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Moreover, this representation reveals already the QR factorization of the semisepar- able matrix S: the Givens transformations appearing in the representation of the matrix are exactly the same as the Givens transformations appearing in the Q factor of the QR factorization. More information can be found in [42, 34]. In [42, 44] other algorithms, based on the Givens-vector representation can be found. The algorithms for retrieving the representation and for constructing the full semiseparable matrix, given the Givens-vector representation, can be deduced easily with the inform- ation in the previous section.

Here an O n implementation of a matrix vector multiplication will be given. The formulas will be given for nonsymmetric semiseparable matrices, who have two se- quences of Givens transformations and two vectors. To deduce the algorithm, we have to decompose the matrix into a strict uppertriangular, and a lower triangular part of S.

Rewriting the formulas reveals the order O n algorithm for the multiplication. Combining the last 2 equalities of all the yi one can derive an order n algorithm to perform the multiplication of S1 and v. The multiplication of S2 and v can be derived in a completely analogous way. We have given an altern- ative definition and a corresponding representation which keeps the interesting proper- ties of the standard representation but does not exhibit the disadvantages.

Appendix A Our main result here will be to prove, as in [21, Chapter II] , that the inverse of a one-pair matrix is an unreducible tridiagonal matrix and vice versa. First some notation has to be introduced. Proposition 7. This proves This matrix will then appear to be a one-pair matrix.

This means that S i1 ,. Theorem 7. Scand, —60, Barrett and P. Inverses of banded matrices. Linear Algebra and Its Applications, —, Chandrasekaran, P. Dewilde, M. Gu, T. Pals, X. Sun, A. Fast stable solvers for sequentially semi—separable linear sys- tems of equations and least squares problems. Pals, and A. Fast stable solver for sequentially seni-separable linear systems of equations.

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Jun 14, at 1 Thank you so much. And there was a point where he said: When implementing the normal equation in octave we want to use the 'pinv' function rather than 'inv. Jun 15, at 1 RishabhKumarSingh indeed!

But this is a mathematical property of of the pinv, it has nothing to do with precision. If inv exists, then pinv will necessarily be equal to inv.

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