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A mathematician associated with Broyden's method and Quasi-Newton methods in numerical analysis. An institutional portrait is catalogued by the University of Essex Library Services: www.flickr.com/photos/149590996@N07/31601668406/

Convexity of the components of a vector field is not the decisive criterion for a root-finding problem. Neither method becomes globally reliable merely because each component is convex.

Prefer the Newton root-finding iteration when the Jacobian matrix is available accurately and at reasonable cost. Solve $J({\mathbf{x}}_k){\mathbf{s}}_k=-\mathbf{f}({\mathbf{x}}_k)$ rather than explicitly forming an inverse matrix. Near a root with a nonsingular Jacobian matrix and a locally Lipschitz Jacobian matrix, the usual exact iteration has quadratic convergence.

Prefer Broyden's method when evaluating or differentiating the vector field makes a fresh Jacobian matrix expensive. It learns an approximate Jacobian matrix through secant updates using function differences. Under appropriate local regularity and initialization assumptions it has superlinear convergence, generally slower asymptotically than the quadratic convergence of exact Newton steps, but often with cheaper iterations.^[ref]

Dimension matters through storage and solves. A full dense Newton factorization costs order $d^3$ arithmetic operations; a dense Broyden approximation uses order $d^2$ storage, and maintained inverse or factorization updates can cost order $d^2$. A naive fresh dense solve can still cost order $d^3$. Sparsity can favour Newton, while very large $d$ can make either dense formulation unsuitable. Neither local rate guarantees convergence from an arbitrary starting point.

The NEOS Guide's mathematical account of Quasi-Newton methods, deriving the secant condition from an integrated Hessian matrix and describing low-rank updating, including BFGS. The related nonlinear-equation account is neos-guide.org/guide/algorithms/broyden/ and discusses Broyden's method.