# Symmetric rank-one

The **Symmetric Rank 1** (**SR1**) method is a quasi-Newton method to update the second derivative (Hessian)
based on the derivatives (gradients) calculated at two points. It is a generalization to the secant method for a multidimensional problem.
This update maintains the *symmetry* of the matrix but does *not* guarantee that the update be *positive definite*.

The sequence of Hessian approximations generated by the SR1 method converges to the true Hessian under mild conditions, in theory; in practice, the approximate Hessians generated by the SR1 method show faster progress towards the true Hessian than do popular alternatives (BFGS or DFP), in preliminary numerical experiments.^{[1]} The SR1 method has computational advantages for sparse or partially separable problems.

A twice continuously differentiable function has a gradient () and Hessian matrix : The function has an expansion as a Taylor series at , which can be truncated

- ;

its gradient has a Taylor-series approximation also

- ,

which is used to update . The above secant-equation need not have a unique solution . The SR1 formula computes (via an update of rank 1) the symmetric solution that is closest to the current approximate-value :

- ,

where

- .

The corresponding update to the approximate inverse-Hessian is

- .

The SR1 formula has been rediscovered a number of times. A drawback is that the denominator can vanish. Some authors have suggested that the update be applied only if

- ,

where is a small number, e.g. .^{[2]}

## See also[edit]

- Quasi-Newton method
- Newton's method in optimization
- Broyden-Fletcher-Goldfarb-Shanno (BFGS) method
- L-BFGS method

## Notes[edit]

## References[edit]

- Byrd, Richard H. (1996) Analysis of a Symmetric Rank-One Trust Region Method.
*SIAM Journal on Optimization*6(4) - Conn, A. R.; Gould, N. I. M.; Toint, Ph. L. (March 1991). "Convergence of quasi-Newton matrices generated by the symmetric rank one update".
*Mathematical Programming*. Springer Berlin/ Heidelberg.**50**(1): 177–195. doi:10.1007/BF01594934. ISSN 0025-5610. PDF file at Nick Gould's website. - Khalfan, H. Fayez (1993) A Theoretical and Experimental Study of the Symmetric Rank-One Update.
*SIAM Journal on Optimization*3(1) - Nocedal, Jorge & Wright, Stephen J. (1999).
*Numerical Optimization*. Springer-Verlag. ISBN 0-387-98793-2.