# A Superfast Toeplitz Solver with Improved Numerical Stability

@article{Stewart2003AST, title={A Superfast Toeplitz Solver with Improved Numerical Stability}, author={Michael Stewart}, journal={SIAM J. Matrix Anal. Appl.}, year={2003}, volume={25}, pages={669-693} }

This paper describes a new O(n log3(n)) solver for the positive definite Toeplitz system Tx=b. Instead of computing generators for the inverse of T, the new algorithm adjoins b to T and applies a superfast Schur algorithm to the resulting augmented matrix. The generators of this augmented matrix and its Schur complements are used by a divide-and-conquer block back-substitution routine to complete the solution of the system. The goal is to avoid the well-known numerical instability inherent in…

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## 36 Citations

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## References

SHOWING 1-10 OF 20 REFERENCES

### A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 2001

A stabilized superfast solver for nonsymmetric Toeplitz systems Tx=b is presented, expressed in such a way that the matrix-vector product T^-1b can be calculated via FFTs and Hadamard products.

### Asymptotically fast solution of toeplitz and related systems of linear equations

- Computer Science
- 1980

### On the stability of the Bareiss and related Toeplitz factorization algorithms

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 1995

A numerical stability analysis of factorization algorithms for computing the Cholesky decomposition of symmetric positive definite matrices of displacement rank 2 shows that the Bareiss algorithm for factorizing a symmetricpositive definite Toeplitz matrix is in the class and hence the BareISS algorithm is stable.

### Stabilizing the Generalized Schur Algorithm

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 1996

A perturbation analysis is used to indicate the best accuracy that can be expected from a finite-precision algorithm that uses the generator matrix as the input data and shows that the modified Schur algorithm is backward stable for a large class of structured matrices.

### Superfast solution of real positive definite toeplitz systems

- Computer Science
- 1988

An implementation of the generalized Schur algorithm for the superfast solution of real positive definite Toeplitz systems of order n where n = 2^\nu and t is the number of positive definite systems.

### Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants

- Computer ScienceJ. Algorithms
- 1980

### On Hyperbolic Triangularization: Stability and Pivoting

- Mathematics
- 1998

This paper treats the problem of triangularizing a matrix by hyperbolic Householder transformations. The stability of this method, which finds application in block updating and fast algorithms for…

### Hyperbolic householder transforms

- Computer Science
- 1988

A class of transformation matrices is developed with a nonorthogonal property designed to permit the efficient deletion of data from least-squares problems, which are shown to effect deletion, or simultaneous addition and deletion, of data with much less sensitivity to rounding errors than for techniques based on normal equations.

### Stability Issues in the Factorization of Structured Matrices

- Computer Science
- 1997

An error analysis of the generalized Schur algorithm of Kailath and Chun is provided, finding that if this algorithm is implemented with hyperbolic transformations in the factored form which is well known to provide numerical stability in the context of Cholesky downdating, then the generalizedSchur algorithm will be stable.