Convergence rate is a measurement of how quickly a sequence or series of values approaches a limit or converges to a specific value.
Convergence rate is important because it indicates the efficiency and accuracy of an algorithm or numerical method. A faster convergence rate means the method can achieve a more accurate solution in fewer steps.
There are several factors that can affect the convergence rate, including the initial guess or starting point, the choice of algorithm or method, and the complexity of the problem being solved.
Convergence rate is typically calculated by measuring the ratio of the difference between two consecutive iterations to the difference between two iterations further away from each other. It can also be expressed as the number of decimal places of accuracy gained per iteration.
Some common convergence rate patterns include linear, geometric, and exponential. Linear convergence means the error decreases by a constant amount with each iteration, while geometric convergence means the error decreases by a constant ratio. Exponential convergence is the fastest, with the error decreasing by a constant power of the iteration number.