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The symmetric rank-one update method is well-known in optimization for itsapplications in the quasi-Newton algorithm. In particular, Conn, Gould, andToint proved in 1991 that the matrix sequence resulting from this methodapproximates the Hessian of the minimized function. Expanding their idea, weprove that the symmetric rank-one update algorithm can be used to approximateany sequence of symmetric invertible matrices, thereby adding a variety ofapplications to more general problems, such as the computation of constrainedgeodesics in shape analysis imaging problems. We also provide numericalsimulations for the method and some of these applications.

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