Noah H. Rhee

A Shifted Cyclic Reduction Algorithm for Quasi-Birth-Death Problems

The problem of the computation of the stochastic matrix G associated with discrete-time quasi-birth-death (QBD) Markov chains is analyzed. We present a shifted cyclic reduction algorithm and show that the speed of convergence of the latter modified algorithm is always faster than that of the original cyclic reduction.

Homotopy algorithm for symmetric eigenvalue problems

SummaryThe homotopy method can be used to solve eigenvalue-eigenvector problems. The purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results. In practice, it is rerely of any interest to compute all the eigenvalues. The homotopy method, having the order preserving property, can provide any specific eigenvalue without calculating any other eigenvalues. Besides this advantage, we note that the homotopy algorithm is to a large degree a parallel algorithm. Numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices.

Eigenvalue inequalities and equalities

Abstract We consider cases of equality in three basic inequalities for eigenvalues of Hermitian matrices: Cauchys interlacing inequalities for principal submatrices, Weyls inequalities for sums, and the residual theorem. Several applications generalize and sharpen known results for eigenvalues of irreducible tridiagonal Hermitian matrices.

On the global and cubic convergence of a quasi-cyclic Jacobi method

SummaryIn this paper we consider the global and the cubic convergence of a quasi-cyclic Jacobi method for the symmetric eigenvalue, problem. The method belongs to a class of quasi-cyclic methods recently proposed by W. Mascarenhas. Mascarenhas showed that the methods from his class asymptotically converge cubically per quasi-sweep (one quasi-sweep is equivalent to 1.25 cyclic sweeps) provided the eigenvalues are simple. Here we prove the global convergence of our method and derive very sharp asymptotic convergence bounds in the general case of multiple eigenvalues. We discuss the ultimate cubic convergence of the method and present several numerical examples which all well comply with the theory.

A modified piecewise linear Markov approximation of Markov operators

In this paper, we propose a modified piecewise linear Markov method for approximating Markov operators defined on L^1(0,1). We present numerical results with the new method, as compared to those using the original piecewise linear Markov method.

Numerical Stability of the Parallel Jacobi Method

In this paper we study numerical stability of the parallel Jacobi method for computing the singular values and singular subspaces of an invertible upper triangular matrix that is obtained from QR decomposition with column pivoting. We show that in this case the parallel Jacobi method locates singular values and singular subspaces to full machine accuracy.

A Note on Relative Perturbation Bounds

In this paper we provide an actual bound for the distance between the original and the perturbed right singular vector subspaces of a general matrix with full column rank. We also provide actual relative componentwise bounds for perturbed eigenvectors of a positive definite matrix.

On the cubic convergence of the Paardekooper method

We show a simple way how asymptotic convergence results can be conveyed from a simple Jacobi method to a block Jacobi method. Our pilot methods are the well known symmetric Jacobi method and the Paardekooper method for reducing a skew-symmetric matrix to the real Schur form. We show resemblance in the quadratic and cubic convergence estimates, but also discrepances in the asymptotic assumptions. By numerical tests we confirm that our asymptotic assumptions for the Paardekooper method are most general.

Piecewise linear least squares approximations of Frobenius–Perron operators

Abstract We propose a piecewise linear numerical method based on least squares approximations for computing stationary density functions of Frobenius–Perron operators associated with piecewise C2 and stretching mappings of the unit interval. We prove the weak convergence of the method for a class of Frobenius–Perron operators, and the numerical results show that it is also norm convergent and has a better convergence rate than the piecewise linear Markov approximation method.

A matrix pair of an almost diagonal skew-symmetric matrix and a symmetric positive definite matrix

Abstract Let A be skew-symmetric, B be symmetric positive definite, and the pair ( A , B ) have multiple eigenvalues. If A is close to Murnaghan form and B is close to diagonal form, then certain principal submatrices of A and B are specially related. In this paper we describe this relationship and quantify it under the usual asymptotic conditions. If B = I , the identity matrix, the result describes a special structure of a skew-symmetric matrix almost in Murnaghan form. We then briefly discuss how the Paardekooper method for skew-symmetric matrices asymptotically behaves in the presence of multiple eigenvalues.