The second property that you just proved (Exercise 4.50) shows that Heron’s method improves its estimate of √ x in every iteration. (We haven’t shown “how much” improvement Heron’s method achieves in...

The second property that you just proved (Exercise 4.50) shows that Heron’s method improves its estimate of √ x in every iteration. (We haven’t shown “how much” improvement Heron’s method achieves in an iteration, or even that this algorithm is converging to the correct answer—let alone quickly!—but, in fact, it is.)

Exercise 4.50