## On sparse and symmetric matrix updating subject to a linear equation

HTML articles powered by AMS MathViewer

- by Ph. L. Toint PDF
- Math. Comp.
**31**(1977), 954-961 Request permission

## Abstract:

A procedure for symmetric matrix updating subject to a linear equation and retaining any sparsity present in the original matrix is derived. The main feature of this procedure is the reduction of the problem to the solution of an*n*dimensional sparse system of linear equations. The matrix of this system is shown to be symmetric and positive definite. The method depends on the Frobenius matrix norm. Comments are made on the difficulties of extending the technique so that it uses more general norms, the main points being shown by a numerical example.

## References

- C. G. Broyden,
*A class of methods for solving nonlinear simultaneous equations*, Math. Comp.**19**(1965), 577–593. MR**198670**, DOI 10.1090/S0025-5718-1965-0198670-6
W. C. DAVIDON, - R. Fletcher and M. J. D. Powell,
*A rapidly convergent descent method for minimization*, Comput. J.**6**(1963/64), 163–168. MR**152116**, DOI 10.1093/comjnl/6.2.163 - Donald Goldfarb,
*A family of variable-metric methods derived by variational means*, Math. Comp.**24**(1970), 23–26. MR**258249**, DOI 10.1090/S0025-5718-1970-0258249-6 - J. Greenstadt,
*Variations on variable-metric methods. (With discussion)*, Math. Comp.**24**(1970), 1–22. MR**258248**, DOI 10.1090/S0025-5718-1970-0258248-4 - H. Y. Huang,
*Unified approach to quadratically convergent algorithms for function minimization*, J. Optim. Theory Appl.**5**(1970), 405–423. MR**288939**, DOI 10.1007/BF00927440 - M. J. D. Powell,
*A new algorithm for unconstrained optimization*, Nonlinear Programming (Proc. Sympos., Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1970, pp. 31–65. MR**0272162**
J. K. REID, - L. K. Schubert,
*Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian*, Math. Comp.**24**(1970), 27–30. MR**258276**, DOI 10.1090/S0025-5718-1970-0258276-9

*Variable Metric Method for Minimization*, Report #ANL-5990 (Rev.), A.N.L. Research and Development Report, 1959.

*Two Fortran Subroutines for Direct Solution of Linear Equations Whose Matrix is Sparse, Symmetric and Positive Definite*, Report AERE-R. 7119, Harwell, 1972.

## Additional Information

- © Copyright 1977 American Mathematical Society
- Journal: Math. Comp.
**31**(1977), 954-961 - MSC: Primary 65F30
- DOI: https://doi.org/10.1090/S0025-5718-1977-0455338-4
- MathSciNet review: 0455338