Newton’s method for finding roots

Near the current estimate xk, the tangent approximates the function by f(xk)+f′(xk)(x−xk). Set this line equal to zero and solve for x to obtain its x-axis intercept. This gives xk+1=xk−f(xk)/f′(xk). Geometrically, it is a new root candidate; algebraically, it is a step based on linearization. If f′(xk)=0, division is impossible. A very small derivative can also produce an excessively large step. For example, for f=x²−2 at x₀=1.5, f=0.25 and f′=3, so x₁≈1.41667. The tangent intercept moves closer to √2. This step illustrates the approximation, not an error bound.

No. Under favorable local conditions, especially near a simple root with nonzero derivative, the method converges quickly. A poor starting value can instead jump into an unsuitable region, oscillate or diverge. At multiple roots the usual convergence order decreases; if f′=0 the step is undefined. Even a continuous function with a root does not guarantee success from every starting point. Choose a justified start, monitor both residual and step size, and set a maximum iteration count. For robustness, combine Newton’s method with a bracketed method such as bisection.

A practical stopping rule combines two quantities: a small residual |f(xk)| and a small change |xk−xk−1| relative to the desired tolerance. A small step alone can mean the method is stalled rather than near a root. Conversely, for poorly scaled functions a small residual may hide substantial uncertainty in x. Check scaling, use both absolute and relative tolerances, and limit the number of iterations. The reported precision should also not imply more accuracy than the input data support. Report a stop caused by a stalled residual as failure, not as successful convergence. Safeguards may still be needed.

Before each step, check whether f′(x_k) is numerically large enough; a nearly zero denominator can produce a huge jump. If needed, restrict the step to a plausible interval or combine Newton with bisection: while a sign change brackets the root, accept a fast Newton step only if it stays inside the bracket. Otherwise halve the interval. This is slower but preserves a meaningful bound for the solution. Monitor |f(x_k)| and step size separately. A Newton step outside the permitted range is not progress, even when its formula was evaluated correctly.

Quadratic convergence typically holds near a simple root, where f′ does not vanish at the root. At an m-fold root, the derivative also vanishes, and standard Newton iteration generally becomes only linear. For f(x)=(x−1)², the step is x_k+1=(x_k+1)/2: the distance to the root is halved each iteration instead of being approximately squared. If multiplicity m is known, the modified iteration x_k+1=x_k−m f(x_k)/f′(x_k) can restore fast convergence. This correction requires the correct multiplicity; choosing m incorrectly can make iteration worse. A residual check can reveal the slowdown before many extra iterations are spent.