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45
Double Bruhat Cells And Total Positivity
"... this paper we extend the results of [3, 4] to the whole variety G0 . We will try to make the point that the natural framework for the study of G0 is provided by the decomposition of G into the disjoint union of double Bruhat cells G = BuB " B \Gamma vB \Gamma ; here B and B \Gamma are two opposit ..."
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Cited by 90 (18 self)
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this paper we extend the results of [3, 4] to the whole variety G0 . We will try to make the point that the natural framework for the study of G0 is provided by the decomposition of G into the disjoint union of double Bruhat cells G = BuB " B \Gamma vB \Gamma ; here B and B \Gamma are two opposite Borel subgroups in G, and u and v belong to the Weyl group W of G. We believe these double cells to be a very interesting object of study in its own right. The term "cells" might be misleading: in fact, the topology of G is in general quite nontrivial. (In some special cases, the "real part" of G was studied in [20, 21]. V. Deodhar [9] studied the intersections BuB " B \Gamma vB whose properties are very different from those of G .) We study a family of birational parametrizations of G , one for each reduced expression i of the element (u; v) in the Coxeter group W \Theta W . Every such parametrization can be thought of as a system of local coordinates in G call these coordinates the factorization parameters associated to i. They are obtained by expressing a generic element x 2 G as an element of the maximal torus H = B " B \Gamma multiplied by the product of elements of various oneparameter subgroups in G associated with simple roots and their negatives; the reduced expression i prescribes the order of factors in this product. The main technical result of this paper (Theorem 1.9) is an explicit formula for these factorization parameters as rational functions on the double Bruhat cell G . Theorem 1.9 is formulated in terms of a special family of regular functions \Delta fl ;ffi on the group G. These functions are suitably normalized matrix coefficients corresponding to pairs of extremal weights (fl; ffi ) in some fundamental representation of G. Again, we b...
Computing the Singular Value Decomposition with High Relative Accuracy
 Linear Algebra Appl
, 1997
"... We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the a ..."
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Cited by 55 (12 self)
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We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, whichin general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as #nite element problems and quantum mechanics, it is the smallest singular values that havephysical meaning, and should be determined accurately by the data. Many recent papers have identi#ed special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite di#erent, motivating us to seek a co...
Total positivity: tests and parametrizations
 Math. Intelligencer
"... A matrix is totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative) real numbers. The first systematic study of these classes of matrices was undertaken in the 1930s by F. R. Gantmacher and M. G. Krein [20, 21, 22], who established their remarkable spectral pr ..."
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Cited by 43 (10 self)
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A matrix is totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative) real numbers. The first systematic study of these classes of matrices was undertaken in the 1930s by F. R. Gantmacher and M. G. Krein [20, 21, 22], who established their remarkable spectral properties (in particular,
Graph Colorings and Related Symmetric Functions: Ideas and Applications
, 1998
"... this paper we will report on further work related to this symmetric function ..."
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Cited by 35 (2 self)
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this paper we will report on further work related to this symmetric function
Numerical Evidence For A Conjecture In Real Algebraic Geometry
, 1998
"... Homotopies for polynomial systems provide computational evidence for a challenging instance of a conjecture about whether all solutions are real. The implementation of SAGBI homotopies involves polyhedral continuation, flat deformation and cheater's homotopy. The numerical difficulties are overcome ..."
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Cited by 23 (4 self)
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Homotopies for polynomial systems provide computational evidence for a challenging instance of a conjecture about whether all solutions are real. The implementation of SAGBI homotopies involves polyhedral continuation, flat deformation and cheater's homotopy. The numerical difficulties are overcome if we work in the true synthetic spirit of the Schubert calculus by selecting the numerically most favorable equations to represent the geometric problem. Since a wellconditioned polynomial system allows perturbations on the input data without destroying the reality of the solutions we obtain not just one instance, but a whole manifold of systems that satisfy the conjecture. Also an instance that involves totally positive matrices has been verified. The optimality of the solving procedure is a promising first step towards the development of numerically stable algorithms for the pole placement problem in linear systems theory.
Accurate eigenvalues and SVDs of totally nonnegative matrices
 SIAM J. Matrix Anal. Appl
, 2005
"... Abstract. We consider the class of totally nonnegative (TN) matrices—matrices all of whose minors are nonnegative. Any nonsingular TN matrix factors as a product of nonnegative bidiagonal matrices. The entries of the bidiagonal factors parameterize the set of nonsingular TN matrices. We present new ..."
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Cited by 23 (10 self)
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Abstract. We consider the class of totally nonnegative (TN) matrices—matrices all of whose minors are nonnegative. Any nonsingular TN matrix factors as a product of nonnegative bidiagonal matrices. The entries of the bidiagonal factors parameterize the set of nonsingular TN matrices. We present new O(n 3) algorithms that, given the bidiagonal factors of a nonsingular TN matrix, compute its eigenvalues and SVD to high relative accuracy in floating point arithmetic, independent of the conventional condition number. All eigenvalues are guaranteed to be computed to high relative accuracy despite arbitrary nonnormality in the TN matrix. We prove that the entries of the bidiagonal factors of a TN matrix determine its eigenvalues and SVD to high relative accuracy. We establish necessary and sufficient conditions for computing the entries of the bidiagonal factors of a TN matrix to high relative accuracy, given the matrix entries. In particular, our algorithms compute all eigenvalues and the SVD of TN Cauchy, Vandermonde, Cauchy–Vandermonde, and generalized Vandermonde matrices to high relative accuracy.
Probabilistic bounds on the coefficients of polynomials with only real zeros
 J. Combin. Theory Ser. A
, 1997
"... The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and sec ..."
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Cited by 20 (0 self)
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The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and second kinds, and Eulerian numbers. Assuming the ak are nonnegative, A(1)> 0 and that A(z) is not constant, it is known that A(z) has only real zeros i the normalized sequence (a 0=A(1);;an=A(1)) is the probability distribution of the Research supported in part by N.S.F. Grant MCS9404345 1 number of successes in n independent trials for some sequence of success probabilities. Such sequences (a 0;;an) are also known to be characterized by total positivity of the in nite matrix (ai,j) indexed by nonnegative integers i and j. This papers reviews inequalities and approximations for such sequences, called Polya frequency sequences which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.
A shape preserving representation with an evaluation algorithm of linear complexity
 Computer Aided Geometric Design
, 2003
"... linear complexity ..."
Immanants of totally positive matrices are nonnegative
 Bull. London Math. Soc
, 1991
"... Let Mn{k) denote the algebra of n x n matrices over some field k of characteristic zero. For each A>valued function / on the symmetric group Sn, we may define a corresponding matrix function on Mn(k) in which (fljji— •> X \ w)&i (i)' " a () ' (U ..."
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Cited by 10 (0 self)
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Let Mn{k) denote the algebra of n x n matrices over some field k of characteristic zero. For each A>valued function / on the symmetric group Sn, we may define a corresponding matrix function on Mn(k) in which (fljji— •> X \ w)&i (i)' " a () ' (U