  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.
