By A. Aricò, M. Donatelli, J. Nagy (auth.), Paul Van Dooren, Shankar P. Bhattacharyya, Raymond H. Chan, Vadim Olshevsky, Aurobinda Routray (eds.)
The goal of Numerical Linear Algebra in signs, structures and regulate is to give an interdisciplinary publication, mixing linear and numerical linear algebra with 3 significant parts of electric engineering: sign and snapshot Processing, and keep an eye on structures and Circuit conception. Numerical Linear Algebra in signs, structures and Control will comprise articles, either the cutting-edge surveys and technical papers, on idea, computations, and functions addressing major new advancements in those parts. The target of the quantity is to supply authoritative and available debts of the fast paced advancements in computational arithmetic, clinical computing, and computational engineering tools, functions, and algorithms. The cutting-edge surveys will profit, particularly, starting researchers, graduate scholars, and people considering to begin a brand new course of study in those parts. A extra basic target is to foster potent communications and trade of knowledge among a variety of medical and engineering groups with mutual pursuits in techniques, computations, and practicable, trustworthy practices.
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50) j ¼ 1: Thus a well of size one is exactly a column of zeros above the diagonal, and some nonzero entry to the right of that column, exactly as in Definition 4. With the class of ðH; mÞ-well-free matrices defined, we next present a theorem containing the classifications to be proved in this section. Theorem 8 Suppose A is a strongly upper Hessenberg n Â n matrix. Then the following are equivalent. (i) A is ðH; mÞ-well-free. (ii) There exists a set of generators of Definition 2 corresponding to A such that bk are companion matrices for k ¼ 2; .
3 (H, m)-well-free Matrices: Generator Classification Theorem 9 An ðH; mÞ-quasiseparable matrix is ðH; mÞ-well-free if and only if there exists a choice of generators fpk ; qk ; dk ; gk ; bk ; hk g of the matrix that are of the form 2 3 2 3 0 0 ÁÁ Á 0 nk;1 1 . . 6 607 .. 7 6 1 0 . .. 7 6 7 .. 7 . .. ;n À 1; hk ¼ 6 bk ¼ 6 6 .. 7; k ¼ 2;. ;n: ð2:51Þ . 0 1 . 6 7 7 6 6. . 7 4 ... 5 4 .. . . 0 n 5 k;mÀ1 0 0 Á ÁÁ 0 1 n k;m 2 Classifications of Recurrence Relations 47 Proof Let A ¼ ðaij Þ be an ðH; mÞ-well-free matrix.
Van Dooren et al. V. 2011 23 24 T. Bella et al. 1 Classical Three-term and Two-term Recurrence Relations and Their Generalizations It is well known that real-orthogonal polynomials frk ðxÞg satisfy three-term recurrence relations of the form rk ðxÞ ¼ ðak x À dk ÞrkÀ1 ðxÞ À ck Á rkÀ2 ðxÞ; ak 6¼ 0; ck [ 0: ð2:1Þ It is also well known that Szegö polynomials f/# k ðxÞg; or polynomials orthogonal not on a real interval but on the unit circle, satisfy slightly different three-term recurrence relations of the form !