Fast Algorithms For The Inverse Of Special Several Matrices

In this paper,by using the matrix LU decomposition,the method for solving linear equation,the definition of inverse matrix,the expansion of the matrix for getting the inverse matrices of a variety of special diagonal matrices,and according to the routine from easy to difficult to propel.At first,employing the matrix LU decomposition to obtain the inverse matrices of a relative simple tridiagonal Toeplitz matrix and a periodic tridiagonal Toeplitz matrix.Then,using the expansion of the matrix for gaining the inverse matrices of a relative complicated heptadiagonal matrix and a periodic heptadiagonal matrix.In the case of verifying the above methods are effective,the matrix LU decomposition was also used for solving the inverse matrices of a more difficult periodic k-tridiagonal matrix and a k-pentadiagonal matrix again finally.In our work,the following aspects are studied in detail:1.The algorithms for the inverses of a tridiagonal Toeplitz matrix and a periodic tridiagonal Toeplitz matrix.The idea of the algorithms is: in line with the special structure of a tridiagonal Toeplitz matrix and a periodic tridiagonal Toeplitz matrix,using the matrix LU decomposition as well as the method for solving linear equation to give the inverse matrices.The algorithm complexities are just O(n2),in detail,the addition and subtraction computational cost of the algorithm for solving the inverse of a tridiagonal Toeplitz matrix is 2n2-n-1,and the multiplication and division is 3n2+n-3;Similarly,the addition and subtraction computational cost of the algorithm for solving the inverse of a periodic tridiagonal Toeplitz matrix is2n2+3n-6,and the multiplication and division is 3n2+9n-20.Finally,both the effectiveness and the stability of the algorithms were verified by numerical examples.2.The algorithms for the inverses of a heptadiagonal matrix and a periodic heptadiagonal matrix.By the expansion of the matrix,the n×n heptadiagonal matrix and theperiodic heptadiagonal matrix were extended to n×(n+3)matrices to obtain the inverse matrices.And the algorithm complexities are just O(n2).The algorithms were verified to be effective by numerical examples last.3.The algorithms for the inverses of a periodic k-tridiagonal matrix and a k-pentadiagonal matrix.The algorithms were similar to the algorithms in 1,they all used the matrix LU decomposition and the definition of inverse matrix.It was worth mentioning that the algorithms in 1 were special cases of these new algorithms.The ideal results: the algorithm complexities for inverting a periodic k-tridiagonal matrix and a k-pentadiagonal matrix both were O(n2).By this way,the condition that each principal minor sequence of the matrix must nonzero is unnecessary.And the algorithms were suited for implementation using computer algebra systems.4.The algorithms for the inverses of a few kinds of special anti diagonal matrix.Based on the known inverses of original diagonal matrices,and the relation of original diagonal matrices with their corresponding anti diagonal matrices,the inverses of anti diagonal matrices were gained quickly.In this paper,taking a heptadiagonal matrix and a periodic heptadiagonal matrix as examples,the inverse matrices of an anti-heptadiagonal matrix and a periodic anti-heptadiagonal matrix were given.