Toeplitz/Hankel Matrix Structure and Polynomial Computations
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Toeplitz matrices, Hankel matrices, and matrices with similar structures (such as Frobenius, Sylvester, and subresultant matrices) are probably the most studied structured matrices. Among their numerous important areas of application, we select fundamental polynomial computations, including multiplication, and the Extended Euclidean Algorithm, together with their several extensions and applications. In this chapter we reveal the correlation among computations with polynomials and structured matrices of Toeplitz and Hankel types (see Figures 2.1 and 2.2) and show superfast algorithms for these computations. In the next chapter we similarly study matrices of Vandermonde and Cauchy types. Apart from the introductory material of the next section and the estimates for the arithmetic cost of multiplying Toeplitz, Hankel, Vandermonde, and Cauchy matrices by vectors, the results of these two chapters are little used in Chapters 4–7. Some sample pseudocodes for the algorithms of Sections 2.4 and 2.5 are collected in Section 2.15.
KeywordsDiscrete Fourier Transform Toeplitz Matrix Toeplitz Matrice Circulant Matrix Euclidean Algorithm
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