An Iterative Algorithm for Computing the Best Estimate of an Orthogonal Matrix
Åke Björck, C. Bowie · SIAM Journal on Numerical Analysis · 1971
The closest unitary matrix, measured in Euclidean norm, to a given rectangular matrix A is known to be the unitary factor in the polar decomposition of A. The paper gives a family of iterative methods of order of convergence $p + 1,\, p = 1,2,3, \cdots $, for computing this matrix. The methods are especially efficient when the columns of A are not too far from being orthonormal. The choice of order of convergence to minimize the amount of computation is discussed. Global convergence properties for the methods of order $ \leqq 4$ are studied and sufficient conditions for convergence in terms of $\| {I - A^H A} \|$ are given.