Newton devised an iterative process, called Newton's Method for finding the roots of functions.
(A root of a function f(x) is a number x* for which f(x*) = 0. That is, the graph of y = f(x) crosses the x-axis at x*.)
Here is the method.
| * Guess a number x0 If f(x0) = 0, we're finished. If not, |
| * Draw the vertical line from (x0,0) to (x0,f(x0)). |
| * Draw the line tangent to the graph of y = f(x) at the point (x0,f(x0)). |
| * Usually, this tangent line intersects the x-axis at a point (x1,0). |
| * Repeat the process with x1 taking the place of x0. |
| * Continue until f(xi) is close enough to 0. |
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| Click the picture to animate. |
Being the inventor of calculus, Newton was able to find a compact expression for this construction.
| x1 = x0 - f(x0)/f '(x0) |
| x2 = x1 - f(x1)/f '(x1) |
| ... |
Writing Nf(x) = x - f(x)/f '(x), Newton's method can be encapsulated in a single iterative scheme:
Return to the complex Newton's method.