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Riemann-Stieltjes integration is an optional topic for a first course in real analysis. In this paper, we examine some of the pedagogical reasons in favor of its inclusion and some of the technical anachronisms associated with it.
In this paper, a homotopy algorithm for finding some or all finite eigenvalues and corresponding eigenvectors of a real symmetric matrix pencil $(A,B)$ is presented, where $A$ is a symmetric tridiagonal matrix and $B$ is a diagonal matrix with $b_i \ge 0$, $i = 1, 2, \ldots , n$. It is shown that there are exactly $m$ ($m$ is the number of finite eigenvalues of $(A,B)$) disjoint, smooth homotopy paths connecting the trivial eigenpairs to the desired eigenpairs. And the eigenvalue curves are monotonic and easy to follow. The performance of the parallel version of our algorithm is presented.