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An Interval Newton Method Based on the Bernstein Form for Bounding the Zeros of Polynomial Systems

Interval Newton methods are widely used to find reliable enclosures for the roots of the polynomial systems. But the computation process needs the evaluation of an enclosure for the derivative of the polynomial and evaluation of the polynomial at a particular point. Again, the subdivisions (if any are needed) will require repeated evaluation of the function values. We propose an alternative approach using Bernstein coefficients to find the value of the Newton operator. This Bernstein Newton operator doesn’t require evaluation of the function at any particular point. Instead the function value is obtained directly from the vertex points. Again the computation of the derivative is very simple using the Bernstein coefficient approach. In addition to this the range enclosures obtained using Bernstein coefficients are much sharper than the range enclosures obtained using many other interval forms. We compare the performance of the proposed Bernstein Newton Contractor with that of Interval Newton contractor using numerical examples, proving the superiority of the proposed approach.