Approximation of Sequences of Symmetric Matrices with the Symmetric Rank-One Algorithm and Applications
Sylvain Arguillère · SIAM Journal on Matrix Analysis and Applications · 2015
The symmetric rank-one update method is well known in optimization for its applications in quasi-Newton algorithms. In particular, Conn, Gould, and Toint proved in 1991 that the matrix sequence resulting from this method approximates the Hessian of the minimized function under a suitable linear-independence assumption. Extending their idea, we prove that symmetric rank-one updates can be used to approximate any sequence of symmetric invertible matrices, which has applications to more general problems, such as the computation of constrained geodesics in shape analysis imaging problems. We also provide numerical simulations for the method and some of these applications.