Symmetric rank-one


The Symmetric Rank 1 method is a quasi-Newton method to update the second derivative
based on the derivatives calculated at two points. It is a generalization to the secant method for a multidimensional problem.
This update maintains the symmetry of the matrix but does not guarantee that the update be positive definite.
The sequence of Hessian approximations generated by the SR1 method converges to the true Hessian under mild conditions, in theory; in practice, the approximate Hessians generated by the SR1 method show faster progress towards the true Hessian than do popular alternatives, in preliminary numerical experiments. The SR1 method has computational advantages for sparse or partially separable problems.
A twice continuously differentiable function has a gradient and Hessian matrix : The function has an expansion as a Taylor series at, which can be truncated
its gradient has a Taylor-series approximation also
which is used to update. The above secant-equation need not have a unique solution .
The SR1 formula computes the symmetric solution that is closest to the current approximate-value :
where
The corresponding update to the approximate inverse-Hessian is
The SR1 formula has been rediscovered a number of times. A drawback is that the denominator can vanish. Some authors have suggested that the update be applied only if
where is a small number, e.g..