Newton’s Method Newton’s method is a powerful tool for solving equations of the form f (x) = 0. Example: Solve x2 = 5. √ We’re going to use Newton’s method to ﬁnd a numerical approximation for 5. Any equation that you understand can be solved this way. In order to use Newton’s method, we deﬁne f (x) = x2 − 5. By ﬁnding the value of x for which f (x) = 0 we solve the equation x2 = 5. Our goal is to discover where the graph √ crosses √ the x-axis. We start with an initial guess — we’ll guess x0 = 2, since 5 ≈ 4 = 2. This is not a very good guess; f (2) = −1, and we’re looking for a number x for which f (x) = 0. We’ll try to improve our guess. We pretend that the function is linear, and look for the point where the tangent line to the function at x0 crosses the x-axis: see Fig. 1. This point (x1 , 0) gives us a new guess at our solution: x1 . x0 x2 x1 Figure 1: Illustration of Newton’s Method The equation for the tangent line is: y − y0 = m(x − x0 ) When the tangent line intercepts the x-axis y = 0, and the x coordinate of that point is our new guess x1 . −y0 y0 − m x1 = m(x1 − x0 ) = x1 − x0 = x0 − y0 m In terms of f : y0 = m = f (x0 ) f � (x0 ) because m is the slope of the tangent line to y = f (x) at the point (x0 , y0 ). Therefore, f (x0 ) x1 = x0 − � f (x0 ) The point of Newton’s method is that we can improve our new guess by repeating this process. To get our (n + 1)st guess we apply this formula to our nth guess: f (xn ) xn+1 = xn − � . f (xn ) In our example, x0 = 2 and f (x) = x2 − 5. We ﬁrst calculate f � (x) = 2x. Thus, x1 = x1 = (x20 − 5) 1 5 = x0 − x0 + 2x0 2 2x0 1 5 x0 + 2 2x0 x0 − The main idea is to repeat (iterate) this process: 1 5 x1 + 2 2x1 1 5 x3 = x2 + 2 2x2 √ and so on. The procedure approximates 5 extremely well. x2 = Let’s see how well this works: x1 = = = x2 = = = = x3 n xn √ 5 − xn 0 2 2 × 10−1 1 9 4 10−2 2 161 72 4 × 10−5 3 1 161 2 72 + 5 72 2 161 = 1 5 2+ 2 2·2 5 1+ 4 9 4 19 5 + 2 4 2 94 9 54 + 8 29 9 10 + 8 9 161 72 1 161 5 72 + 2 72 2 161 10−10 Notice that the number of digits of accuracy doubles with each iteration; x2 is as good an approximation as you’ll ever need, and x3 is as good an approxi mation as the one displayed by your calculator.