Backward Stability of Iterations for Computing the Polar Decomposition
نویسندگان
چکیده
منابع مشابه
Backward Stability of Iterations for Computing the Polar Decomposition
Among the many iterations available for computing the polar decomposition the most practically useful are the scaled Newton iteration and the recently proposed dynamically weighted Halley iteration. Effective ways to scale these and other iterations are known, but their numerical stability is much less well understood. In this work we show that a general iteration Xk+1 = f(Xk) for computing the...
متن کاملA 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...
متن کاملComputing the Polar Decomposition—with Applications
A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate of convergence and it is shown how reliable estimates of the optimal parameters may be computed in practice. To add to the known best approximation property of the unitary polar factor, the Hermit...
متن کاملA Parallel Algorithm for Computing the Polar Decomposition
The polar decomposition A = UH of a rectangular matrix A, where U is unitary and H is Hermitian positive semidefinite, is an important tool in various applications, including aerospace computations, factor analysis and signal processing. We consider a pth order iteration for computing U that involves p independent matrix inversions per step and which is hence very amenable to parallel computati...
متن کاملOptimizing Halley's Iteration for Computing the Matrix Polar Decomposition
We introduce a dynamically weighted Halley (DWH) iteration for computing the polar decomposition of a matrix, and prove that the new method is globally and asymptotically cubically convergent. For matrices with condition number no greater than 1016, the DWH method needs at most 6 iterations for convergence with the tolerance 10−16. The Halley iteration can be implemented via QR decompositions w...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: SIAM Journal on Matrix Analysis and Applications
سال: 2012
ISSN: 0895-4798,1095-7162
DOI: 10.1137/110857544