716 research outputs found
Certified Algorithms for proving the structural stability of two dimensional systems possibly with parameters
International audienceIn [1], a new method for testing the structural stability of multidimensional systems has been presented. The key idea of this method is to reduce the problem of testing the structural stability to that of deciding if an algebraic set has real points. Following the same idea, we consider in this work the specific case of two-dimensional systems and focus on the practical efficiency aspect. For such systems, the problem of testing the stability is reduced to that of deciding if a bivariate algebraic system with finitely many solutions has real ones. Our first contribution is an algorithm that answers this question while achieving practical efficiency. Our second contribution concerns the stability of two dimensional systems with parameters. More precisely, given a two-dimensional system depending on a set of parameters, we present a new algorithm that computes regions of the parameter space in which the considered system is structurally stable
A local construction of the Smith normal form of a matrix polynomial
We present an algorithm for computing a Smith form with multipliers of a
regular matrix polynomial over a field. This algorithm differs from previous
ones in that it computes a local Smith form for each irreducible factor in the
determinant separately and then combines them into a global Smith form, whereas
other algorithms apply a sequence of unimodular row and column operations to
the original matrix. The performance of the algorithm in exact arithmetic is
reported for several test cases.Comment: 26 pages, 6 figures; introduction expanded, 10 references added, two
additional tests performe
Modular quantum signal processing in many variables
Despite significant advances in quantum algorithms, quantum programs in
practice are often expressed at the circuit level, forgoing helpful structural
abstractions common to their classical counterparts. Consequently, as many
quantum algorithms have been unified with the advent of quantum signal
processing (QSP) and quantum singular value transformation (QSVT), an
opportunity has appeared to cast these algorithms as modules that can be
combined to constitute complex programs. Complicating this, however, is that
while QSP/QSVT are often described by the polynomial transforms they apply to
the singular values of large linear operators, and the algebraic manipulation
of polynomials is simple, the QSP/QSVT protocols realizing analogous
manipulations of their embedded polynomials are non-obvious. Here we provide a
theory of modular multi-input-output QSP-based superoperators, the basic unit
of which we call a gadget, and show they can be snapped together with LEGO-like
ease at the level of the functions they apply. To demonstrate this ease, we
also provide a Python package for assembling gadgets and compiling them to
circuits. Viewed alternately, gadgets both enable the efficient block encoding
of large families of useful multivariable functions, and substantiate a
functional-programming approach to quantum algorithm design in recasting QSP
and QSVT as monadic types.Comment: 15 pages + 9 figures + 4 tables + 45 pages supplement. For codebase,
see https://github.com/ichuang/pyqsp/tree/bet
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