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We describe rank structures in generalized inverses of possibly rectangular banded matrices. In particular, we show that various kinds of generalized inverses of rectangular banded matrices have submatrices whose rank depends on the bandwidth and on the nullity of the matrix. Moreover, we give an explicit representation formula for some generalized inverses of strictly banded matrices.
Abstract: We describe a stable algorithm, having linear complexity, for the solution of banded-plus-semiseparable linear systems. The algorithm exploits the structural properties of the inverse of a semiseparable matrix. Stability is achieved by combining these properties with partial pivoting techniques. Several numerical experiments are shown to confirm the effectiveness of the proposed approach.
Abstract: Mixed and componentwise condition numbers are useful in understanding stability properties of algorithms for solving structured linear systems. The DFT (discrete Fourier transform) is an essential building block of these algorithms. We obtain estimates of mixed and componentwise condition numbers of the DFT. To this end, we explicitly compute certain special vectors that share with their...
A classical result of structured numerical linear algebra states that the inverse of a nonsingular semiseparable matrix is a tridiagonal matrix. Such a property of a semiseparable matrix has been proved to be useful for devising linear complexity solvers, for establishing recurrence relations among its columns or rows and, moreover, for efficiently evaluating its characteristic polynomial. In this...
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