By Raf Vandebril
The basic homes and mathematical constructions of semiseparable matrices have been awarded in quantity 1 of Matrix Computations and Semiseparable Matrices. In quantity 2, Raf Vandebril, Marc Van Barel, and Nicola Mastronardi talk about the idea of based eigenvalue and singular price computations for semiseparable matrices. those matrices have hidden houses that let the improvement of effective tools and algorithms to safely compute the matrix eigenvalues.
This thorough research of semiseparable matrices explains their theoretical underpinnings and includes a wealth of knowledge on enforcing them in perform. some of the exercises featured are coded in Matlab and will be downloaded from the internet for additional exploration.
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Extra resources for Matrix Computations and Semiseparable Matrices - Eigenvalue and Singular Value Methods
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0 2 (k) (k) T (k) G A G j−1 j j − −−−−−−−−−−−− → × 6. 6. 6. 6 6× 6 6 6 6 6 60 6 6. 4 .. 0 j j+1 ↓ ↓ ··· × × 0 0 . . .. .. .. ··· × × 0 0 ··· × 0 0 ··· ··· 0 ⊗ . . . . . . ··· 0 ⊗ ··· × . .. .. ··· × ··· ··· ··· 0 0 . . . 1. The transformation Aj = Gj entries of rows (columns) j and j + 1 proportional. (k) ··· ··· ··· ··· ··· .. ··· 0 ··· . . 0 ··· 0 ··· ··· ··· . . . ··· . . . 5 . 7 7 07 7 07 7. 5 . makes the first j transformations, we obtain the following matrix:11 (k) T (k) Aj−1 = Gj−1 ⎡ × ⎢ ..
D. Bindel, J. W. Demmel, W. Kahan, and O. A. Marques. On computing Givens rotations reliably and efficiently. ACM Transactions on Mathematical Software, 28(2):206–238, June 2002. 2 Orthogonal similarity transformations of symmetric matrices As already mentioned in the introduction of this chapter, this section focuses on orthogonal similarity transformations of symmetric matrices to easier forms. The aim of the proposed reduction methods is to exploit the structure of the reduced matrices in the eigenvalue computations.
0 ··· 0 × × In the tridiagonalization procedure, the first step is finished, and all the desired zeros are created in the bottom row. Because we want to obtain a semiseparable matrix in this case, we have to apply another, extra transformation. (1) T (1) Multiplying An−2 to the left by Gn−1 , leads to the following situation: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ → ⎣ → × × .. × 0 × .. ··· .. ··· ··· . × 0 × .. × × × ⎤ ⎡ 0 ⎢ .. ⎥ ⎢ . ⎥ ⎥ (1) T (1) ⎢ ⎥G ⎢ n−1 An−2 ⎢ 0 ⎥ ⎥−−−−−−−−→⎢ ⎣ ⊗ ⎦ × (1) An−2 (1) T × × ..