This article shows that the distributive laws of Beck in the bicategory of sets and matrices wherein monads are categories determine strict factorization systems on their composite monads Conversely it is shown that strict factorization systems on categories give rise to distributive laws Moreover these processes are shown to be mutually inverse in a precise sense Strict factorization systems a...

Various scienti€c applications use Gaussian elimination or Cholesky or QR factorization to solve dense linear systems. For an important class of problems, a relatively large number of small size systems is generated and must be solved. Typically, the order of these linear systems is up to a few hundred, and their number is from a few thousand to millions. For example, subsurface transportation ...

The minimal 2-norm solution to an underdetermined system Ax b of full rank can be computed using a QR factorization of AT in two different ways. One method requires storage and reuse of the orthogonal matrix Q, while the method of seminormal equations does not. Existing error analyses show that both methods produce computed solutions whose normwise relative error is bounded to first order by ca...

Incomplete factorization has been shown to be a good preconditioner for the conjugate gradient method on a wide variety of problems. It is well known that allowing some ll-in during the incomplete factorization can signiicantly reduce the number of iterations needed for convergence. Allowing ll-in, however, increases the time for the factorization and for the triangular system solves. In additi...

We say that a symmetric matrix K is quasi-definite if it has the form K = [ −E AT A F ] where E and F are symmetric positive definite matrices. Although such matrices are indefinite, we show that any symmetric permutation of a quasi-definite matrix yields a factorization LDLT . We apply this result to obtain a new approach for solving the symmetric indefinite systems arising in interior-point m...