A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability

We propose a scaling scheme for Newton's iteration for calculating the polar decomposition. The scaling factors are generated by a simple scalar iteration in which the initial value depends only on estimates of the extreme singular values of the original matrix, which can, for example, be the Frobenius norms of the matrix and its inverse. In exact arithmetic, for matrices with condition number no greater than 10¹⁶, with this scaling scheme no more than 9 iterations are needed for convergence to the unitary polar factor with a convergence tolerance roughly equal to 10⁻¹⁶. It is proved that if matrix inverses computed in finite precision arithmetic satisfy a backward-forward error model, then the numerical method is backward stable. It is also proved that Newton's method with Higham's scaling or with Frobenius norm scaling is backward stable.