## Separable Type Representations of Matrices and Fast Algorithms: Volume 1 Basics. Completion Problems. Multiplication and Inversion AlgorithmsThis two-volume work presents a systematic theoretical and computational study of several types of generalizations of separable matrices. The main attention is paid to fast algorithms (many of linear complexity) for matrices in semiseparable, quasiseparable, band and companion form. The work is focused on algorithms of multiplication, inversion and description of eigenstructure and includes a large number of illustrative examples throughout the different chapters. The first volume consists of four parts. The first part is of a mainly theoretical character introducing and studying the quasiseparable and semiseparable representations of matrices and minimal rank completion problems. Three further completions are treated in the second part. The first applications of the quasiseparable and semiseparable structure are included in the third part where the interplay between the quasiseparable structure and discrete time varying linear systems with boundary conditions play an essential role. The fourth part contains factorization and inversion fast algorithms for matrices via quasiseparable and semiseparable structure. The work is based mostly on results obtained by the authors and their coauthors. Due to its many significant applications and the accessible style the text will be useful to engineers, scientists, numerical analysts, computer scientists and mathematicians alike. |

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𝐴 arithmetic operations band matrices block entries block lower triangular block matrix block upper triangular columns consider Corollary defined Determine the matrices diagonal entries d(k diagonal plus semiseparable Eidelman entries of sizes equalities equation Fast Algorithms Gohberg Green matrix Hence input-output operator inverse matrix Inversion Algorithms invertible Lemma linear algebraic lower quasiseparable lower semiseparable lower triangular matrix main diagonal matrices of sizes matrix multiplications matrix of order matrix Q matrix with block minimal completion rank minimal rank completion Moreover Operator Theory orders rk partially specified block partially specified matrix partition positive definite Proof quasiseparable generators p(i quasiseparable order quasiseparable representations rank A(k rank factorization relations Representations of Matrices respectively scalar matrix semiseparable representation sizes mix strictly lower triangular tridiagonal matrix unitary matrices upper quasiseparable upper rank numbers upper semiseparable upper triangular matrix vector