By unwinding the matrix recurrence for B_k , the DFP formula can be expressed as a
compact matrix representation. Specifically, defining S_k = \begin{bmatrix} s_0 & s_1 & \ldots & s_{k-1} \end{bmatrix}, Y_k = \begin{bmatrix} y_0 & y_1 & \ldots & y_{k-1} \end{bmatrix}, and upper triangular and diagonal matrices \big(R_k\big)_{ij} := \big(R^{\text{SY}}_k\big)_{ij} = s^T_{i-1}y_{j-1}, \quad \big(R^{\text{YS}}_k\big)_{ij} = y^T_{i-1}s_{j-1}, \quad (D_k)_{ii} := \big(D^{\text{SY}}_k\big)_{ii} = s^T_{i-1}y_{i-1} \quad \quad \text{ for } 1 \le i \le j \le k the DFP matrix has the equivalent formula B_k = B_0 + J_k N^{-1}_k J^T_k, J_k = \begin{bmatrix} Y_k & Y_k - B_0 S_k \end{bmatrix} N_k = \begin{bmatrix} 0_{k \times k} & R^{\text{YS}}_k \\ \big( R^{\text{YS}}_k \big)^T & R_k+R^T_k-(D_k+S^T_k B_0 S_k) \end{bmatrix} The inverse compact representation can be found by applying the
Sherman-Morrison-Woodbury inverse to B_k. The compact representation is particularly useful for limited-memory and constrained problems. ==See also==