Find Function Roots With Iteration and Convergence

Solving f(x) = 0 when there's no closed form is the use case. Bisection brackets a root between a and b with f(a)·f(b) < 0 and halves the interval each step. Every iteration buys roughly one bit of precision, so reaching 10⁻⁸ from a unit-width bracket takes about 27 steps. Slow, but unconditionally convergent as long as the bracket is valid.

Newton's method uses the derivative: x_{n+1} = x_n − f(x_n) / f'(x_n). Convergence is quadratic near the root, so error roughly squares each step and machine precision lands in 5–7 iterations from a decent starting guess. The cost: it can diverge, oscillate, or land on the wrong root if you start too far away or near a critical point of f. Brent's method is the standard hybrid in practice, combining bisection's safety with secant-style speed; it's what scipy.optimize.brentq calls under the hood. The page runs Newton, bisection, and Brent on the same f and reports the iteration trace for each, so you can see convergence behavior directly rather than just trusting a "converged" flag.

Bisection brackets the root between two points where f changes sign. Each iteration halves the interval, guaranteeing convergence but taking 20–50 iterations for typical tolerances. It never fails if you start with a valid bracket (f(a) and f(b) have opposite signs).

Newton-Raphson uses the derivative to leap toward the root: x_{n+1} = x_n − f(x_n)/f′(x_n). When it works, convergence is quadratic—error roughly squares each step, reaching machine precision in 5–10 iterations. But it requires a good initial guess and can fail if f′ is zero or nearly so.

Use bisection when you're unsure of starting conditions or when the function is poorly behaved. Use Newton when you have a reasonable guess and the derivative is available. A hybrid approach—bisection to narrow down, then Newton to finish—combines reliability with speed.

For bisection, pick [a, b] so f(a) and f(b) have opposite signs. The Intermediate Value Theorem guarantees at least one root inside. If both values share a sign, either there's no root in that interval or an even number of roots that cancel out. Narrow the bracket visually by plotting first.

For Newton, start near where you expect the root. A guess too far away can send iterations off to infinity or to a different root. Plot f(x) to identify approximate crossings. If the function has multiple roots, different starting guesses find different roots.

Critical points (where f′ = 0) trap Newton. If your guess lands near a local extremum, the tangent line is nearly horizontal, and the next step jumps far away. Avoid starting exactly at extrema. Bisection doesn't care about extrema inside the bracket.

The iteration table shows how each step refines the approximation. For Newton, watch the error shrink rapidly—if error is 10⁻² at step 3, it might be 10⁻⁴ at step 4, then 10⁻⁸ at step 5. That's quadratic convergence in action.

Bisection's table shows the interval halving. Error decreases by a factor of 2 each iteration. To get from 10⁻¹ to 10⁻⁶ requires about 17 halvings (since 2¹⁷ ≈ 130,000). Predictable but slow compared to Newton's geometric leaps.

Convergence stops when |f(x)| drops below the tolerance or when successive approximations differ by less than the tolerance. If iterations max out without meeting the tolerance, the method didn't converge—check your starting conditions or function behavior.

A repeated root (like x² = 0 at x = 0) slows Newton to linear convergence because f and f′ both vanish at the root. The formula x − f/f′ becomes 0/0 in the limit. Special modifications (like modified Newton) handle this, but basic implementations struggle.

Flat slopes (f′ ≈ 0 away from the root) cause wild jumps. If the tangent line is nearly horizontal, the x-intercept lies far away. One bad step can throw iterations off to infinity or into oscillation between distant values.

Bisection fails if the interval lacks a sign change—either no root exists inside, or an even number of roots cancel out (f dips below zero then returns above without crossing back). Always verify that f(a) × f(b) < 0 before relying on bisection.

The graph shows f(x) with the x-axis. Roots appear where the curve crosses zero. Plotting before computing tells you roughly where to start and how many roots exist in your range. Multiple crossings mean multiple roots—each requires a separate run.

Iteration points can be overlaid on the plot. For Newton, see how each tangent line jumps to the next approximation. For bisection, watch the interval shrink around the crossing. Visual feedback clarifies why convergence succeeds or fails.

Near-miss roots (where f gets close to zero but doesn't cross) can trick visual inspection. Zoom in to confirm the curve actually crosses or merely grazes the axis. A grazing touch means a repeated root or no root at all in the strict sense.