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Mapping computations

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Abstract

Let Z be a set of integers and Z n×n be a ring for any integer n. We define \({\hat{s}}\in \mathbf{Z}^{n}\) as a latter point. Hom(Z n,Z m) denotes as a homomorphism of Z n into Z m. For any element \({\hat{q}}\) in Z n, we define S+T:Z nZ m as \((S+T)({\hat{q}})=S({\hat{q}})+T({\hat{q}})\) . As a result, S+T become a homomorphism of Z n into Z m. We also define kU:Z nZ m as \((kU)({\hat{q}})=k(U({\hat{q}}))\) . Consequently, kU become a homomorphism of Z n into Z m. Moreover, Hom (Z n,Z m) is isomorphic to Z n×m. A novel class of the structured matrices which is a set of elements of Hom (Z n,Z n) over a ring of integers with a displacement structure, referred to as a C-Cauchy-like matrix, will be formulated and presented.

Using the displacement approach, which was originally discovered by Kailath, Kung, and Morf (J. Math. Anal. Appl. 68:395–407, 1979), a new superfast algorithm for the multiplication of a C-Cauchy-like matrix of the size n×n over a field with a vector will be designed. The memory space for storing a C-Cauchy-like matrix of the size n×n over a field is O(n) versus O(n 2) for a general matrix of the size n×n over a field. The arithmetic operations of a product of a C-Cauchy-like matrix and a vector is reduced dramatically to O(n) from O(n 2), which can be used to transform a latter point \({\hat{s}}\in Z^{n}\) to another latter point \({\hat{t}}\in Z^{n}\) such that \({\hat{t}}=C{\hat{s}}\) .

Moreover, the displacement structure can also be extended to a Kronecker matrix W Z. A new class of the Kronecker-like matrices with the displacement rank r, r<n will be also discovered. The memory space for storing a Kronecker-like matrix of the size (n×1)(1×n) over a field is decreased to O(rn). The arithmetic operations for a product of a Kronecker-like matrix with the displacement rank r and a vector is also accelerated to O(rn).

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Chen, Z. Mapping computations. J Supercomput 48, 152–162 (2009). https://doi.org/10.1007/s11227-008-0209-x

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