7 Iterative methods for matrix equations（7为矩阵方程的迭代方法）.pdf

G1BINM Introduction to Numerical Methods 7–1 7 Iterative methods for matrix equations 7.1 The need for iterative methods We have seen that Gaussian elimination provides a method for finding the exact solution (if rounding errors can be avoided) of a system of equations Ax = b. However Gaussian 3 elimination requires approximately n /3 operations (where n is the size of the system), which may become prohibitively time-consuming if n is very large. Another weakness is that Gaussian elimination requires us to store all the components of the matrix A. In many real applications (especially the numerical solution of differential equations), the matrix A is sparse, meaning that most of its elements are zero, in which case keeping track of the whole matrix is wasteful. In situations like these it may be preferable to adopt a method which produces an approximate rather than exact solution. We will describe three iterative methods, which start from an initial guess x and produce successively better approximations x , x , . . . . 0 1 2 The iteration can be halted as soon as an adequate degree of accuracy is obtained, and the hope is that this takes a significantly shorter time than the exact method of Gaussian elimination would require. 7.2 Splitting the matrix All the methods we will consider involve splitting the matrix A into the difference between two new matrices S and T : A = S − T . Thus the equation Ax = b gives Sx = T x + b, based on which we can try the iteration Sxk+1 = T xk + b. (7.1) Now if th