Broyden's method is a generalization of the secant method to nonlinear systems. Roughly speaking, the secant method replaces the derivative by a finite difference
Broyden gives a systems generalization of this formula
Instead of using the Jacobian matrix, Broyden uses a matrix 
satisfying the
same formula. Note the system provides only N equations to determine the
NXN matrix. The ``best possible'' choice for is a minimal modification
of 
, 
. This
leads to a uniquely defined matrix. Set 
Then
Broyden's method then iterates
Of course, once again we are faced with finding an inverse matrix, something we
would like to avoid. rather, one can solve
and then update
.
Broyden also suggested using
the Sherman-Morrison formula (sometimes called the `bad Broyden' method)
which tells us how to compute the inverse
in an efficient manner. The formula says that, for a nonsingular matrix A
and vectors u, v, such that 
, then
To implement the Sherman-Morrison formula in Broyden, then,
let 
.
A count shows that this formula leads to an update of the inverse costing only
operations.
Broyden's method is not as fast as the quadratic convergence of Newton's method. But the smaller operation count per iteration is often worth the trade-off.