BFGS法
(1970), “The convergence of a class of double-rank minimization algorithms”, Journal of the Institute of Mathematics and Its Applications 6: 76–90, doi:10.1093/imamat/6.1.76
Fletcher, R. (1970), “A New Approach to Variable Metric Algorithms”, Computer Journal 13 (3): 317–322, doi:10.1093/comjnl/13.3.317
Fletcher, Roger (1987), Practical methods of optimization (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-91547-8
Goldfarb, D. (1970), “A Family of Variable Metric Updates Derived by Variational Means”, Mathematics of Computation 24 (109): 23–26, doi:10.1090/S0025-5718-1970-0258249-6
α = 1 y k T s k {\displaystyle \alpha ={\frac {1}{\mathbf {y} _{k}^{T}\mathbf {s} _{k}}}}
β = − 1 s k T B k s k {\displaystyle \beta =-{\frac {1}{\mathbf {s} _{k}^{T}B_{k}\mathbf {s} _{k}}}}
上の α {\displaystyle \alpha } および β {\displaystyle \beta } を B k + 1 = B k + α u u ⊤ + β v v ⊤ {\displaystyle B_{k+1}=B_{k}+\alpha \mathbf {u} \mathbf {u} ^{\top }+\beta \mathbf {v} \mathbf {v} ^{\top }} に代入して書き下すと、次のようになる。
B k + 1 = B k + y k y k T y k T s k − B k s k s k T B k T s k T B k s k {\displaystyle B_{k+1}=B_{k}+{\frac {\mathbf {y} _{k}\mathbf {y} _{k}^{\mathrm {T} }}{\mathbf {y} _{k}^{\mathrm {T} }\mathbf {s} _{k}}}-{\frac {B_{k}\mathbf {s} _{k}\mathbf {s} _{k}^{\mathrm {T} }B_{k}^{\mathrm {T} }}{\mathbf {s} _{k}^{\mathrm {T} }B_{k}\mathbf {s} _{k}}}}