# Zhang Neural Networks for Fast and Accurate Computations of the Field of Values

## TL;DR

This paper introduces Zhang Neural Networks (ZNN), a novel approach for rapidly and accurately computing the field of values of matrices, achieving high precision with a single step ahead finite difference scheme.

## Contribution

The paper presents a new neural network method, ZNN, tailored for matrix field of values problems, demonstrating superior accuracy and speed compared to existing techniques.

## Key findings

- ZNN achieves over 15 accurate digits of the FoV boundary.
- Uses a 6th order error finite difference scheme for high precision.
- Records fast computation times for hermitean matrix flows.

## Abstract

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 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 in record time when applied to hermitean matrix flows $A(t)$.