Mapping computations

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 n →Z 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 n →Z 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).