Broyden

Broyden refers to a family of numerical methods for solving systems of nonlinear equations F(x) = 0 and, more broadly, for unconstrained optimization. Named after C. G. Broyden, these methods are part of the quasi-Newton class and aim to approximate the Jacobian matrix J(x) that describes how F changes with x, updating this approximation at each iteration to produce Newton-like steps without recomputing the full Jacobian.

In a typical iteration, one maintains an estimate J_k of the Jacobian at x_k. After taking a trial step to x_{k+1}, one forms s_k = x_{k+1} - x_k and y_k = F(x_{k+1}) - F(x_k). The update to J_k is chosen as a rank-one correction J_{k+1} = J_k + (y_k - J_k s_k) u_k^T, where u_k is selected so that the secant equation J_{k+1} s_k = y_k holds and the modification is minimized in a chosen norm. This yields the Broyden family of updates. There is also a corresponding update for the inverse Jacobian, leading to related formulations. Special instances are referred to as Good Broyden (updates to the Jacobian) and Bad Broyden (updates to the inverse Jacobian).

Broyden's method reduces to Newton's method when the Jacobian is exact. In general, it offers robustness and efficiency for moderately large systems, especially when forming or factorizing the Jacobian is expensive. Global convergence is not guaranteed, and practical implementations often incorporate line searches or trust-region strategies. The method is widely used in computational mathematics and serves as a foundation for other quasi-Newton schemes, including optimization-oriented variants such as BFGS.

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