Hermite Normal Form Research Articles

TRANSFORMATION OF NESTED LOOPS WITH MODULO INDEXING TO AFFINE RECURRENCES

For multi-dimensional (M-D) signal and data processing systems, transformation of algorithmic specifications is a major instrument both in code optimization and code generation for parallelizing compilers and in control flow optimization as a preprocessor for architecture synthesis. State-of-the-art transformation techniques are limited to affine index expressions. This is however not sufficient for many important applications in image, speech and numerical processing. In this paper, a novel transformation method is introduced, oriented to the subclass of algorithm specifications that contains modulo expressions of affine functions to index M-D signals. The method employs extensively the concept of Hermite normal form. The transformation method can be carried out in polynomial time, applying only integer arithmetic.

Achieving diagonal interactor matrix for multivariable linear systems with uncertain parameters

The notion of interactor matrix or equivalently the Hermite normal form, is a generalization of relative degree to multivariable systems, and is crucial in problems such as decoupling, inverse dynamics, and adaptive control. In order for a system to be input-output decoupled using static state feedback, the existence of a diagonal interactor matrix must first be established. For a multivariable linear system which does not have a diagonal interactor matrix, dynamic precompensation or dynamic state feedback is required for achieving a diagonal interactor matrix for the compensated system. Such precompensation often depends on the parameters of system, and is thus difficult to implement with accuracy when the system is subject to parameter uncertainty. In this paper we characterize a class of linear systems which can be precompensated to achieve a diagonal interactor matrix without the exact knowledge of the system parameters. More precisely, we present necessary and sufficient conditions on the transfer matrix of the system under which there exists a diagonal dynamic precompensator such that the compensated system has a diagonal interactor matrix. These conditions are associated with the so-called (non)generic singularity of certain matrix related to the system structure but independent of the system parameters. The result of this paper is expected to be useful in robust and adaptive designs.

Bricklaying and the Hermite Normal Form

Bricklaying and the Hermite Normal Form

Bricklaying and the Hermite Normal Form

An integer lattice in n-dimensional space is the set of all integer linear combinations of n linearly independent vectors with integer entries. The translation group of the lattice consists of all the translations of n-dimensional space that take lattice points to lattice points. This group is isomorphic to the set of lattice points under vector addition. A fundamental brick will be a rectangular brick with one vertex at the origin and sides along each positive coordinate axis. A brickwork will consist of a tiling of n-space by n-dimensional bricks that are translations of a fundamental brick. The set of these translations form the translation group of the brickwork.

Asymptotically Fast Triangularization of Matrices over Rings

This paper considers the problems of triangularizing and diagonalizing matrices over rings, with particular emphasis on the integral case. It begins with a description of fast algorithms for the computation of Hermite and Smith normal forms of integer matrices. Then it shows how to apply fast matrix multiplication techniques to the problem of triangularizing a matrix over a ring using elementary column operations. These general results lead to an algorithm for triangularizing integer matrices that has a faster running time than the known Hermite normal form algorithms. The triangular matrix that is computed has small entries like the Hermite normal form, and will suffice for many applications.

An application of the Hermite normal form in integer programming

A new algorithm for solving integer programming problems is developed. It is based on cuts which are generated from the Hermite normal form of the basis matrix at a continuous extreme point.

Hermite canonical form and Smith canonical form of a matrix over a principal ideal domain

This paper presents computational aspects of Smith and Hermite normal forms for matrices over general principal ideal domains. After reviewing the standard definitions and properties of Hermite and Smith normal forms, we ask when there is an effective algorithm which will find a unique canonical form for all matrices in the same equivalence class. We show that the ability to compute canonical members of an associate class and canonical representatives of a residue class in the underlying ring are necessary and sufficient conditions for an effective procedure to find unique Hermite and Smith forms. We then present a uniform technique for reducing intermediate expression swell during the computation of Smith and Hermite normal forms for matrices over general principal ideal domains. This generalizes previous work by handling non-square matrices which are not of full column rank. We also present examples which show that the heuristic of directly computing modulo the determinant as proposed by some can yield the wrong answer. We instead compute with the matrix augmented by the determinant times the identity matrix. This always yields the correct result and has the desired effect of allowing one to keep all elements which occur during the computation bounded by the determinant.

Residual hermite normal form computations

This paper extends the class of Hermite normal form solution procedures that use modulo determinant arithmetic. Given any relatively prime factorization of the determinant value, integral congruence relations are used to compute the Hermite normal form. A polynomial-time complexity bound that is a function of the length of the input string exists for this class of procedures. Computational results for this new approach are given.

