Question

Describe what the Sherman-Morrison-Woodbury formula is for, and why it is important in the BFGS algorithm.

Answer 1

Sherman- Morrison formula: The special case if woodbury formula is called Sherman- Marrison formula In linear algebra, Jack Sherman and Winifred. J. Morrison compute the inverse of the sum of an invertible matrix . A and outer product uv^T of vector u and v so that is names a Sherman- Morrison formula consider the following statement consider A is an invertible squarer matrix and column vector are u and v Therefore if v^T A^-1 u notequal 0 Therefore, the sherman- Morrison formula is as follow (A + uv^T)^-1 = A^-1 - A^-1 uv^T A^-1/1 + v^T A^-1 u = 1/2 where, an invertible matrix A, outer product uv^T of vector u & V Sherman- Morrison_Woodbury when a factorization of A is already exists then solution of At middot uv^t is described by Sherman + Morrison formula The Sherman- Morrison - Woodbury formula is the generalization to rank k modification to A (At uv^T)^-1 = A^-1 - A^-1