By combining Newton's method for the matrix sign function with a squaring 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, such as the well-known inversion-free Schulz method, which requires two matrix multiplications per step. However, by avoiding a complete computation of the matrix sign and instead concentrating only on the negative invariant subspace, the final Newton steps can be replaced by steps which require only one matrix squaring each. This efficiency is attained without sacrificing the quadratic convergence of Newton's method.
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