A hyperbolic tangent identity and the geometry of Padé sign function iterations

The rational iterations obtained from certain Padé approximations associated with computing the matrix sign function are shown to be equivalent to iterations involving the hyperbolic tangent and its inverse. Using this equivalent formulation many results about these Padé iterations, such as global convergence, the semi-group property under composition, and explicit partial fraction decompositions can be obtained easily. In the second part of the paper it is shown that the behavior of points under the Padé iterations can be expressed, via the Cayley transform, as the combined result of a completely regular iteration and a chaotic iteration. These two iterations are decoupled, with the chaotic iteration taking the form of a multiplicative linear congruential random number generator where the multiplier is equal to the order of the Padé approximation.

[1]  E. Denman,et al.  The matrix sign function and computations in systems , 1976 .

[2]  S. K. Park,et al.  Random number generators: good ones are hard to find , 1988, CACM.

[3]  L. Shieh,et al.  Computation of the matrix sign function using continued fraction expansion , 1994, IEEE Trans. Autom. Control..

[4]  James Demmel,et al.  Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I , 1993, PPSC.

[5]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[6]  N. Higham.  The matrix sign decomposition and its relation to the polar decomposition , 1994 .

[7]  Alan F. Beardon,et al.  Iteration of Rational Functions , 1991 .

[8]  A. Laub,et al.  Rational iterative methods for the matrix sign function , 1991 .

[9]  Judith Gardiner,et al.  A generalization of the matrix sign function solution for algebraic Riccati equations , 1985, 1985 24th IEEE Conference on Decision and Control.

[10]  R. B. Leipnik,et al.  Rapidly convergent recursive solution of quadratic operator equations , 1971 .

[11]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[12]  J. L. Howland.  The sign matrix and the separation of matrix eigenvalues , 1983 .

[13]  Z. Kovarik.  Some Iterative Methods for Improving Orthonormality , 1970 .

[14]  J. Ritt.  Periodic functions with a multiplication theorem , 1922 .

[15]  J. D. Roberts,et al.  Linear model reduction and solution of the algebraic Riccati equation by use of the sign function , 1980 .

[16]  Einar Hille,et al.  Introduction to complex analysis , 1983 .

[17]  Alan J. Laub,et al.  A Parallel Algorithm for the Matrix Sign Function , 1990, Int. J. High Speed Comput..

[18]  Gerald Bierman.  Computational aspects of the matrix sign function solution to the ARE , 1984, The 23rd IEEE Conference on Decision and Control.

[19]  A. Laub,et al.  Matrix-sign algorithms for Riccati equations , 1992 .