Rate of convergence

In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, the concept of rate of convergence is of practical importance when working with a sequence of successive approximations for an iterative method, as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may even make the difference between needing ten or a million iterations. In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, the concept of rate of convergence is of practical importance when working with a sequence of successive approximations for an iterative method, as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may even make the difference between needing ten or a million iterations. Similar concepts are used for discretization methods. The solution of the discretized problem converges to the solution of the continuous problem as the grid size goes to zero, and the speed of convergence is one of the factors of the efficiency of the method. However, the terminology in this case is different from the terminology for iterative methods. Series acceleration is a collection of techniques for improving the rate of convergence of a series discretization. Such acceleration is commonly accomplished with sequence transformations. Suppose that the sequence ( x k ) {displaystyle (x_{k})} converges to the number L {displaystyle L} . The sequence is said to converge linearly to L {displaystyle L} , if there exists a number μ ∈ ( 0 , 1 ) {displaystyle mu in (0,1)} such that where the number μ {displaystyle mu } is called the rate of convergence. The sequence is said to converge superlinearly (i.e. faster than linearly) to L {displaystyle L} , if The sequence is said to converge sublinearly (i.e. slower than linearly) to L {displaystyle L} , if If the sequence converges sublinearly and additionally