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Vladimir Rokhlin, Arthur K Watson Professor of Computer Science & Mathematics, Yale University

Finding scattering resonances via generalized colleague matrices

Locating scattering resonances is a standard task in certain areas of physics and engineering. This often can be reduced to finding zeros of complex analytic functions. In this talk, I will discuss a scheme for finding all roots of a complex analytic function in a square domain in C. The scheme can be viewed as a generalization of the classical approach to finding roots of a function on an interval by first approximating it by a polynomial in the Chebyshev basis, followed by diagonalizing the so-called “colleague matrices.” This extension to the complex domain is based on several observations that enable the construction of polynomial bases that satisfy three-term recurrences and are reasonably well-conditioned, giving rise to “generalized colleague matrices.” We also introduce a special-purpose QR algorithm for finding eigenvalues of the resulting structured matrices stably and efficiently. I will demonstrate the effectiveness of the approach via several numerical examples.