New quasi-Newton method for solving systems of nonlinear equations. (English) Zbl 1458.65069

Lukšan, Ladislav; Vlček, Jan

Appl. Math., Praha 62, No. 2, 121-134 (2017).

Summary: We propose a new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR or LU decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires \(O(n^2)\) arithmetic operations per iteration in contrast with the Newton method, which requires \(O(n^3)\) operations per iteration. Computational experiments confirm the high efficiency of the new method.

MSC:

65K10

Numerical optimization and variational techniques

Keywords:

nonlinear equation; system of equations; trust-region method; quasi-Newton method; adjoint Broyden method; numerical algorithm; numerical experiment

Software:

UFO