A NEWTON-SQUARING ALGORITHM FOR COMPUTING THE NEGATIVE INVARIANT SUBSPACE OF A MATRIX

By combining Newton's method for the matrix sign function with a squar ing procedure, a basis for the negative invariant subspace of a matrix can be computed efficiently. The algorithm presented is a variant of multiplication-rich schemes for computing the matrix sign function suc h as the well-known inversion-free Schulz method which requires two ma trix multiplications per step. However, by avoiding a complete computa tion of the matrix sign and instead concentrating only on the negative invariant subspace, the final Newton steps can be replaced by steps w hich require only one matrix squaring each. This efficiency is attaine d without sacrificing the quadratic convergence of Newton's method.