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In this paper, a new and different neural network, called Zhang Neural Network (ZNN) is appropriated from discrete time-varying matrix problems and applied to the angle parameter-varying matrix field of values (FoV) problem. This problem acts as a test bed for newly discovered convergent 1-step ahead finite difference formulas of high truncation error orders. The ZNN method that uses a look-ahead finite difference scheme of error order 6 gives us 15+ accurate digits of the FoV boundary points in record time when applied to hermitian matrix flows A (t). [ABSTRACT FROM AUTHOR]

We revisit an earlier idea of Marcus and Pesce from 1987 for generating fields of values ofbymatricesvia 2 by 2 matrix compressions and their easily constructed elliptical field of values. This approach is used to reduce the cost of finding the FOV boundary curve of a matrixand makes it more accurate at the same time. The new algorithm succeeds by using fewer eigenanalyses and constructing 2 by 2 matrix compression ellipses offor approximatinginstead of only eigenanalyses. An application to verify a counterexample to theZemanek Conjecture for companion matrices of monic polynomials and their normalized derivatives inand, respectively, in dimension 4 is given. [ABSTRACT FROM PUBLISHER]