Worst-Case Complexity Bounds on Algorithms for Computing the Canonical Structure of Finite Abelian Groups and the Hermite and Smith Normal Forms of an Integer Matrix

An $O(s^5 M(s^2 ))$ algorithm for computing the canonical structure of a finite Abelian group represented by an integer matrix of size s (this is the Smith normal form of the matrix) is presented. Moreover, an $O(s^3 M(s^2 ))$ algorithm for computing the Hermite normal form of an integer matrix of size s is given. The upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by Kannan and Bachem in [SIAM J. Comput., 8 (1979), pp. 499–507] and Chou and Collins in [SIAM J. Comput., 11 (1982), pp. 687–708].

Hermite Normal Form Computation Using Modulo Determinant Arithmetic

This paper describes a new class of Hermite normal form solution procedures which perform modulo determinant arithmetic throughout the computation. This class of procedures is shown to possess a polynomial time complexity bound which is a function of the length of the input string. Computational results are also given.

Generating sets for rings of algebraic integers

Let R be the ring of integers of some finite algebraic extension of the rationals Q of degree n. A necessary and sufficient condition for s elements of R to be an R-basis is given, in terms of the Hermite normal form of a certain n ×ns integral matrix depending on the elements, and on the structure constants of R.

Algorithms for the Solution of Systems of Linear Diophantine Equations

Two algorithms for the solution of linear Diophantine systems, which well restrain intermediate expression swell, are presented. One is an extension and improvement of Kannan and Bachem’s algorithm for the Smith and the Hermite normal forms of a nonsingular square integral matrix. The complexity of this algorithm is investigated and a polynomial time bound is derived. Also a much better coefficient bound is obtained compared to Kannan and Bachem’s analysis. The other is based on ideas of Rosser, which were originally used in finding a general solution with smaller coefficients to a linear Diophantine equation and in computing the exact inverse of a nonsingular square integral matrix. These algorithms are implemented by using the infinite precision integer arithmetic capabilities of the SAC-2 system. Their performances are compared. Finally future studies are mentioned.

Hermite normal forms and the rank of a matrix

Three of the more fundamental facts of linear algebra concerning Hermite normal forms and rank are given similar, efficient proofs based on the principle that any set of positive integers contains a least element. These proofs avoid much of the detail regarding indices encountered in standard proofs and provide interesting alternatives to such proofs.

On the transformation of linear multivariable systems to canonical forms

An algorithm is presented which enables the transformation of the state equations of linear multivariable systems to any of six canonical forms. It is based on transformation of a matrix to the Hermite normal form, and does not require matrix inversion.

Exact Inversion of a Rational Polynomial Matrix Using Finite Field Transforms

A unique code (called Hensel’s code) is derived for a single variable rational polynomial function with integral coefficients, by converting its numerator and denominator into polynomials over a finite field and truncating the infinite expansion of the quotient. The four basic polynomial arithmetic algorithms for these codes are described; their application for the reduction of a rational polynomial matrix to the Hermite normal form (by a sequence of elementary transformations) and hence for the exact (generalized) inversion of the rational polynomial matrix is demonstrated.

Efficient algorithm for irreducible realization of a rational matrix

Abstract A computationally efficient algorithm is presented for irreducible realization of a given rational transfer function matrix. It is shown that either the Hankel matrix for the system or a reduced form of it can be systematically transformed to the Hermite normal form using outer products. The dynamical equations then follow immediately. This algorithm requires less computation than the existing algorithms.

Algorithms for Hermite and Smith Normal Matrices and Linear Diophantine Equations

New algorithms for constructing the Hermite normal form (triangular) and Smith normal form (diagonal) of an integer matrix are presented. A new algorithm for determining the set of solutions to a system of linear diophantine equations is presented. A modification of the Hermite algorithm gives an integer-preserving algorithm for solving linear equations with real-valued variables. Rough bounds for the number of operations are cubic polynomials involving the order of the matrix and the determinant of the matrix. The algorithms are valid if the elements of the matrix are in a principal ideal domain.

