The norm of a vector || x|| tells us how big the vector is as a whole (as opposed to how large each element of the vector is). For this example, the true solution is x = (1, 2, −1). However, to better understand the behavior of an iterative method, it is enlightening to use the method to solve a system A x = b for which we do know the true solution and analyze how quickly the approximations are converging to the true solution. Obviously, we don't usually know the true solution x. We are interested in the error e at each iteration between the true solution x and the approximation x (k): e ( k) = x − x (k). We show the results in the table below, with all values rounded to 3 decimal places. We iterate this process to find a sequence of increasingly better approximations x (0), x (1), x (2), …. So, if our initial guess x (0) = ( x 1 (0), x 2 (0), x 3 (0)) is the zero vector 0 = (0,0,0) - a common initial guess unless we have some additional information that leads us to choose some other - then we find x (1) = ( x 1 (1), x 2 (1), x 3 (1) ) by solving Let's apply Jacobi's Method to the systemĪt each step, given the current values x 1 ( k), x 2 ( k), x 3 ( k), we solve for x 1 ( k+1), x 2 ( k+1), and x 3 ( k+1) in Then Jacobi’s Method can be written in matrix-vector notation as
If we write D, L, and U for the diagonal, strict lower triangular and strict upper triangular and parts of A, respectively, X ( k) = ( x 1 ( k), x 2 ( k), …, x i ( k), …, x n ( k) ),Īnd superscript k corresponds to the particular iteration (not the k th power of x i ). To be clear, the subscript i means that x i ( k) is the i th element of vector That is, given current values x ( k) = ( x 1 ( k), x 2 ( k), …, x n ( k)), find new values by solving for X ( k) = ( x 1 ( k), x 2 ( k), x 3 ( k), …, x n ( k))įor x, the strategy of Jacobi's Method is to use the first equation and the current values of x 2 ( k), x 3 ( k), …, x n ( k) to find a new value x 1 ( k+1), and similarly to find a new value x i ( k) using the i th equation and the old values of the other variables. Still, it is a good starting point for learning about more useful, but more complicated, iterative methods. Note that the simplicity of this method is both good and bad: good, because it is relatively easy to understand and thus is a good first taste of iterative methods bad, because it is not typically used in practice (although its potential usefulness has been reconsidered with the advent of parallel computing). Perhaps the simplest iterative method for solving A x = b is Jacobi’s Method.
0 kommentar(er)
