Frobenius Form Research Articles

On the inverse problem for even-order ordinary differential equations in the higher-order calculus of variations

A paper of Anderson and Thompson demonstrates that the inverse problem in the calculus of variations for systems of fourth-order ordinary differential equations gives rise to a system of PDEs in Frobenius form for the multiplier functions, and therefore has a structure much simpler than that of the corresponding problem for second-order equations. We show that a similar simplification holds for systems of equations of order 2 k ( k>2).

Synthesis of Dynamical Systems by Similarity Transformation

For non-linear time-varying systems the canonical transformations providing Frobenius form for the closed-loop system's matrix of the non-linear systems are introduced. Continious and discrete cases are considered. Stabilization of the nonlinear system in Frobenious form is considered. The same problems for the system with the perturbation of the certain class are solved.

On the computation of minimal polynomials, cyclic vectors, and frobenius forms

Various algorithms connected with the computation of the minimal polynomial of an n × n matrix over a field K are presented here. The complexity of the first algorithm, where the complete factorization of the characteristic polynomial is needed, is O(√ nn 3). It produces the minimal polynomial and all characteristic subspaces of a matrix of size n. Furthermore, an iterative algorithm for the minimal polynomial is presented with complexity O( n 3 + n 2 m 2), where m is a parameter of the shift Hessenberg matrix used. It does not require knowledge of the characteristic polynomial. Important here is the fact that the average value of m or m A is O( log n). Next we are concerned with the topic of finding a cyclic vector for a matrix. We first consider the case where its characteristic polynomial is square-free. Using the shift Hessenberg form leads to an algorithm at cost O( n 3 + m 2 n 2). A more sophisticated recurrent procedure gives the result in O( n 3) steps. In particular, a normal basis for an extended finite field of size q n will be obtained with deterministic complexity O( n 3 + n 2 log q). Finally, the Frobenius form is obtained with asymptotic average complexity O( n 3 log n). All algorithms are deterministic. In all four cases, the complexity obtained is better than for the heretofore best known deterministic algorithm.

Nearly Optimal Algorithms for Canonical Matrix Forms

A Las-Vegas-type probabilistic algorithm is presented for finding the Frobenius canonical form of an $n \times n$ matrix T over any field $\mathfrak{K}$. The algorithm requires $O^{\sim}(\operatorname{MM}(n)) = \operatorname{MM}(n) \cdot (\log n)^{O(1)}$ operations in $\mathfrak{K}$, where $O(\operatorname{MM}(n))$ operations in K are sufficient to multiply two $n \times n$ matrices over $\mathfrak{K}$. This nearly matches the lower bound of $\Omega (\operatorname{MM}(n))$ operations in $\mathfrak{K}$ for this problem and improves on the $O(n^{4})$ operations in $\mathfrak{K}$ required by the previous best-known algorithms. A fast parallel implementation of the algorithm is also demonstrated for the Frobenius form, which is processor-efficient on a PRAM. As an application we give an algorithm to evaluate a polynomial $g \in {\mathfrak{K}}[x]$ at T which requires only $O^{\sim}(\operatorname{MM}(n))$ operations in $\mathfrak{K}$ when $\deg g \leq n^{2}$. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix (for testing whether two matrices are similar), and matrix powering, which are substantially faster than those previously known.

Factorized Sparse Approximate Inverse Preconditionings I. Theory

This paper considers construction and properties of factorized sparse approximate inverse preconditionings well suited for implementation on modern parallel computers. In the symmetric case such preconditionings have the form $A \to G_L AG_L^T $, where $G_L $ is a sparse approximation based on minimizing the Frobenius form $\| I - G_L L_A \|_F $ to the inverse of the lower triangular Cholesky factor $L_A $ of A, which is not assumed to be known explicitly. These preconditionings preserve symmetry and/or positive definiteness of the original matrix and, in the case of M-, H-, or block H-matrices, lead to convergent splittings.

A uniform algorithm for the transformation of multivariable systems into canonical forms

The main purpose of the paper is to present a uniform algorithm for transforming time-invariant, linear, dynamical systems into various canonical forms. The first part deals with the single-input case. The method is based on the factorization of the transformation matrix. This can be expressed as the product of the controllability matrix and of a certain nonsingular upper triangular or lower secondary triangular matrix corresponding to the Frobenius, bidiagonal, tridiagonal, Schwarz, Routh, subdiagonal, or first Cauer canonical form. In the second part block canonical forms for multivariable systems are given. They represent the decoupling of the given control system into weakly connected subsystems. Two levels of classification of multivariable canonical forms are suggested, based on Kronecker and Hermite invariants and on the inner structure of the subsystems.

Computation of matrix fraction descriptions of linear time-invariant systems

In this paper, algorithms are presented for obtaining relatively prime matrix fraction descriptions (MFD's) of linear time-invariant systems. Orthogonal coordinate transformations are used to reduce a given state-space model to one whose state matrix is in "block Hessenberg" form. The observability (controllability) indexes of the system are also obtained in the process. A recursive algorithm is then described for obtaining a relatively prime MFD from the block Hessenberg representation. It is then shown that some simplification can be made in the recursive algorithm by first reducing the block Hessenberg form, by means of a nonsingular (triangular) coordinate transformation, to a more compact form, such as the "block Frobenius" form. A permutation of the state variables of the block Frobenius form yields a canonical representation similar to the Luenberger canonical form. Some numerical and other properties of the algorithms are discussed, and the use of the algorithms is illustrated by numerical examples. The numerical performance of the algorithms is also compared with that of the structure algorithm of Wolovich and Falb.

An Algorithm for the Exact Reduction of a Matrix to Frobenius Form Using Modular Arithmetic. I

An Algorithm for the Exact Reduction of a Matrix to Frobenius Form Using Modular Arithmetic. I

An Algorithm for the Exact Reduction of a Matrix to Frobenius form Using Modular Arithmetic. II

An Algorithm for the Exact Reduction of a Matrix to Frobenius form Using Modular Arithmetic. II

An algorithm for the exact reduction of a matrix to Frobenius form using modular arithmetic. I

This paper is in two parts. Part I contains a description of the Danilewski algorithm for reducing a matrix to Frobenius form using rational arithmetic. This algorithm is modified for use over the field of integers modulo p. The modified algorithm yields exact integral factors of the characteristic polynomial. A description of the single-modulus algorithm is given. Part II contains a description of the multiple-modulus algorithm. Since different moduli may yield different factorizations, an algorithm is given for determining which factorizations are not correct factorizations over the integers of the characteristic polynomial.

An algorithm for the exact reduction of a matrix to Frobenius form using modular arithmetic. II

Part I contained a description of the single-modulus algorithm for reducing a matrix to Frobenius form, obtaining exact integral factors of the characteristic polynomial. Part II contains a description of the multiple-modulus algorithm. Since different moduli may yield different factorizations, an algorithm is given for determining which factorizations are not correct factorizations over the integers of the characteristic polynomial. Part II also contains a discussion of the selection of the moduli and numerical examples.