we are trying to solve $$ f(x) = 0 $$ Where \( f(x) \) is a vector valued function with each element the value of one of the benchmark instruments, and \( x \) is the column vector of curve pillar point values (either discount factors or rates)

The QuantSA version proceeds by approximating \(f(x)\) as linear: $$ f(x+\delta) \approx f(x) + J(x)\delta $$ Where \(J\) is the Jacobian matrix with $$J_{ij} = \frac{\partial f_i}{\partial x_j}$$ since we require $$f(x+\delta)=0$$ we estimate $$\delta = -J(x)^{-1}f(x)$$