In this talk, we provide an upper bound of the condition number of a Vandermonde-like matrix which is the eigenvector matrix for a non-symmetric time scheme matrix arising from a direct parallel-in-time numerical method of second-order accuracy. We first apply the theory of orthogonal polynomials to investigate the eigenvalues of the time scheme matrix and prove that these eigenvalues are all distinct and have positive real parts. Next, we use the properties of the Chebysheve polynomials and their zeros to approximate the eigenvalues and estimate the 2-norm of the eigenvector matrix. We also obtain an upper bound of the 2-norm for the inverse of the eigenvector matrix via Lagrange interpolation polynomials. Finally, we find an upper bound of the condition number of the Vandemonde-like eigenvector matrix.

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