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21
Supertropical algebra
, 2007
"... Abstract. We develop the algebraic polynomial theory for “supertropical algebra, ” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of ghost elements, which play the key role in our structure theory. Here, we work in a slightly more gener ..."
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Cited by 17 (10 self)
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Abstract. We develop the algebraic polynomial theory for “supertropical algebra, ” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of ghost elements, which play the key role in our structure theory. Here, we work in a slightly more general situation over an arbitrary semiring, and develop a theory which contains the analogs of the basic theorems of classical commutative algebra (such as the Euclidean algorithm and the Hilbert Nullstellensatz), as well as some results without analogs in the classical theory, such as generation of prime ideals of the polynomial semiring by binomials. Examples are also given to show how this theory differs from classical commutative algebra.
Tropical arithmetic & algebra of tropical matrices
, 2005
"... The purpose of this paper is to study the tropical algebra – the algebra over the tropical semiring. We start by introducing a new approach to arithmetic over the maxplus semiring which generalizes the former concept in use. Regarding this new arithmetic, matters of tropical matrices are discusse ..."
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Cited by 14 (9 self)
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The purpose of this paper is to study the tropical algebra – the algebra over the tropical semiring. We start by introducing a new approach to arithmetic over the maxplus semiring which generalizes the former concept in use. Regarding this new arithmetic, matters of tropical matrices are discussed and the properties of these matrices are studied. These are the preceding phases toward the characterization of the tropical inverse matrix which is eventually attained. Further development yields the notion of tropical normalization and the principle of basis change in the tropical sense.
Tropical Scaling of Polynomial Matrices
, 905
"... Abstract The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in p ..."
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Cited by 5 (2 self)
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Abstract The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in particular to this companion form. This scaling, which is inspired by an earlier work of Akian, Bapat, and Gaubert, relies on the computation of “tropical roots”. We give explicit bounds, in a typical case, indicating that these roots provide accurate estimates of the order of magnitude of the different eigenvalues, and we show by experiments that this scaling improves the accuracy (measured by normwise backward error) of the computations, particularly in situations in which the data have various orders of magnitude. In the case of quadratic polynomial matrices, we recover in this way a scaling due to Fan, Lin, and Van Dooren, which coincides with the tropical scaling when the two tropical roots are equal. If not, the eigenvalues generally split in two groups, and the tropical method leads to making one specific scaling for each of the groups.
The level set method for the twosided maxplus eigenproblem
, 2010
"... We consider the maxplus analogue of the eigenproblem for matrix pencils, Ax = λBx. We show that the spectrum of (A,B) (i.e., the set of possible values of λ) is a finite union of intervals, which can be computed by a pseudopolynomial number of calls to an oracle that computes the value of a mean ..."
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Cited by 4 (0 self)
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We consider the maxplus analogue of the eigenproblem for matrix pencils, Ax = λBx. We show that the spectrum of (A,B) (i.e., the set of possible values of λ) is a finite union of intervals, which can be computed by a pseudopolynomial number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between Ax and λBx. The spectrum is obtained as the zero level set of this function.
THEME Modeling, Optimization, and Control of Dynamic SystemsTable of contents
"... 4.3. Switched systems 5 5. Software................................................................................. 6 6. New Results.............................................................................. 6 6.1. New results: geometric control 6 6.2. New results: quantum control 8 6.3. New res ..."
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4.3. Switched systems 5 5. Software................................................................................. 6 6. New Results.............................................................................. 6 6.1. New results: geometric control 6 6.2. New results: quantum control 8 6.3. New results: neurophysiology 8 6.4. New results: switched systems 9
unknown title
, 2007
"... Dynamic and static limitation in multiscale reaction networks, revisited ..."
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Theme: Modeling, Optimization, and Control of Dynamic Systems
, 2010
"... c t i v i t y te p o r ..."
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TROPICAL ROOTS AS APPROXIMATIONS TO EIGENVALUES OF MATRIX POLYNOMIALS∗
"... Abstract. The tropical roots of t×p(x) = max0≤j≤ ` ‖Aj‖xj are points at which the maximum is attained at least twice. These roots, which can be computed in only O(`) operations, can be good approximations to the moduli of the eigenvalues of the matrix polynomial P (λ) = j=0 λ jAj, in particular whe ..."
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Abstract. The tropical roots of t×p(x) = max0≤j≤ ` ‖Aj‖xj are points at which the maximum is attained at least twice. These roots, which can be computed in only O(`) operations, can be good approximations to the moduli of the eigenvalues of the matrix polynomial P (λ) = j=0 λ jAj, in particular when the norms of the matrices Aj vary widely. Our aim is to investigate this observation and its applications. We start by providing annuli defined in terms of the tropical roots of t×p(x) that contain the eigenvalues of P (λ). Our localization results yield conditions under which tropical roots offer order of magnitude approximations to the moduli of the eigenvalues of P (λ). Our tropical localization of eigenvalues are less tight than eigenvalue localization results derived from a generalized matrix version of Pellet’s theorem but they are easier to interpret. Tropical roots are already used to determine the starting points for matrix polynomial eigensolvers based on scalar polynomial root solvers such as the EhrlichAberth method and our results further justify this choice. Our results provide the basis for analyzing the effect of Gaubert and Sharify’s tropical scalings for P (λ) on (a) the conditioning of linearizations of tropically scaled P (λ) and (b) the backward stability of eigensolvers based on linearizations of tropically scaled P (λ). We anticipate that the tropical roots of t×p(x), on which the tropical scalings are based, will help designing polynomial eigensolvers with better numerical properties than standard algorithms for polynomial eigenvalue problems such as that implemented in the MATLAB function polyeig.
unknown title
, 2006
"... Algèbres maxplus et mathématiques de la décision/Maxplus algebras and mathematics of decision ..."
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Algèbres maxplus et mathématiques de la décision/Maxplus algebras and mathematics of decision