Algorithms for the matrix <Emphasis Type="Italic">p</Emphasis>th root

New theoretical results are presented about the principal matrix pth root. In particular, we show that the pth root is related to the matrix sign function and to the Wiener–Hopf factorization, and that it can be expressed as an integral over the unit circle. These results are used in the design and analysis of several new algorithms for the numerical computation of the pth root. We also analyze the convergence and numerical stability properties of Newtons method for the inverse pth root. Preliminary computational experiments are presented to compare the methods. AMS subject classification 15A24, 65H10, 65F30Numerical Analysis Report 454, Manchester Centre for Computational Mathematics, July 2004.Dario A. Bini: This work was supported by MIUR, grant number 2002014121.Nicholas J. Higham: This work was supported by Engineering and Physical Sciences Research Council grant GR/R22612 and by a Royal Society – Wolfson Research Merit Award.     

A way to manage the thermal flexibility of ligand candidates for bioassays

An original way to manage both stereochemistry and conformational constraints in ligand candidates for bioassays is presented with reference to a group of model N,N′-tetrasubstituted o-phthalamides and thioamides. The study shows that a scale of thermal flexibility in solution can be envisaged, the divisions of which are represented by compounds sharing quite similar geometrical features. NMR spectroscopy and powder X-ray analysis were used for the physical chemical investigation. An attempt to exploit the conformational instability of a model thioamide in the medium of a bioassay was also performed.     

Improved cyclic reduction for solving queueing problems

The cyclic reduction technique (Buzbee et al., 1970), rephrased in functional form (Bini and Meini, 1996), provides a numerically stable, quadratically convergent method for solving the matrix equation X = ∑^{+ ∞} _i=0 X_i A_i, where the A_i's are nonnegative k × k matrices such that ∑^{+ ∞} _i=0 A_i is column stochastic. In this paper we propose a further improvement of the above method, based on a point-wise evaluation/interpolation at a suitable set of Fourier points, of the functional relations defining each step of cyclic reduction (Bini and Meini,1996). This new technique allows us to devise an algorithm based on FFT having a lower computational cost and a higher numerical stability. Numerical results and comparisons are provided. This revised version was published online in August 2006 with corrections to the Cover Date.     

Relaxed functional iteration techniques for the numerical solution of M/G/1 type Markov chains

We introduce a new iterative method for the computation of the minimal nonnegative solutionG of the matrix equation , arising in the numerical solution of M/G/1 type Markov chains. The idea consists in applying a relaxation technique to customarily used functional iteration formulas. The proposed method is easy to implement and outperforms, in terms of number of iterations and execution time, the standard functional iteration techniques.     

Efficient computation of the extreme solutions of

We propose a new quadratically convergent algorithm, having a low computational cost per step and good numerical stability properties, which allows the simultaneous approximation of the extreme solutions of the matrix equations and . The algorithm is based on the cyclic reduction method.