Computing the LU factorization of a square n × n matrix A requires about (2/3)*n³ flops to compute. In some applications like optimization, solving systems that are low-rank modifications of A are also needed. Although the modified matrix (A+uv^T) in general has no elements equal to those of A, the inverse of it can be mathematically written using the Sherman-Morrison formula as

As usual,

• Make sure the sizes match up in the formula above. In particular, writing something as a fraction means the denominator must be a scalar. Is it above?

• Similarly, anything subtracted from A^-1 must also be a n × n matrix; is the expression above the right size?

• Computationally, never explicitly form the inverse matrix A^-1. Instead, wherever it appears in a formula, instead think of solving a linear system with coefficient matrix A = LU. So given the LU factorization of A, how should the computations be arranged to solve a linear system (A+uv^T)x = y

  using the Sherman-Morrison formula?

• Assuming the matrix A is numerically OK, where could the S-M formula break down? So what test should be inserted before blindly applying it?

Other "Small" Modifications of A

• Another common modification of A is a sequence of rank-1 modifications. In that case, the S-M formula can be used recursively, going back to A for which the LU factorization was initially computed. At some point well before the number of rank-1 modifications becomes n, it will be more efficient to bite the bullet and refactor the latest modified A, and restart. At what number of rank-1 modifications is that break-point, in terms of number of flops?

• If instead of a sequence of rank-1 modifications of A, a single rank-k modification is done, the Sherman-Morrison-Woodbury formula generalizes the S-M formula to handle (A+UCV^T)^-1, where U and V are n × k and C is a k × k matrix. The formula is easily looked-up, but see if you can guess the form from the S-M one. Hint: it won't take the form of a matrix divided by a scalar; in this case the denominator in S-M becomes the matrix (C - VA^-1U)^-1

• If only a row or column of A is modified, it can be written as a rank-1 or rank-2 modification of A using well-chosen vectors u and v. What are those?