Fast Algorithm Of Scaled Factor Circulant Matrices

For years, circulant matrices, which are an important component of the matrix theory and applied mathematics. Because of many good properties and structures of such matrices , they have become one of the most important and active research fields of applied mathematics and computation mathematics. So many kinds of circulant matrices have been discussed, such as the scaled factor circulant matrices, which have a widely range of application in signal processing, code theory, image processing, and so on.In paper [29],J.L.Stuart and J.R.Weaver gave the definition of the scaled factor circulant matrices and discussed the basic properties of such matrices. Cen Jianmiao[25] discussed the spectral decomposition of the scaled factor circulant matrices and its applications . Jiang Zhaolin and Liu Sanyang presented a fast algorithm of calculating the inverse and the generalized inverse of scaled factor circulant matrices by the fast algorithm for computing polynomials [16]and Euclid algorithm[18] and a fast algorithm for calculating the reverse by using only interpolation methods. [19].In this paper, several fast algorithms for scaled factor circulant matrices have been derived. As theoretical and numerical experiments results show, these fast algorithms is very useful.In chapter 1, we give a brief introduction to the importance and the current research situation on circulant matrices. Moreover, some definition and properties of the special structured matrices which are discussed in our paper, have been given.In chapter 2 , we present several fast algorithms for computing the inverse , self-reflective g-inverse , the group inverse and the Moore- Penrose inverse ofa scaled factor circulant matrix, using FFT.In chapter 3, we derived a new fast algorithm for solving linear systems of the real scaled factor circulant type by FHT. As we know, DFT is defined in complex field. However, in many practical applications, many data are real. So when we solve a linear systems of scaled factor circulant matrices using FFT, we must convert the real numbers to the complex numbers firstly, which will decrease the efficiency of the algorithm. Because the discrete Hartley transformation is defined in real field and has a fast algorithm which is similar to FFT, solving a linear systems of scaled factor circulant matrices using FHT must be more efficient.In chapter 4, we researched the radication of scaled factor circulant matrices. By using FFT, we presented a fast algorithm for radication of such matrices of order n and the computation time complexity is O(n2) for calculating one radical matrix.