Many iterative algorithms for optimization calculations use a second derivative approximation, B say, in order to calculate the search direction d=¸-B’-1∇f(x). In order to avoid inverting B the paper works with matrices Z, whose columns satisfy the conjugacy relations ZTBZ=1. It presents an update of Z that is compatible with members of the Broyden family that generate positive definite second derivative approximations. The algorithm requires only 3n2+O(n) flops for the update of Z and the calculation of d. The columns of the resultant Z matrices have interesting conjugacy and orthogonality properties with respect to previous second derivative approximations and function gradients, respectively. The update also provides a simple proof of Dixon’s theorem. For the BFGS method the paper adapts the algorithm in order to obtain a null space method for linearly constrained calculations.
