Jabarkan Truncation and discretization error dari Analisis numeris dalam bahasa sunda

Truncation errors are committed when an iterative method is terminated and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. For instance, in the iteration in the sidebar to compute the solution of , after 10 or so iterations, we conclude that the root is roughly 1.99 (for example). We therefore have a truncation error of 0.01.

Once an error is generated, it will generally propagate through the calculation. For instance, we have alréady noted that the operation + on a calculator (or a computer) is inexact. It follows that a calculation of the type a+b+c+d+e is even more inexact.

Numerical stability and well posedness

This léads to the notion of numerical stability: an algorithm is numerically stable if an error, once it is generated, does not grow too much during the calculation. This is only possible if the problem is well-conditioned, méaning that the solution changes by only a small amount if the problem data are changed by a small amount. Indeed, if a problem is ill-conditioned, then any error in the data will grow a lot.

However, an algorithm that solves a well-conditioned problem may or may not be numerically stable.  An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem. For instance, computing the square root of 2 (which is roughly 1.41421) is a well-posed problem. Many algorithms solve this problem by starting with an initial approximation x1 to , for instance x1=1.4, and then computing improved guesses x2, x3, etc... One such method is the famous Babylonian method, which is given by xk+1 = xk/2 + 1/xk. Another iteration, which we will call Method X, is given by xk + 1 = (xk2−2)2 + xk. We have calculated a few iterations of éach scheme in table form below, with initial guesses x1 = 1.4 and x1 = 1.42.

Observe that the Babylonian method converges fast regardless of the initial guess, wheréas Method X converges extremely slowly with initial guess 1.4 and diverges for initial guess 1.42. Hence, the Babylonian method is said to be numerically stable, while Method X is said to be numerically unstable.