If and are non-negative real numbers, we can form their arithmetic mean as well as their geometric mean . This procedure can be repeated to form a sequence of arithmetic and geometic means and . By the arithmetic-geometric means inequality we have (with equality holding only when ), hence these sequences converge to a number between and , with the rate of convergence being superlinear. The arithmetic-geometric mean of and is defined as this limit
The AGM lies between the arithmetic and geometric means of and ,
As a numerical method, the arithmetic-geometric mean has much to recommend it. By its nature, it automatically provides upper and lower bounds for the answer, so one does not have to separately estimate error. To compute the arithmetic-geometric mean to a certain accuracy, we only need to carry out the computation until the difference between and is smaller than the desired accuracy.
Because convergence is superlinear, only a few iterations are necessarry to obtain the answer. For instance, if we compute with less than a billion, we already obtain at least fifteen-place accuracy after eight iterations, as the following computation of shows:
The fact that relatively few iterations are necessarry to obtain a highly accurate result also means that one does not have to worry much about the cumulative effect of roundoff errors in the various steps of the computation.
|Date of creation||2013-03-22 14:23:46|
|Last modified on||2013-03-22 14:23:46|
|Last modified by||rspuzio (6075)|