On a fast and nearly division-free algorithm for the characteristic polynomial

Johansson, Fredrik

doi:10.48550/arxiv.2011.12573

Cited by 3 publications

(5 citation statements)

References 22 publications

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“…Berkowitz algorithm [49] is faster), it is a rather simple and general way to solve the inverse of a polynomial matrix problem. Despite the recursive nature of FLA, it can be easily modified to carry out the N matrix multiplications in parallel [46,47,50,51].…”

Section: Discussionmentioning

confidence: 99%

“…For larger matrix dimensions or higher degree polynomials that the ones analyzed in this paper, FLA might suffer from numerical instability in the computation of the polynomial coefficients due to accumulated errors in the trace in Eq. (10) and from the recursive nature of the successive polynomial coefficients [45,46]. However in Ref.…”

Section: Comparison With Recursive Approachesmentioning

confidence: 99%

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2D topological matter from a boundary Green's functions perspective: Faddeev-LeVerrier algorithm implementation

Alvarado

Yeyati

2022

SciPost Phys.

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Since the breakthrough of twistronics a plethora of topological phenomena in correlated systems has appeared. These devices can be typically analyzed in terms of lattice models using Green's function techniques. In this work we introduce a general method to obtain the boundary Green's function of such models taking advantage of the numerical Faddeev-LeVerrier algorithm to circumvent some analytical constraints of previous works. We illustrate our formalism analyzing the edge features of a Chern insulator, the Kitaev square lattice model for a topological superconductor and the Checkerboard lattice hosting topological flat bands. The efficiency and accuracy of the method is demonstrated by comparison to recursive Green's function calculations.

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Section: Discussionmentioning

confidence: 99%

Section: Comparison With Recursive Approachesmentioning

confidence: 99%

2D topological matter from a boundary Green's functions perspective: Faddeev-LeVerrier algorithm implementation

Alvarado

Yeyati

2022

SciPost Phys.

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“…Berkowitz algorithm [41] is faster), it is a rather simple and general way to solve the inverse of a polynomial matrix problem. Despite the recursive nature of FLA, it can be easily modified to carry out the N matrix multiplications in parallel [38,39,[42][43][44].…”

Section: Discussionmentioning

confidence: 99%

Section: ĝRcmentioning

confidence: 99%

2D topological matter from a boundary Green's functions perspective: Faddeev-LeVerrier algorithm implementation

Alvarado¹,

Yeyati²

2021

Preprint

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“…This algorithm is poorly known and has been rediscovered several times (see e.g. [Joh20]). We adapt this algorithm for secure MPC using classical techniques.…”

Section: Our Contributionsmentioning

confidence: 99%

Secure Multi-Party Linear Algebra with Perfect Correctness

Maire,

Vergnaud

2024

IACR CiC

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We present new secure multi-party computation protocols for linear algebra over a finite field, which improve the state-of-the-art in terms of security. We look at the case of unconditional security with perfect correctness, i.e., information-theoretic security without errors. We notably propose an expected constant-round protocol for solving systems of m linear equations in n variables over Fq with expected complexity O(k n^2.5 + k m) (where complexity is measured in terms of the number of secure multiplications required) with k > m(m+n)+1. The previous proposals were not error-free: known protocols can indeed fail and thus reveal information with probability Omega(poly(m)/q). Our protocols are simple and rely on existing computer-algebra techniques, notably the Preparata-Sarwate algorithm, a simple but poorly known “baby-step giant-step” method for computing the characteristic polynomial of a matrix, and techniques due to Mulmuley for error-free linear algebra in positive characteristic.

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On a fast and nearly division-free algorithm for the characteristic polynomial

Cited by 3 publications

References 22 publications

2D topological matter from a boundary Green's functions perspective: Faddeev-LeVerrier algorithm implementation

2D topological matter from a boundary Green's functions perspective: Faddeev-LeVerrier algorithm implementation

2D topological matter from a boundary Green's functions perspective: Faddeev-LeVerrier algorithm implementation

Secure Multi-Party Linear Algebra with Perfect Correctness

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