Computation of unimodular matrices

Bruno, A D; Azimov, A. A.

doi:10.20948/prepr-2022-46

KIAM Prepr.

2022

DOI: 10.20948/prepr-2022-46

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Computation of unimodular matrices

A D Bruno¹,

A. A. Azimov²

Abstract: Here we give an algorithm for solving the following problem. Let m<n integer vectors be given in the n-dimensional real space. Their linear span forms a linear subspace L in R<sup>n</sup>. It is required to calculate such an unimodular matrix that a linear transformation with it transforms the subspace L into a coordinate one. Also, programs that implement the algorithms and power transformations, for which they are needed, are given.

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2022

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Cited by 2 publications

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Computation of asymptotic forms of solutions to system of nonlinear partial differential equations

Bruno¹,

Batkhin²

2022

KIAM Prepr.

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Here we considerably develop the methods of power geometry for a system of partial differential equations and apply them to two different fluid dynamics problems: computing the boundary layer on a needle in the first approximation and computing the asymptotic forms of solutions to problem of evolution of the turbulent flow. For each equation of the system, its Newton polyhedron and its hyperfaces with their normals and truncated equations are calculated. To simplify the truncated systems, power-logarithmic transformations are used and the truncated systems are further extracted. Here we propose algorithms for computing unimodular matrices of power transformations for differential equations. Results: 1) the boundary layer on the needle is absent in liquid, while in gas it is described in the first approximation; 2) the solutions to the problem of evolution of turbulent flow have 8 asymptotic forms presented explicitly.

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Computation of asymptotic forms of solutions to system of nonlinear partial differential equations

Bruno¹,

Batkhin²

2022

KIAM Prepr.

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Strong resonances in nonlinear Hamiltonian system

Batkhin¹,

Khaydarov²

2022

KIAM Prepr.

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To investigate formal stability of an equilibrium of a multi-parameter Hamiltonian system with three degrees of freedom in the case of common position conditions for the existence of resonances of the third and fourth orders of multiplicity are found. These conditions are formulated as zeroes of polynomials from the coefficients of the characteristic polynomial of the linear part Hamilton system. We describe the partition of the set of stability in the space of coefficients of the characteristic polynomial into such parts where strong resonances are absent and where Bruno’s Theorem can be applied to determine the formal stability. We also consider some values of the coefficients of the characteristic polynomial at which the multiplicity of resonances is equal to two. Some example of a resonant set description is considered for a system with two parameters.

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Computation of unimodular matrices

Cited by 2 publications

References 4 publications

Computation of asymptotic forms of solutions to system of nonlinear partial differential equations

Computation of asymptotic forms of solutions to system of nonlinear partial differential equations

Strong resonances in nonlinear Hamiltonian system

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