Abstract
A new rank-two variable-metric method is derived using Greenstadt’s variational approach [<italic>Math. Comp.</italic>, this issue]. Like the Davidon-Fletcher-Powell (DFP) variable-metric method, the new method preserves the positive-definiteness of the approximating matrix. Together with Greenstadt’s method, the new method gives rise to a one-parameter family of variable-metric methods that includes the DFP and rank-one methods as special cases. It is equivalent to Broyden’s one-parameter family [<italic>Math. Comp.</italic>, v. 21, 1967, pp. 368–381]. Choices for the inverse of the weighting matrix in the variational approach are given that lead to the derivation of the DFP and rank-one methods directly.
Keywords
MathematicsRank (graph theory)Metric (unit)Variable (mathematics)Positive definitenessApplied mathematicsWeightingMatrix (chemical analysis)Hessian matrixInverseCombinatoricsPositive-definite matrixMathematical analysisEigenvalues and eigenvectorsGeometry
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Publication Info
- Year
- 1970
- Type
- article
- Volume
- 24
- Issue
- 109
- Pages
- 23-26
- Citations
- 3044
- Access
- Closed
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Cite This
Donald Goldfarb
(1970).
A family of variable-metric methods derived by variational means.
Mathematics of Computation
, 24
(109)
, 23-26.
https://doi.org/10.1090/s0025-5718-1970-0258249-6
Identifiers
- DOI
- 10.1090/s0025-5718-1970-0258249-6