Stable iterations for the matrix square root

Any matrix with no nonpositive real eigenvalues has a unique square root for which every eigenvalue lies in the open right half-plane. A link between the matrix sign function and this square root is exploited to derive both old and new iterations for the square root from iterations for the sign function. One new iteration is a quadratically convergent Schulz iteration based entirely on matrix multiplication; it converges only locally, but can be used to compute the square root of any nonsingular M-matrix. A new Padé iteration well suited to parallel implementation is also derived and its properties explained. Iterative methods for the matrix square root are notorious for suffering from numerical instability. It is shown that apparently innocuous algorithmic modifications to the Padé iteration can lead to instability, and a perturbation analysis is given to provide some explanation. Numerical experiments are included and advice is offered on the choice of iterative method for computing the matrix square root.     

Horning-crown macrocycles: novel hybrids of calixarenes and crown ethers

[reaction: see text] Novel macrocycles possessing ether linkages and 2,6-disubstituted phenolics were produced in one step and with 100% atom economy by isoaromatization of chameleon macrocyclic precursors possessing 2,6-diarylidenecyclohexanone moieties. Intramolecular hydrogen bonding of the phenolic hydrogen atoms influenced the shape of the macrocycles and dictated host-guest behavior.     

Computing the Matrix Cosine

An algorithm is developed for computing the matrix cosine, building on a proposal of Serbin and Blalock. The algorithm scales the matrix by a power of 2 to make the -norm less than or equal to 1, evaluates a Padé approximant, and then uses the double angle formula cos(2A)=2cos(A)²–I to recover the cosine of the original matrix. In addition, argument reduction and balancing is used initially to decrease the norm. We give truncation and rounding error analyses to show that an [8,8] Padé approximant produces the cosine of the scaled matrix correct to machine accuracy in IEEE double precision arithmetic, and we show that this Padé approximant can be more efficiently evaluated than a corresponding Taylor series approximation. We also provide error analysis to bound the propagation of errors in the double angle recurrence. Numerical experiments show that our algorithm is competitive in accuracy with the Schur–Parlett method of Davies and Higham, which is designed for general matrix functions, and it is substantially less expensive than that method for matrices of -norm of order 1. The dominant computational kernels in the algorithm are matrix multiplication and solution of a linear system with multiple right-hand sides, so the algorithm is well suited to modern computer architectures.     

Time-stepping and preserving orthonormality

Certain applications produce initial value ODEs whose solutions, regarded as time-dependent matrices, preserve orthonormality. Such systems arise in the computation of Lyapunov exponents and the construction of smooth singular value decompositions of parametrized matrices. For some special problem classes, there exist time-stepping methods that automatically inherit the orthonormality preservation. However, a more widely applicable approach is to apply a standard integrator and regularly replace the approximate solution by an orthonormal matrix. Typically, the approximate solution is replaced by the factorQ from its QR decomposition (computed, for example, by the modified Gram-Schmidt method). However, the optimal replacement—the one that is closest in the Frobenius norm—is given by the orthonormal polar factor. Quadratically convergent iteration schemes can be used to compute this factor. In particular, there is a matrix multiplication based iteration that is ideally suited to modern computer architectures. Hence, we argue that perturbing towards the orthonormal polar factor is an attractive choice, and we consider performing a fixed number of iterations. Using the optimality property we show that the perturbations improve the departure from orthonormality without significantly degrading the finite-time global error bound for the ODE solution. Our analysis allows for adaptive time-stepping, where a local error control process is driven by a user-supplied tolerance. Finally, using a recent result of Sun, we show how the global error bound carries through to the case where the orthonormal QR factor is used instead of the orthonormal polar factor. This work was supported by Engineering and Physical Sciences Research Council grants GR/H94634 and GR/K80228.     

Coupled mode perturbation theory of range dependence

The conventional coupled mode solution is combined with perturbation theory to give a fast, accurate range-dependent normal mode solution for deep water acoustic propagation. Perturbation theory is used to calculate the new normal modes at each range step. The new modes are obtained as a linear combination of the modes for the previous step without requiring a numerical solution of the depth-separated wave equation. The process may be repeated for many steps and yields normal modes and eigenvalues which are sufficiently accurate for solution of practical problems in deep water. The method is applied to long-range propagation through oceanic fronts.     

On pth roots of stochastic matrices

In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic pth root of a stochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of matrix pth roots, and in particular on the existence of stochastic pth roots of stochastic matrices. Our contributions include characterization of when a real matrix has a real pth root, a classification of pth roots of a possibly singular matrix, a sufficient condition for a pth root of a stochastic matrix to have unit row sums, and the identification of two classes of stochastic matrices that have stochastic pth roots for all p. We also delineate a wide variety of possible configurations as regards existence, nature (primary or nonprimary), and number of stochastic roots, and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix.     

Taming functionality: easy-to-handle chiral phosphiranes

Enantiopure chiral phosphiranes possessing a binaphthyl backbone demonstrate remarkable thermal stability, are highly resistant to air-oxidation and are effective ligands in catalytic asymmetric hydrosilylations.