# On Matrices With Displacement Structure: Generalized Operators and Faster Algorithms

## Abstract

*generator*of $A$, this product is classically computed with a cost ranging from $O(\alpha^2 \mathscr{M}(n))$ to $O(\alpha^2 \mathscr{M}(n)\log(n))$ arithmetic operations, depending on the type of structure of $A$; here, $\mathscr{M}$ is a cost function for polynomial multiplication. In this paper, we first generalize classical displacement operators, based on block diagonal matrices with companion diagonal blocks, and then design fast algorithms to perform the task above for this extended class of structured matrices. The cost of these algorithms ranges from $O(\alpha^{\omega-1} \mathscr{M}(n))$ to $O(\alpha^{\omega-1} \mathscr{M}(n)\log(n))$, with $\omega$ such that two $n \times n$ matrices can be multiplied using $O(n^\omega)$ ring operations. By combining this result with classical randomized regularization techniques, we obtain faster Las Vegas algorithms for structured inversion and linear system solving.

### Keywords

### MSC codes

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*Evaluating polynomials at fixed sets of points*, SIAM J. Comput., 4 (1975), pp. 533--539.

*Polynomial and Matrix Computations, Volume 1: Fundamental Algorithms*, Birkhäuser, Boston, 1994.

*Asymptotically fast solution of Toeplitz and related systems of linear equations*, Linear Algebra Appl., 34 (1980), pp. 103--116.

*Solving structured linear systems with large displacement rank*, Theoret. Comput. Sci., 407 (2008), pp. 155--181.

*Complexity issues in bivariate polynomial factorization*, in Proceedings of ISSAC'04, ACM Press, 2004, pp. 42--49.

*Tellegen's principle into practice*, in Proceedings of ISSAC'03, ACM Press, 2003, pp. 37--44.

*Polynomial evaluation and interpolation on special sets of points*, J. Complexity, 21 (2005), pp. 420--446.

*Algebraic Complexity Theory*, Springer, New York, 1997.

*Faster algorithms for multivariate interpolation with multiplicities and simultaneous polynomial approximations*, IEEE Trans. Inform. Theory, 61 (2015), pp. 2370--2387.

*New inversion formulas for matrices classified in terms of their distance from Toeplitz matrices*, Linear Algebra Appl., 27 (1979), pp. 31--60.

*Modern Computer Algebra*, 3rd ed., Cambridge University Press, Cambridge, 2013.

*Complexity of multiplication with vectors for structured matrices*, Linear Algebra Appl., 202 (1994), pp. 163--192.

*Fast algorithms with preprocessing for matrix-vector multiplication problems*, J. Complexity, 10 (1994), pp. 411--427.

*The middle product algorithm I*, Appl. Algebra Engrg. Comm. Comput., 14 (2004), pp. 415--438.

*Computing specified generators of structured matrix inverses*, in Proceedings of ISSAC'10, ACM Press, 2010, pp. 281--288.

*Displacement ranks of a matrix*, Bull. Amer. Math. Soc. (New Series), 1 (1979), pp. 769--773.

*Displacement ranks of matrices and linear equations*, J. Math. Anal. Appl., 68 (1979), pp. 395--407.

*Asymptotically fast solution of Toeplitz-like singular linear systems*, in Proceedings of ISSAC'94, ACM Press, 1994, pp. 297--304.

*Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems*, Math. Comp., 64 (1995), pp. 777--806.

*On Wiedemann's method of solving sparse linear systems*, in AAECC-9, Lecture Notes in Comput. Sci. 539, Springer, Berlin, 1991, pp. 29--38.

*The Theory of Matrices, 2nd. ed.*, Computer Science and Applied Mathematics, Academic Press, New York, 1985.

*Powers of tensors and fast matrix multiplication*, in Proceedings of ISSAC'14, ACM Press, 2014, pp. 296--303.

*Fast Algorithms for Multivariable Systems*, Ph.D. thesis, Stanford University, Stanford, 1974.

*Doubling algorithms for Toeplitz and related equations*, in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, 1980, pp. 954--959.

*A unified superfast algorithm for boundary rational tangential interpolation problems and for inversion and factorization of dense structured matrices*, in Proceedings of 39th FOCS, IEEE, Washington, DC, 1998, pp. 192--201.

*A unified superfast algorithm for confluent tangential interpolation problem and for structured matrices*, in Advanced Signal Processing Algorithms, Architectures, and Implementations, ASPAAI'IX, SPIE, Bellingham, WA, 1999, pp. 312--323.

*Matrix-vector product for confluent Cauchy-like matrices with application to confluent rational interpolation*, in Proceedings of STOC'00, ACM Press, 2000, pp. 573--581.

*Trilinear aggregating with implicit canceling for a new acceleration of matrix multiplication*, Comp. & Maths. with Appls., 8 (1982), pp. 23--34.

*On computations with dense structured matrices*, Math. Comp., 55 (1990), pp. 179--190.

*A unified superfast divide-and-conquer algorithm for structured matrices*. MSRI Preprint 1999-033, Mathematical Sciences Research Institute, Berkeley, CA, April 1999.

*Nearly optimal computations with structured matrices*, in Proceedings of SODA'00, ACM Press, 2000, pp. 953--962.

*Structured Matrices and Polynomials*, Birkhäuser Boston, 2001.

*Transformations of matrix structures work again*, Linear Algebra Appl., 465 (2015), pp. 107--138.

*Inversion of displacement operators*, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 660--677.

*Superfast algorithms for Cauchy-like matrix computations and extensions.*, Linear Algebra Appl., 310 (2000), pp. 83--108.

*Schnelle Multiplikation von Polynomen über Körpern der Charakteristik*2, Acta Inform., 7 (1977), pp. 395--398.

*Schnelle Multiplikation großer Zahlen*, Computing, 7 (1971), pp. 281--292.

*Fast algorithms for elementary operations with complex power series*, Discrete Math. Appl., 20 (2010), pp. 25--60.

*Gaussian elimination is not optimal*, Numer. Math., 13 (1969), pp. 354--356.

## Information & Authors

### Information

#### Published In

#### Copyright

#### History

**Submitted**: 23 February 2016

**Accepted**: 27 February 2017

**Published online**: 1 August 2017

#### Keywords

#### MSC codes

### Authors

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

- Stability Analysis of the Matrix-vector Product (via FFT) for a Toeplitz-like MatrixWSEAS TRANSACTIONS ON MATHEMATICS, Vol. 21 | 25 February 2022
- Computing the Characteristic Polynomial of Generic Toeplitz-like and Hankel-like MatricesProceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation | 18 July 2021
- Subquadratic-Time Algorithms for Normal Basescomputational complexity, Vol. 30, No. 1 | 2 March 2021
- Fast Encoding of AG Codes Over C ab CurvesIEEE Transactions on Information Theory, Vol. 67, No. 3 | 1 Mar 2021
- Algorithms for simultaneous Hermite–Padé approximationsJournal of Symbolic Computation, Vol. 102 | 1 Jan 2021
- Change of Basis for m-primary Ideals in One and Two VariablesProceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation | 8 July 2019
- On Computing the Resultant of Generic Bivariate PolynomialsProceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation | 11 July 2018
- Algorithms for Structured Linear Systems Solving and Their ImplementationProceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation | 23 July 2017