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...

We construct interpolation operators for functions taking values in a symmetric space—a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the generalized polar decomposition—a generalization of the well-known factorizat...

In this investigation, the interaction of C20 and Si2H2 molecules was explored in the M06-2X/6-311++G(d,p)level of theory in gas solution phases. The obtained interaction energy values with standard method werecorrected by basis set superposition error (BSSE) during the geometry optimization for all molecules atthe same level of theory. Also, the bonding interaction between th...

Let B be an m × n (m ≥ n) complex (or real) matrix. It is known that there is a unique polar decomposition B = QH, where Q * Q = I, the n × n identity matrix, and H is positive definite, provided B has full column rank. If B is perturbed to B, how do the polar factors Q and H changes? This question has been investigated quite extensively, but most work so far was on how the perturbation changed...

We constructively prove (i.e., in ZF set theory) a decomposition theorem for certain order bounded disjointness preserving operators between any two Riesz spaces, real or complex, in terms of the absolute value of another order bounded disjointness preserving operator. In this way, we constructively generalize results by Abramovich, Arensen and Kitover (1992), Grobler and Huijsmans (1997), Hart...

For unbounded operators t, s between Hilbert C*-modules which admit the polar decompositions V|t|, W|s|, respectively, we obtain an explicit upper bound estimate for the gap between t and s in terms of the norm of the bounded operators V − W , C|t| − C|s| and C|t∗| − C|s∗|, where C|t| and C|s| are the Cayley transforms of |t| and |s|. The result are used to drive a criterion for continuity of t...