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 {\cal  _A(t) = \cos(t) H + \sin(t) K if this is possible. Here H\!K{\cal  _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 {\cal  _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 {\cal  _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 {\cal  _A(t_k) for every chosen t_k \in {[} 0,2\pi{]}  .
This paper describes and develops a fast and accurate path following algorithm that computes the field of values boundary curve for every conceivable complex or real square matrix A . It relies on a matrix flow decomposition method that finds a proper block-diagonal flow representation for the associated hermitean matrix flow {\cal F _A(t) = \cos(t) H + \sin(t) K . Here{\cal F _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 {\cal F _A(t), the algorithm decomposes a given dense general matrix A unitarily into block-diagonal form U^*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 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 {\cal 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 very fast when compared with Johnson's Francis QR eigenvalue and Bendixon rectangle based method that computes complete eigenanalyses of {\cal F _A(t_k) for every chosen t_k \in {[} 0,2\pi{]} more slowly .
A
path following
A
or real
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, 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.
.
A
proper
A
associated
D
H\!K
A
F
R
if this is possible. Here H\!K
. Here
A
F
R
1-parameter varying
1-parameter-varying
D
H\!K
A
F
R
the
a
A
block-diagonal form
A
j > 1
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that conform in size with those of H\!K_A
whose individual sizes add up to the size of A
R
sub field
sub-fields
D
H\!K
A
F
R
global
Francis
R
Bedixon
Bendixon
D
H\!K
A
F
A
more slowly