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September 2014
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13 Reads
In this chapter we formulate the main polynomial root finding problem in terms of the eigenvalue problem for the companion matrix. We will easily see that the companion matrix belongs to the class of upper Hessenberg matrices which are rank one perturbations of unitary matrices. We will study this class in detail here.
September 2014
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6 Reads
In this chapter we discuss an alternative method which is based on the using of the QR factorization A =ΘR of an upper Hessenberg matrix A from the class HN, with a unitary upper Hessenberg matrix T and an upper triangular matrix R.
September 2014
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8 Reads
In this chapter we study in detail the divide and conquer method for Hermitian matrices with quasiseparable representations. For Hermitian quasiseparable of order one matrices we obtain a complete algorithm to compute the eigenvalue decomposition of a matrix.
September 2014
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13 Reads
In this chapter we extend the notion of rank numbers introduced in Chapter 4 to wider sets of submatrices. Lower rank numbers for a square matrix relative to the diagonal i – j are introduced as the ranks of the maximal submatrices entirely located under that diagonal, and the upper rank numbers relative to a diagonal are defined correspondingly. If the given matrix is invertible, a strong link exists between these numbers for the matrix and its inverse. In particular, the lower and upper rank numbers relative to the main diagonal are the same for a matrix with square blocks on the main diagonal and for its inverse matrix. This implies that for such a square matrix the lower and upper quasiseparable orders coincide with the ones of the inverse matrix.
September 2014
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8 Reads
It is clear from the preceding chapter that any matrix has quasiseparable representations. By padding given quasiseparable generators with zero matrices of large sizes one can arrange that they have arbitrarily large orders. However, one is looking for generators of minimal orders, because they will give better computational complexity in applications.
September 2014
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13 Reads
The divide step consists in splitting a single problem into two smaller independent problems with half the size of the original problem. This is done recursively, until the obtained problems are of a convenient size which is small enough so that they can be solved by standard techniques.
September 2014
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10 Reads
Generators with orders one are complex numbers. One can then apply the linear complexity inversion algorithm from Theorem 18.2 suggested under the conditions of invertibility of the principal leading submatrices of the matrix. In this chapter an algorithm to determine quasiseparable generators of the inverse without any restrictions on the principal leading submatrices is obtained.
September 2014
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8 Reads
Here we consider completions to Green matrices under some conditions on the specified band. If all the submatrices B k of the form (8.1) are invertible, then the unique completion to a Green matrix is a positive (definite) matrix if and only if all the matrices D k of the form (9.1) are positive (definite).
September 2014
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20 Reads
One of the methods used in the solution of the eigenvalue problem is to reduce at first a matrix to a unitarily similar one in the upper Hessenberg form. The subsequent QR iterations for an upper Hessenberg matrix can be performed in an efficient way.
September 2014
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8 Reads
Here we consider matrices with the lower quasiseparable and upper semiseparable representations discussed in Section §14.2. We show that such representations correspond to discrete systems without backward recursions, i.e., to one-direction systems. We study such systems in detail and derive inversion algorithms for matrices of their input-output operators.
... The third mass m3 represents the human head that has no contact with the backrest, however the head oscillations along y-direction are stabilised by the damper (dsky) suspended in the inertial reference point (sky-hook damper control [22]). In order to describe the dynamics of multi-input and multi-output (MIMO) system presented in Fig. 1, the statespace model is engaged as a set of first-order differential equations [23]: ...
January 2006
... One of the main developers of the state space realization theory was R. Kalman based on a major result he proved in 1963 [55] (and elaborated on in his 1965 paper [54]) that bridged linear control theory and the concepts of controllability and observability with minimal realizability and the construction of minimal realizations of a given rational function from nonminimal realizations of it. He also proved the important 'state space similarity theorem' which describes how two minimal realizations of the same rational function are related by similarity (see [55,Theorem 8], [54,Proposition 2], [16,Theorem 3.1], and [15,Theorem 7.7]). Shortly thereafter, in 1965 [54] he showed how these concepts were related to the notion of the McMillan degree of a rational matrix-valued function (also called the McMillan-Duffin-Hazony degree as it was, according to R. Kalman [54], first introduced by B. McMillan [65] and further studied by R. Duffin and D. Hazony [35], but credit for an equivalent definition of degree and its usefulness in network realization theory actually seems to belong to B. Tellegen based on his 1948 paper, see [8,81]). ...
January 2008
... In other words, one should be able to substitute concrete values of Planck's constant into the deformed product. The first successful attempt to develop a theory satisfying this requirement was made by M. Rieffel [55][56][57][58]. In his approach, a deformation is a continuous field (or a bundle) of C * -algebras endowed with an additional structure. ...
January 2008
... The Jordan canonical form is a powerful tool for solving several problems arising in the theory of matrices and matrix computations since it contains the whole information about the algebraic structure of the underlying linear transformation and the corresponding matrix [1][2][3][4]. With the aid of the Jordan form, important theoretical and applied problems are solved including the stability of linear time-invariant systems, computation of matrix functions, and solution of matrix equations. ...
January 2006
... Computing with structured matrices typically involves developing specialized algorithms that take advantage of the underlying structure of the matrix to reduce the computational complexity of matrix operations. Many fast algorithms have been developed for computing with structured matrices [1][2][3][4][5]. These algorithms can significantly reduce the computation complexity when implementing operations with such matrices. ...
January 2014
... Matrices having a low quasiseparable order are representable with a linear or almost linear amount of parameters [9,41], and to efficiently store and operate with such matrices it is often a good idea to rely on hierarchical matrix formats. In this work, we rely on a specific subclass of the set of hierarchical representations called hierarchically semiseparable matrices (HSS) [13]. ...
January 2014
... Earlier results, that go back to [19,20] ( [9] provides an overview), concern analytic operator-valued functions that take on contractive values on the open unit disk; in this case, a realization (1) exists where now A, B, C, and D are Hilbert space operators. The theory of realizations is of importance in control and systems theory and in interpolation problems, and it provides a useful tool in operator theory in general; see, e.g., the monographs [11,10]. ...
January 2008
... Matrices over commutative rings [52,10] have also received considerable attention. Matrices of polynomials [26,23], Laurent polynomials [15,37], and analytic functions [11,73] arise in multichannel broadband signal processing [61,57]. The study of matrices over the commutative ring of vectors with circular convolution as multiplication was initiated in [8,41,40,39] and extended to infinite-dimensional vectors in [56]. ...
January 2009
... The theory of linear systems has a long history and we refer for instance to the books [29,34,38,41,42,43] for information, background and 1 references. In view of the associated state space method (see for instance [21], [30]), the notion of rational function and the realization theory of matrix-valued rational functions play a key role in linear system theory. Certain families of matrix-valued rational functions, connected to the notion of dissipativity, are of special interest. ...
January 2010
... In order to compare the stability of our algorithm with that of algorithm tailored for polynomial rootfinding [7,15] we computed also the backward error in terms of the coefficient of the polynomial. In particular, let p(x) = n i=0 p i x i = n i=1 (x − λ i ) be our monic test polynomial with roots λ i , and denote byλ i the computed roots obtained with our algorithm applied to the companion matrix A. We denote byp(x) the polynomial havingλ i as exact roots, i.e.,p(x) = (x −λ i ). ...
December 2012
Indagationes Mathematicae
