On the order of convergence of Broyden’s method

Abstract

We present two theoretical results and two surprising conjectures concerning convergence properties of Broyden’s method for smooth nonlinear systems of equations. First, we show that when Broyden’s method is applied to a nonlinear mapping $$F:\mathbb {R}^n\rightarrow \mathbb {R}^n$$ F : R n → R n with $$n-d$$ n - d affine component functions and the initial matrix $$B_0$$ B 0 is chosen suitably, then the generated sequence $$(u^k,F(u^k),B_k)_{k\ge 1}$$ ( u k , F ( u k ) , B k ) k ≥ 1 can be identified with a lower-dimensional sequence that is also generated by Broyden’s method. This property enables us to prove, second, that for such mixed linear–nonlinear systems of equations a proper choice of $$B_0$$ B 0 ensures 2d-step q-quadratic convergence, which improves upon the previously known 2n steps. Numerical experiments of high precision confirm the faster convergence and show that it is not available if $$B_0$$ B 0 deviates from the correct choice. In addition, the experiments suggest two surprising possibilities: It seems that Broyden’s method is $$(2d-1)$$ ( 2 d - 1 ) -step q-quadratically convergent for $$d>1$$ d > 1 and that it admits a q-order of convergence of $$2^{1/(2d)}$$ 2 1 / ( 2 d ) . These conjectures are new even for $$d=n$$ d = n .