Newton's Method Across Multiple Root Intervals

One Equation, Several Possible Limits

When an equation has more than one root, Newton’s method does not come with a rule saying which root a particular starting value will find. The outcome depends on the function and on the starting value: starts in different regions can approach different roots, and some starts may lead to an undefined step or leave the function’s domain.

To track that behavior, it helps to regard Newton’s formula as a function of the current iterate. If \(f\) is differentiable at \(x_n\) and \(f'(x_n)\ne 0\), the next iterate is

The formula only defines a step when \(f(x_n)\) and \(f'(x_n)\) exist and the derivative is nonzero. For a sequence of steps, this check must be repeated at every iterate, not just at the initial value.

Different Root Intervals Can Lead to Different Roots

A simple example makes the role of starting values explicit. Consider \(f(x)=(x-1)(x-3)=x^2-4x+3\), whose roots are \(1\) and \(3\). Its derivative is \(f'(x)=2x-4\), so Newton’s step is undefined at \(x=2\). At every other real number the iteration map is

In this example, the point \(2\) divides the real line into two intervals. It is not itself a valid starting point, because \(f'(2)=0\). The intervals on either side have different limiting roots. The behavior can be proved directly from the iteration map.

Why the Two Intervals Have Different Limits

We justify the theorem by examining how the iteration map moves points. For \(x\ne2\), algebra gives

If \(x<2\) and \(x\ne1\), the first expression is negative, so \(N(x)<1\). If also \(x<1\), then \(f(x)>0\) and \(f'(x)<0\). Thus \(f(x)/f'(x)<0\), and \(N(x)=x-f(x)/f'(x)>x\). Consequently, once an iterate is below \(1\), the next iterate is greater than the current one but still below \(1\). This includes every start below \(2\), after at most one step; the root \(x=1\) itself remains fixed.

For a start below \(1\), its iterates therefore increase and are bounded above by \(1\), so they have a limit \(L\le1\). The iteration map is continuous on this region, where its denominator is nonzero. Taking limits in \(x_{n+1}=N(x_n)\) gives \(L=N(L)\). The equation \(L=N(L)\) implies \(f(L)=0\), since \(f'(L)\ne0\). The only root at or below \(1\) is \(1\), so \(L=1\).

On the other side, if \(x>2\) and \(x\ne3\), the second expression shows \(N(x)>3\). For \(x>3\), both \(f(x)>0\) and \(f'(x)>0\), so \(N(x)<x\). A start between \(2\) and \(3\) moves above \(3\) in one step; after that, the iterates decrease while remaining above \(3\). They have a limit \(L\ge3\). As before, continuity and \(L=N(L)\) imply \(f(L)=0\), and the only root at or above \(3\) is \(3\). Hence the limit is \(3\). This proof also shows why a start near the point where the derivative vanishes can make a large first jump.

Root Selection Is Not a Nearest-Root Rule

With several roots, a starting value’s nearest root does not necessarily determine the limit of Newton’s method. The tangent-line step depends on both \(f(x_n)\) and \(f'(x_n)\); it is not a step toward the closest root by definition. The next example gives an exact instance where the iteration goes to a root that is much farther from the start than another root.

This example also distinguishes root selection from mere root proximity. Newton’s method is governed by the geometry of the tangent line at the current iterate. A step can cross other roots or move into a different region; root intervals are useful to investigate, but they must be established for the function in question rather than inferred from distances alone.

Check the Domain at Every Step

Two separate conditions matter when iterating. First, the function must be defined at the current value, and its derivative must exist there and be nonzero. Second, the new value must remain in the domain if another step is to be taken. A starting value belonging to the domain alone does not guarantee that even the first step exists: for example, \(f(x)=x^2\) is defined at \(x_0=0\), but \(f'(0)=0\), so the Newton step is undefined.

Even when the first step exists, it can leave the domain. For \(f(x)=\ln x\), the real domain is \(x>0\), the root is \(1\), and \(f'(x)=1/x\). The following example illustrates why checking the new iterate is essential.

A Reliable Way to Track Multiple-Root Iterations

When an equation has several roots, analyze an iteration by combining algebra, the location of the roots, and domain checks. A useful process is:

A common mistake is to compute one valid step and assume that the iteration can continue. For \(f(x)=\ln x\) and \(x_0=e^2\), the incorrect reasoning would be: “\(x_0>0\), so Newton’s method is defined.” That only checks the starting point. In fact, \(f'(e^2)\ne0\) makes the first update valid, but the update is \(x_1=-e^2<0\), outside the domain. The correction is to check both conditions at every stage: the current point must admit a Newton step, and the resulting point must be admissible for the next one.

Tracking intervals and checking admissibility prevent two different errors: assigning a start to a root merely because that root is nearby, and treating an invalid or domain-leaving sequence as though it were a continuing Newton iteration. The iteration map gives a precise way to investigate which root a start approaches, while the domain and derivative checks determine whether the sequence is defined at all.