1-parameter varying linear combination of the real and skew part matrices H = (A+A^*)/2 and K = (A-A^*)/(2i) of A. For decomposing flows H\!K  (t), the algorithm decomposes the given dense matrix A conformally into diagonal blocks as H\!K in U^*AU = \text { diag} (A_j) for a unitary matrix U . It then computes the field of values boundaries separately for each diagonal block A_j using the fast ZNN parameter varying flow eigenvalue method. Finally it saves the convex hull of all intermediate field of values boundary curves in order to plot the field of values of Aproperly . The algorithm removes standard restrictions for path following FoV methods that generally cannot deal with decomposing matrices A due to possible eigencurve crossings for such A . Tests and numerical comparisons are included   .
This paper describes and develops a fast and accurate algorithm that computes the field of values boundary curve for every conceivable complex square matrix A, i.e., for all matrices, those with or without repeated eigenvalues and with or without Jordan block structures , with ill-conditioned eigenvalues or ill-conditioned ones and so forth . It relies on a matrix flow decomposition method that finds a block-diagonal flow representation for the hermitean matrix flow H\!K _A (t) = \cos(t) H + \sin(t) K if this is possible . Here H\!K _A (t) is a 1-parameter varying linear combination of the real and skew part matrices H = (A+A^*)/2 and K = (A-A^*)/(2i) of A. For decomposing flows H\!K _A (t), the algorithm decomposes the given dense general matrix A unitarily into U^*AU = \text { diag} (A_j) with diagonal blocks A_j that conform in size with those of H\!K_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 field of values boundary points then determines the field of values boundary curve correctly for decomposing and non-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 H\!K_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 very fast when compared with Johnson's global QR eigenvalue and Bedixon rectangle based method that computes complete eigenanalyses of H\!K_A(t_k) for every chosen t_k \in 0,2\pi{] .
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matrix A conformally into diagonal blocks as H\!K in
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0,2\pi{]