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Approximation of sequences of symmetric matrices with the symmetric rank-one algorithm and applications

The symmetric rank-one update method is well-known in optimization for its applications in the quasi-Newton algorithm. In particular, Conn, Gould, and Toint proved in 1991 that the matrix sequence resulting from this method approximates the Hessian of the minimized function. Expanding their idea, we prove that the symmetric rank-one update algorithm can be used to approximate any sequence of symmetric invertible matrices, thereby adding a variety of 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.