Superfast solution of real positive definite toeplitz systems
SIAM Journal on Matrix Analysis and Applications
Divide-and-conquer solutions of least-squares problems for matrices with displacement structure
SIAM Journal on Matrix Analysis and Applications
LAPACK's user's guide
FORTRAN subroutines for general Toeplitz systems
ACM Transactions on Mathematical Software (TOMS)
Generalized Displacement Structure for Block-Toeplitz,Toeplitz-Block, and Toeplitz-Derived Matrices
SIAM Journal on Matrix Analysis and Applications
Fast Gaussian elimination with partial pivoting for matrices with displacement structure
Mathematics of Computation
Displacement structure: theory and applications
SIAM Review
Factorizations of Cauchy matrices
Journal of Computational and Applied Mathematics - Special issue: dedicated to William B. Gragg on the occasion of his 60th Birthday
Stable and Efficient Algorithms for Structured Systems of Linear Equations
SIAM Journal on Matrix Analysis and Applications
Fast reliable algorithms for matrices with structure
Fast reliable algorithms for matrices with structure
An updated set of basic linear algebra subprograms (BLAS)
ACM Transactions on Mathematical Software (TOMS)
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems
SIAM Journal on Matrix Analysis and Applications
A Superfast Toeplitz Solver with Improved Numerical Stability
SIAM Journal on Matrix Analysis and Applications
A Method for Generating Infinite Positive Self-adjoint Test Matrices and Riesz Bases
SIAM Journal on Matrix Analysis and Applications
Fast Solution of Toeplitz- and Cauchy-Like Least-Squares Problems
SIAM Journal on Matrix Analysis and Applications
A Superfast Algorithm for Toeplitz Systems of Linear Equations
SIAM Journal on Matrix Analysis and Applications
smt: a Matlab toolbox for structured matrices
Numerical Algorithms
Extended companion matrix for approximate GCD
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
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We describe a fast solver for linear systems with reconstructible Cauchy-like structure, which requires O(rn 2) floating point operations and O(rn) memory locations, where n is the size of the matrix and r its displacement rank. The solver is based on the application of the generalized Schur algorithm to a suitable augmented matrix, under some assumptions on the knots of the Cauchy-like matrix. It includes various pivoting strategies, already discussed in the literature, and a new algorithm, which only requires reconstructibility. We have developed a software package, written in Matlab and C-MEX, which provides a robust implementation of the above method. Our package also includes solvers for Toeplitz(+Hankel)-like and Vandermonde-like linear systems, as these structures can be reduced to Cauchy-like by fast and stable transforms. Numerical experiments demonstrate the effectiveness of the software.