CODEWITHSUNDEEP
pythonmatrixnumerical-methodsnewtons-method
ShoutOutAndCalculate
ShoutOutAndCalculate

Reputation: 587

Numerical instability of system of equations of large order polynomials

I have a set of non linear polynomials

V_n(Input[N])=0

where n runs from 1 to N, with N of Input[N] unknowns. The V_n() contained the polynomial tern of x**N. I used the newton's methods to find the roots but encountered the numerical stability issues, part of it contributed from the computation of the matrix inverse. i.e. though 2000**2000=1.1481306952742545242E+6602 was easily computed, the elements taken in matrix inverse were separated by 1E200, 1E200**2, were not. This lead to the inverse of matrix multiplication have a large remainders

max( Inverse(Jacobian(V_n(Input[N]))) * V_n(Input[N]) )=1E200

I've been trying to find the normalization procedure for the newton's method, but with limited progresses. Without the normalization, the root finding algorithm with the newton's method could handle N~10, with normalization, this increased N~50. However, I needed to compute the system of linear equations at N~2000. The good news is the maximum of the roots max(Input[N])~N or approximately $N$ itself.

Is there a way to find the roots of this system of equations?

Upvotes: 0

Views: 80

Answers (0)

Related Questions