Separable Type Representations of Matrices and Fast Algorithms, Volume 1: Basics. Completion Problems. Multiplication and Inversion Algorithms

Part 1. Basics on separable, semiseparable and quasiseparable representations of matrices.- 1. Matrices with separable representation and low complexity algorithms.- 2. The minimal rank completion problem.- 3. Matrices in diagonal plus semiseparable form.- 4. Quasiseparable representations: the basics.- 5. Quasiseparable generators.- 6. Rank numbers of pairs of mutually inverse matrices, Asplund theorems.- 7. Unitary matrices with quasiseparable representations.- Part 2. Completion of matrices with specified bands.- 8. Completion to Green matrices.- 9. Completion to matrices with band inverses and with minimal ranks.- 10. Completion of special types of matrices.- 11. Completion of mutually inverse matrices.- 12. Completion to unitary matrices.- Part 3. Quasiseparable representations of matrices, descriptor systems with boundary conditions and first applications.- 13. Quasiseparable representations and descriptor systems with boundary conditions.- 14. The first inversion algorithms.- 15. Inversion of matrices in diagonal plus semiseparable form.- 16. Quasiseparable/semiseparable representations and one-direction systems.- 17. Multiplication of matrices.- Part 4. Factorization and inversion.- 18. The LDU factorization and inversion.- 19. Scalar matrices with quasiseparable order one.- 20. The QR factorization based method.​