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I'm trying to improve the speed of the following iteration to calculate $s_k$:

$$B_k^{-1} = \Bigg( I + \frac{s_{k}s_{k-1}^T}{||s_{k-1}||^2}\Bigg)...\Bigg(I+ \frac{s_1s_0^T}{||s_0||^2}\Bigg) B_0^{-1}\\s_{k+1} = -\frac{B_k^{-1}r_{k+1}}{1+\frac{s_k^TB_k^{-1}r_{k+1}}{||s_k||^2}}$$

If we calculate $B_k^{-1}$ explicitly then each step is roughly $O(n^4)$ to calculate $B_k^{-1}$. If instead we compute $B_k^{-1}r_{k+1}$ by expanding the product from the right, we do $k$ matrix-vector multiplications which is $O(n^3)$, a lot better.

Is there a way to find $s_k$ in the form of a low rank update using the value of previous $s_i$?

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  • $\begingroup$ What do you mean by "find $s_k$ in the form of a low rank update"? $s_k$ is a vector, it has rank 1 (if rank is defined at all for vectors). $\endgroup$
    – Federico Poloni
    Commented Mar 21, 2019 at 8:08
  • $\begingroup$ Also, if I understand correctly, the recursion in BFGS is not the equation that you write, but a two-sided version of it (update matrices I + something on both sides of $B^{-1}$). Are you sure that this is the recursion that you wish to use? Do you realize that this version produces a non-symmetric $B_k^{-1}$, unlike the standard one? $\endgroup$
    – Federico Poloni
    Commented Mar 21, 2019 at 8:35

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You can do this using the Sherman-Morrison Formula and some other tricks, detailed starting on page 124 of Kelley, Iterative Methods for Linear and Nonlinear Equations

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