We present a new transform, triple dqds, to help to compute the eigenvalues of a real tridiagonal... more We present a new transform, triple dqds, to help to compute the eigenvalues of a real tridiagonal matrix C using real arithmetic. The algorithm uses the real dqds transform to shift by a real number and triple dqds to shift by a complex conjugate pair. We present what seems to be a new criteria for splitting the current pair L,U. The algorithm rejects any transform which suffers from excessive element growth and then tries a new transform. Our numerical tests show that the algorithm is about 100 times faster than the Ehrlich-Aberth method of D. A. Bini, L. Gemignani and F. Tisseur. Our code is comparable in performance to a complex dqds code and is sometimes 3 times faster.
We prove convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one... more We prove convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one-point spectrum-the Jordan form is one big Jordan block. First we develop properties of eigenvector matrices. We also show how to deal with the singular case. Mathematics Subject Classification (2000) 65F15 1 Introduction This paper presents a rigorous proof that the LR algorithm, without shifts, applied to an unreduced tridiagonal matrix with a one-point spectrum converges to an upper bidiagonal matrix. The rate of convergence is very slow, like 1/k after k steps, but what is remarkable is that the algorithm actually converges. We hasten to say that this result is not exactly new. In the middle of the 1960s J.H. Wilkinson sketched out the underlying reason for this surprising result, both for LR and QR, but he was not concerned with tridiagonal matrices and he needed assumptions that the column and row eigenvector matrices were completely regular. Moreover, he did not show that a certain universal matrix was also completely regular. So the contribution of this paper is twofold. We show that in the unreduced tridiagonal case the eigenvector matrices are completely regular and we show that the
We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toepl... more We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.
Our contribution is two-folded. First, starting from the known fact that every real skew-Hamilton... more Our contribution is two-folded. First, starting from the known fact that every real skew-Hamiltonian matrix has a real Hamiltonian square root, we give a complete characterization of the square roots of a real skew-Hamiltonian matrix W. Second, we propose a structure-exploiting method for computing square roots of W. Compared to the standard real Schur method, which ignores the structure, our method requires significantly less arithmetic.
We present a new transform, triple dqds, to help to compute the eigenvalues of a real tridiagonal... more We present a new transform, triple dqds, to help to compute the eigenvalues of a real tridiagonal matrix C using real arithmetic. The algorithm uses the real dqds transform to shift by a real number and triple dqds to shift by a complex conjugate pair. We present what seems to be a new criteria for splitting the current pair L,U. The algorithm rejects any transform which suffers from excessive element growth and then tries a new transform. Our numerical tests show that the algorithm is about 100 times faster than the Ehrlich-Aberth method of D. A. Bini, L. Gemignani and F. Tisseur. Our code is comparable in performance to a complex dqds code and is sometimes 3 times faster.
We prove convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one... more We prove convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one-point spectrum-the Jordan form is one big Jordan block. First we develop properties of eigenvector matrices. We also show how to deal with the singular case. Mathematics Subject Classification (2000) 65F15 1 Introduction This paper presents a rigorous proof that the LR algorithm, without shifts, applied to an unreduced tridiagonal matrix with a one-point spectrum converges to an upper bidiagonal matrix. The rate of convergence is very slow, like 1/k after k steps, but what is remarkable is that the algorithm actually converges. We hasten to say that this result is not exactly new. In the middle of the 1960s J.H. Wilkinson sketched out the underlying reason for this surprising result, both for LR and QR, but he was not concerned with tridiagonal matrices and he needed assumptions that the column and row eigenvector matrices were completely regular. Moreover, he did not show that a certain universal matrix was also completely regular. So the contribution of this paper is twofold. We show that in the unreduced tridiagonal case the eigenvector matrices are completely regular and we show that the
We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toepl... more We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.
Our contribution is two-folded. First, starting from the known fact that every real skew-Hamilton... more Our contribution is two-folded. First, starting from the known fact that every real skew-Hamiltonian matrix has a real Hamiltonian square root, we give a complete characterization of the square roots of a real skew-Hamiltonian matrix W. Second, we propose a structure-exploiting method for computing square roots of W. Compared to the standard real Schur method, which ignores the structure, our method requires significantly less arithmetic.
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Papers by Carla Ferreira