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How to See Why Newton's Method Fails
Manesh Jayawardhana
CIO & Co-founder
Newton’s method finds a root in four iterations from a good starting point. From a starting point half a unit away it can shoot off to x = 400, come back, and settle on a completely different root.
Both behaviours come from the same one-line formula, and seeing why is more useful than memorising the convergence conditions.
The method is one geometric idea
At the current estimate, replace the function with its tangent line. The tangent is easy to solve — it is a straight line — so take where it crosses the axis as the next estimate. Repeat.
x_{n+1} = x_n − f(x_n) / f'(x_n)
That is all of it. Everything about the method’s speed and its failures follows from the fact that the tangent line is a good approximation near the point of tangency and an arbitrarily bad one away from it.
Why it is so fast when it works
Near a simple root, convergence is quadratic: the number of correct digits roughly doubles each iteration.
Starting with one correct digit, you get two, then four, then eight, then sixteen. Four or five iterations from a reasonable start reaches the limit of double precision, which is why Newton’s method is the default in numerical libraries despite its failure modes.
That doubling is specific to simple roots. At a repeated root the derivative vanishes along with the function, convergence degrades to linear, and the method becomes unremarkable — noticeably slower and less accurate.
The three ways it fails
A near-zero derivative. At or near a turning point the tangent is almost horizontal, so it crosses the axis a very long way from the current estimate. One iteration can move you further from every root than you started.
Cycling. For some functions and starting points, the iteration bounces between two values indefinitely. It never diverges and never converges, and without an iteration cap it runs forever.
Converging to the wrong root. Perfectly well-behaved convergence, to a root you did not want. The set of starting points leading to each root can be intricate — this is the mechanism behind Newton fractals.
| Symptom | Cause |
|---|---|
| Estimate jumps far away | Near-zero derivative at a turning point |
| Values alternate forever | Cycle |
| Converges slowly | Repeated root — linear, not quadratic |
| Converges to another root | Started in a different basin |
Practical use
Almost nothing uses Newton’s method unguarded. The standard approach is a hybrid: bracket the root with a slow reliable method like bisection, then switch to Newton once you are close enough for its speed to pay.
Alongside that, real implementations cap iterations, check the derivative is not too small before dividing, and verify the result actually satisfies the equation rather than trusting convergence.
None of that is exotic — it is the difference between a textbook formula and something that can be shipped.
The derivative has to come from somewhere
A practical constraint the textbook version glosses over.
Newton’s method needs f’(x) at every iterate. For a function given as a formula that is straightforward. For anything else it is not:
Numerical differentiation approximates the derivative from nearby function values. It works, and it introduces error that grows as the step size shrinks due to floating-point cancellation, which limits achievable accuracy.
Automatic differentiation computes exact derivatives alongside the function value. Excellent where available and it requires the function to be written in a supporting framework.
The secant method avoids derivatives entirely by approximating from the last two iterates. Slightly slower convergence and no derivative required, which for a function that is expensive to evaluate is frequently the better trade.
Common mistakes to avoid
- Starting at or near a turning point.
- Running without an iteration cap, so a cycle hangs.
- Assuming quadratic convergence at a repeated root, where it is linear.
- Trusting the result without substituting it back into the original function.
- Using Newton’s method alone where a bracketing method would guarantee convergence.
How to do it with Newton’s Method Visualizer
The Newton’s Method Visualizer shows each tangent and iterate.
- Enter the function and a starting point.
- Step through and watch the tangent lines — the geometry explains the behaviour better than the formula.
- Try a starting point near a turning point deliberately, to see the overshoot.
- Compare a simple root against a repeated one and watch the convergence rate change.
Other maths tools are in the tools directory.
Frequently asked questions
Why does Newton’s method sometimes fail?
Because the tangent line can point anywhere. Near a turning point the derivative is close to zero, so the next estimate lands far away. Some starting points produce cycles that never converge at all.
How fast is it?
Quadratic near a simple root — correct digits roughly double each step, so four or five iterations often reach machine precision. At a repeated root it degrades to linear.
What makes a good starting point?
One close to the target root and away from turning points. In practice a bracketing method gets you close first, and Newton finishes quickly from there.
Final thought
Try it from a bad starting point once. Watching a tangent throw the estimate four hundred units away explains the method’s failure conditions better than any statement of them.