Question

2. In BFGS, the matrix $B_{k+1}$ is defined by $B_{k+1} = B_k + frac{y_k y_k^T}{y_k^T s_k} - frac{B_k s_k s_k^T B_k}{s_k^T B_k s_k}$ where $B_k$ is an invertible matrix and y and s are two column vectors. Verify that its inverse can be written as $B_{k+1}^{-1} = B_k^{-1} + frac{(s_k^T y_k + y_k^T B_k^{-1} y_k)(s_k s_k^T)}{(s_k^T y_k)^2} - frac{B_k^{-1} y_k s_k^T + s_k y_k^T B_k^{-1}}{s_k^T y_k}$

          2. In BFGS, the matrix $B_{k+1}$ is defined by
$B_{k+1} = B_k + frac{y_k y_k^T}{y_k^T s_k} - frac{B_k s_k s_k^T B_k}{s_k^T B_k s_k}$
where $B_k$ is an invertible matrix and y and s are two column vectors.
Verify that its inverse can be written as
$B_{k+1}^{-1} = B_k^{-1} + frac{(s_k^T y_k + y_k^T B_k^{-1} y_k)(s_k s_k^T)}{(s_k^T y_k)^2} - frac{B_k^{-1} y_k s_k^T + s_k y_k^T B_k^{-1}}{s_k^T y_k}$

$2. In BFGS, the matrix Bk+1 is defined by Bk+1 = Bk + fracyk yk^Tyk^T sk - fracBk sk sk^T Bksk^T Bk sk where Bk is an invertible matrix and y and s are two column vectors. Verify that its inverse can be written as Bk+1^-1 = Bk^-1 + frac(sk^T yk + yk^T Bk^-1 yk)(sk sk^T)(sk^T yk)^2 - fracBk^-1 yk sk^T + sk yk^T Bk^-1sk^T yk$

Added by Holly T.

Question

Please give Ace some feedback