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§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)… ►As , and converge to a common limit called the AGM (Arithmetic-Geometric Mean) of and . …showing that the convergence of to 0 and of and to is quadratic in each case. … ►
§22.20(ii) Arithmetic-Geometric Mean… ►Then as sequences , converge to a common limit , the arithmetic-geometric mean of . … ►
§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)►The AGM, , of two positive numbers and is defined in §19.8(i). … ►
In the general case, given by , we compute the roots , , , say, of the cubic equation ; see §1.11(iii). These roots are necessarily distinct and represent , , in some order.
If , or and are not both real, then we label , , so that the triangle with vertices , , is positively oriented and is its longest side (chosen arbitrarily if there is more than one). In particular, if , , are collinear, then we label them so that is on the line segment . In consequence, , satisfy (with strict inequality unless , , are collinear); also , .
Finally, on taking the principal square roots of and we obtain values for and that lie in the 1st and 4th quadrants, respectively, and , are given by