Newton's method explained
Finding a root of the equation f(x)=0 is the job. Most of the time there is no tidy formula. Newton's method draws the tangent to the curve at the current point and takes the place where that line meets the x-axis as the next guess. The curve is hard, but a line finds 0 easily. The formula is xn+1=xn - f(xn)/f'(xn). Repeat it and the guess slides down toward the root. When it works, the number of correct digits roughly doubles at every step, which is why it is called convergence of order 2. It is not a cure-all. Where the slope is flat the tangent throws the guess far away, and a difficult curve can loop between two points or diverge. A small change in the start can land on a different root. Set beside the safe bisection method, Newton's speed is plain to see. The gold tangent in the picture is that whole step.
The gray curve is f(x)=x²-2, and its root is √2≈1.414. Set the start x0 with the slider and press the step button. A gold tangent is drawn at the current point, and the small circle where that tangent meets the x-axis is the next guess. Each press slides the point down toward the root, and the f(x) below moves closer to 0. The point is to follow the easy 0 of a straight line instead of the curve. Reset returns to the start. When the point sits on the root, the function value there has nearly vanished.
The error of the same equation is shown as bars on a log scale. One bar is |xn - √2| after one more step. On a log scale, shorter means more accurate, and a bar that reaches the floor is nearly perfect. Move the x0 slider and, wherever you start, the bars drop like a cliff in a few steps. The error is roughly squared at every step. An error of 0.1 does not become 0.01. It shrinks toward the square of 0.01. The correct-digit readout jumping by about a factor of two at each step is convergence of order 2.
Fast does not mean safe. You can choose three failures. Overshoot is when x0 is near 0, so f'(x)≈0, the tangent is almost horizontal, and the guess is thrown off the picture. The cycle is f=x³-2x+2 started at 0, which walks between 0 and 1 forever and never reaches a root. Divergence is a curve that flattens far away, like arctan, where the steps grow and walk off. The tangent arrows show where the step goes wrong. Newton's method needs a good start and a safety net.
Now f=x³-x. There are three roots, -1, 0, and +1. Move the start x0 with the slider and the color tells which root Newton falls into. Green is -1, gold is 0, blue is +1, and gray is divergence. The whole band is a map of each starting point's fate. Near the middle it goes cleanly to 0, but around ±1/√5 the colors tangle like hairs. There, moving the start by a tiny amount changes the root you land on. The same rule draws a boundary that fine.
Last, compare the speeds. Solve f=x²-2 on [1,2], and one step button advances Newton (gold, x0=2) and bisection (blue, the interval [1,2]) together. Bisection keeps the half where the sign changes and cuts the interval in half each time. It is safe, but it adds only about 0.3 correct digits per step. Newton's error plunges in the same number of steps. Compare the two errors in the readout and Newton is far ahead of bisection within a few steps. Safety belongs to bisection and speed to Newton, so in practice the two are mixed. Each press of the same button walks both methods forward one notch together.