Donald Goldfarb presents a new rank-two variable-metric method derived using Greenstadt's variational approach. This method, like the Davidon-Fletcher-Powell (DFP) method, preserves the positive-definiteness of the approximating matrix. Together with Greenstadt's method, it forms a one-parameter family of variable-metric methods that includes the DFP and rank-one methods as special cases. The paper also discusses the derivation of these methods from Greenstadt's variational approach and shows that they are equivalent to Broyden's one-parameter family. The correction terms for the DFP and rank-one methods are derived and shown to preserve the positive-definiteness of the approximating matrix, ensuring the stability of the algorithms. The paper concludes by exploring the relationships between different correction terms and their implications for the convergence behavior of variable-metric methods.Donald Goldfarb presents a new rank-two variable-metric method derived using Greenstadt's variational approach. This method, like the Davidon-Fletcher-Powell (DFP) method, preserves the positive-definiteness of the approximating matrix. Together with Greenstadt's method, it forms a one-parameter family of variable-metric methods that includes the DFP and rank-one methods as special cases. The paper also discusses the derivation of these methods from Greenstadt's variational approach and shows that they are equivalent to Broyden's one-parameter family. The correction terms for the DFP and rank-one methods are derived and shown to preserve the positive-definiteness of the approximating matrix, ensuring the stability of the algorithms. The paper concludes by exploring the relationships between different correction terms and their implications for the convergence behavior of variable-metric methods.