Algorithms for Hermite and Smith normal matrices and linear Diophantine equations

New algorithms for constructing the Hermite normal form (triangular) and Smith normal form (diagonal) of an integer matrix are presented. A new algorithm for determining the set of solutions to a system of linear diophantine equations is presented. A modification of the Hermite algorithm gives an integer-preserving algorithm for solving linear equations with real-valued variables. Rough bounds for the number of operations are cubic polynomials involving the order of the matrix and the determinant of the matrix. The algorithms are valid if the elements of the matrix are in a principal ideal domain.

A Generalized Inverse Which Gives all the Integral Solutions to a System of Linear Equations

Previous article Next article A Generalized Inverse Which Gives all the Integral Solutions to a System of Linear EquationsMichiel F. Hurt and Carter WaidMichiel F. Hurt and Carter Waidhttps://doi.org/10.1137/0119053PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Burton W. Jones, The Arithmetic Theory of Quadratic Forms, Carcus Monograph Series, no. 10, The Mathematical Association of America, Buffalo, N. Y., 1950, 61–62 MR0037321 0041.17505 Google Scholar[2] Irving Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc., 66 (1949), 464–491 MR0031470 0036.01903 CrossrefISIGoogle Scholar[3] Marvin Marcus and , Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon Inc., Boston, Mass., 1964xvi+180 MR0162808 0126.02404 Google Scholar[4] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc., 51 (1955), 406–413 MR0069793 0065.24603 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Channel Estimation and Self-Positioning for UAV SwarmIEEE Transactions on Communications, Vol. 67, No. 11 | 1 Nov 2019 Cross Ref Solution of Systems of Linear Diophantine EquationsComputer Algebra in Scientific Computing CASC 2001 | 1 Jan 2001 Cross Ref Applications of Generalized InversesSystems and Management Science by Extremal Methods | 1 Jan 1992 Cross Ref An application of the Hermite normal form in integer programmingLinear Algebra and its Applications, Vol. 140 | 1 Oct 1990 Cross Ref On an integral transformation for integer programmingOperations Research Letters, Vol. 7, No. 4 | 1 Aug 1988 Cross Ref Generalized inverses of matrices: a perspective of the work of PenroseMathematical Proceedings of the Cambridge Philosophical Society, Vol. 100, No. 3 | 24 October 2008 Cross Ref A computational form for finding the integral solutions to a system of linear equationsComputers & Mathematics with Applications, Vol. 7, No. 5 | 1 Jan 1981 Cross Ref Nonnegative integral solution of linear equationsProceedings of the Indian Academy of Sciences - Section A, Vol. 89, No. 1 | 1 Jan 1980 Cross Ref An introduction to the application of the simplest matrix-generalized inverse in systems scienceIEEE Transactions on Circuits and Systems, Vol. 25, No. 9 | 1 Sep 1978 Cross Ref An integer arithmetic method to compute generalized matrix inverse and solve linear equations exactlyProceedings of the Indian Academy of Sciences - Section A, Vol. 87, No. 9 | 1 Sep 1978 Cross Ref Further results on integral generalized inverses of integral matricesLinear and Multilinear Algebra, Vol. 6, No. 3 | 30 May 2007 Cross Ref Finite field computation technique for exact solution of systems of linear equations and interval linear programming problemsInternational Journal of Systems Science, Vol. 8, No. 10 | 30 March 2007 Cross Ref Annotated Bibliography on Generalized Inverses and ApplicationsGeneralized Inverses and Applications | 1 Jan 1976 Cross Ref Applications of Generalized Inverses to Programming, Games and NetworksGeneralized Inverses and Applications | 1 Jan 1976 Cross Ref $AX - XB = C$, Resultants and Generalized InversesRobert E. HartwigSIAM Journal on Applied Mathematics, Vol. 28, No. 1 | 12 July 2006AbstractPDF (1879 KB)On the General Solution to Systems of Mixed-Integer Linear EquationsV. J. Bowman and Claude-Alain BurdetSIAM Journal on Applied Mathematics, Vol. 26, No. 1 | 12 July 2006AbstractPDF (400 KB) Volume 19, Issue 3| 1970SIAM Journal on Applied Mathematics History Submitted:16 December 1969Published online:12 July 2006 InformationCopyright © 1970 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0119053Article page range:pp. 547-550ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

Minimum-state realizations of linear time-varying systems

An algorithm for removing uncontrollable (unobservable) state variables from time-varying systems is presented. The algorithm makes use of the time-variable controllability (observability) matrix which is augmented with the identity matrix and transformed into Hermite normal form. In this new set of coordinates, the uncontrollable (unobservable) state variables are easily identified and removed. An example is given.