= Solution
<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.<A Class of Methods for Solving Nonlinear Simultaneous Equations by Charles George Broyden>[pp. 577–579, cost of Jacobian evaluation and construction of updating methods.]{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.
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