A Traub-like algorithm for Hessenberg-quasiseparable-Vandermonde matrices of arbitrary order

Although Gaussian elimination uses O(n 3) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n 2) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomial-Vandermonde matrices involving real orthogonal polynomials, and the Szego polynomials.

[1]  Thomas Kailath,et al.  Displacement-structure approach to polynomial Vandermonde and related matrices , 1997 .

[2]  Lothar Reichel,et al.  Fast inversion of vandermonde-like matrices involving orthogonal polynomials , 1993 .

[3]  Vadim Olshevsky,et al.  Eigenvector computation for almost unitary Hessenberg matrices and inversion of Szegő-Vandermonde matrices via discrete transmission lines , 1998 .

[4]  Israel Gohberg,et al.  A Björck–Pereyra-type algorithm for Szegö–Vandermonde matrices based on properties of unitary Hessenberg matrices , 2007 .

[5]  I. Gohberg,et al.  A modification of the Dewilde-van der Veen method for inversion of finite structured matrices , 2002 .

[6]  SIAM,et al.  ASSOCIATED POLYNOMIALS AND UNIFORM METHODS FOR THE SOLUTION OF LINEAR PROBLEMS , .

[7]  I︠a︡. L. Geronimus.  Polynomials orthogonal on a circle and their applications , 1954 .

[8]  Charles D. Schaper,et al.  Communications, Computation, Control, and Signal Processing: A Tribute to Thomas Kailath , 1997 .

[9]  F. D. Parker.  Inverses of Vandermonde Matrices , 1964 .

[10]  I. Gohberg,et al.  On a new class of structured matrices , 1999 .

[11]  I. Gohberg,et al.  Linear complexity inversion algorithms for a class of structured matrices , 1999 .

[12]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[13]  W. G. Horner,et al.  A new method of solving numerical equations of all orders, by continuous approximation , 1815 .

[14]  John Maroulas,et al.  Polynomials with respect to a general basis. I. Theory , 1979 .

[15]  Israel Gohberg,et al.  Fast inversion of Chebyshev--Vandermonde matrices , 1994 .

[16]  Lothar Reichel,et al.  Chebyshev-Vandermonde systems , 1991 .

[17]  Israel Gohberg,et al.  The Fast Generalized Parker-Traub Algorithm for Inversion of Vandermonde and Related Matrices , 1997, J. Complex..

[18]  V. Olshevsky.  Pivoting for structured matrices and rational tangential interpolation , 2001 .

[19]  Å. Björck,et al.  Solution of Vandermonde Systems of Equations , 1970 .

[20]  Nicholas J. Higham,et al.  Stability analysis of algorithms for solving confluent Vandermonde-like systems , 1990 .

[21]  Israel Gohberg,et al.  A Fast Björck-Pereyra-Type Algorithm for Solving Hessenberg-Quasiseparable-Vandermonde Systems , 2009, SIAM J. Matrix Anal. Appl..

[22]  T. Constantinescu,et al.  Schur's Algorithm and Several Applications , 1992 .

[23]  Israel Gohberg,et al.  Eigenstructure of order-one-quasiseparable matrices. Three-term and two-term recurrence relations , 2005 .

[24]  Eugene E. Tyrtyshnikov,et al.  Structured matrices: recent developments in theory and computation , 2001 .