Back to the Roots: Polynomial System Solving, Linear Algebra, Systems Theory

Dreesen, Philipp; Batselier, Kim; Moor, Bart L. R. De

doi:10.3182/20120711-3-be-2027.00217

Cited by 36 publications

(58 citation statements)

References 7 publications

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“…This allows us to use standard numerical linear algebra tools to compute all or some of the solutions in a similar way as companion matrix is used in Matlab's command roots [12] to compute the zeros of a scalar univariate polynomial. Similar ideas can be found in [3,17]. An advantage of our approach is that it does not require symbolic computation; a disadvantage are very large matrices.…”

Section: Problemmentioning

confidence: 78%

“…In this case, as shown in [14], if all eigenvalues of (1) are algebraically simple, which is the generic case, they agree with the finite regular eigenvalues of the pair of singular matrix pencils (3). A numerical algorithm from [15], based on the staircase algorithm by Van Dooren [20], may then be used to compute the common regular part of (3) and extract the finite regular eigenvalues.…”

Section: Introductionmentioning

confidence: 90%

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Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

Hochstenbach

Muhič

Plestenjak

2015

Journal of Computational and Applied Mathematics

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Section: Problemmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 90%

Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

Hochstenbach

Muhič

Plestenjak

2015

Journal of Computational and Applied Mathematics

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“…Observe that relations similar to (19) can be written down for multiplication with any variable x i . Indeed, for every variable x i a corresponding row selection matrix can be derived.…”

Section: Extending Stetter's Eigenvalue Problem To the Projective Casementioning

confidence: 99%

“…Problems such as elimination of variables [15], computing an approximate greatest common divisor [16], computing a Gröbner and border basis [17], finding the affine roots of a polynomial system are all solved numerically in this Polynomial Numerical Linear Algebra (PNLA) framework. The Macaulay matrix plays a central role in all these problems [18,19].…”

Section: Introductionmentioning

confidence: 99%

On the null spaces of the Macaulay matrix

Batselier

Dreesen

Moor

2014

Linear Algebra and its Applications

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MSC: 54B05 12E05 65F99 Keywords: Macaulay matrix Multivariate polynomials Row space Null space Syzygies RootsIn this article both the left and right null space of the Macaulay matrix are described. The left null space is shown to be linked with the occurrence of syzygies in its row space. It is also demonstrated how the dimension of the left null space is described by a piecewise function of polynomials. We present two algorithms that determine these polynomials. Furthermore we show how the finiteness of the number of basis syzygies results in the notion of the degree of regularity. This concept plays a crucial role in describing a basis for the right null space of the Macaulay matrix in terms of differential functionals. We define a canonical null space for the Macaulay matrix in terms of the projective roots of a polynomial system and extend the multiplication property of this canonical basis to the projective case. This results in an algorithm to determine the upper triangular commuting multiplication matrices. Finally, we discuss how Stetter's eigenvalue problem to determine the roots of a multivariate polynomial system can be extended to the case where a multivariate polynomial system has both affine roots and roots at infinity.

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“…There various other techniques based on "resultants" (see e.g. Canny and Emiris (1993), Emiris (1996), Emiris and Mourrain (1999), Dreesen et al (2012)), on "homotopy" (e.g. Garcia and Zangwill (1979), Morgan and Sommese (1987)) as well as the " Ritt-Wu" methods (e.g.…”

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confidence: 99%

Efficient Perturbation Methods for Solving Regime-Switching DSGE Models

Maih¹

2015

SSRN Journal

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In an environment where economic structures break, variances change, distributions shift, conventional policies weaken and past events tend to reoccur, economic agents have to form expectations over different regimes. This makes the regime-switching dynamic stochastic general equilibrium (RS-DSGE) model the natural framework for analyzing the dynamics of macroeconomic variables. We present efficient solution methods for solving this class of models, allowing for the transition probabilities to be endogenous and for agents to react to anticipated events. The solution algorithms derived use a perturbation strategy which, unlike what has been proposed in the literature, does not rely on the partitioning of the switching parameters. These algorithms are all implemented in RISE, a flexible object-oriented toolbox that can easily integrate alternative solution methods. We show that our algorithms replicate various examples found in the literature. Among those is a switching RBC model for which we present a third-order perturbation solution.JEL Classification: C6, E3, G1.

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Back to the Roots: Polynomial System Solving, Linear Algebra, Systems Theory

Cited by 36 publications

References 7 publications

Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

On the null spaces of the Macaulay matrix

Efficient Perturbation Methods for Solving Regime-Switching DSGE Models

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