The Davidon-Fletcher-Powell formula (DFP) finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition (see below). It was the first quasi-Newton method which generalize the secant method to a multidimensional problem. This update maintains the symmetry and positive definiteness of the Hessian matrix.

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  • The Davidon-Fletcher-Powell formula (DFP) finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition (see below). It was the first quasi-Newton method which generalize the secant method to a multidimensional problem. This update maintains the symmetry and positive definiteness of the Hessian matrix. Given a function <math>f(x)</math>, its gradient (<math>\nabla f</math>), and positive definite Hessian matrix <math>B</math>, the Taylor series is: <math>f(x_k+s_k)=f(x_k)+\nabla f(x_k)^T s_k+\frac{1}{2} s^T_k {B} s_k </math>, and the Taylor series of the gradient itself (secant equation): <math>\nabla f(x_k+s_k)=\nabla f(x_k)+B s_k</math>, is used to update <math>B</math>. The DFP formula finds a solution that is symmetric, positive definite and closest to the current approximate value of <math>B_k</math>: <math>B_{k+1}= (I-\gamma_k y_k s_k^T) B_k (I-\gamma_k s_k y_k^T)+\gamma_k y_k y_k^T</math>, where <math>y_k=\nabla f(x_k+s_k)-\nabla f(x_k)</math>, <math>\gamma_k =\frac{1}{y_k^T s_k}</math>. and <math>B_k</math> is a symmetric and positive definite matrix. The corresponding update to the inverse Hessian approximation <math>H_k=B_k^{-1}</math> is given by: <math>H_{k+1}=H_{k}-\frac {H_k y_k y_k^T H_k}{ y_k^T H_k y_k}+\frac{s_k s_k^T}{y_k^{T} s_k}</math>. <math>B</math> is assumed to be positive definite, and the vectors <math>s_k^T</math> and <math>y</math> must satisfy the curvature condition: <math>s_k^T y_k=s_k^T B s_k>0</math>. The DFP formula is quite effective, but it was soon superseded by the BFGS formula.
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  • The Davidon-Fletcher-Powell formula (DFP) finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition (see below). It was the first quasi-Newton method which generalize the secant method to a multidimensional problem. This update maintains the symmetry and positive definiteness of the Hessian matrix.
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  • Davidon-Fletcher-Powell formula
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