Calculus I · Unit 2B · exploration

Newton's Method: Convergence and Failure Modes

A Tangent-Line Iteration as a Dynamical System

Newton's iteration is

N(x)=x-\frac{f(x)}{f'(x)}.

A root $r$ is a fixed point because $N(r)=r$ . Near a simple root, Taylor expansion shows

e_{n+1}\approx \frac{f''(r)}{2f'(r)}e_n^2, \qquad e_n=x_n-r.

Squaring the error explains quadratic convergence.

Read this graph as text

Near a simple root, Newton's error is approximately squared each step. For a well-chosen starting point, the number of correct digits can roughly double from one iteration to the next. The table visualizes this rapid error collapse for x 2-2=0 . The error does not merely shrink by a fixed percentage. Once the iterates are close to the simple root, an error around 10 -3 is followed by one around 10 -6 , then around 10 -12 . That is the practical signature of quadratic convergence.

Every relationship in near a simple root, newton's error is approximately squared each step is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.

Why it matters: The table makes "quadratic convergence" visible without requiring a full proof. It also creates a benchmark against which slow convergence or failure can be recognized. Students should see that Newton's method is spectacular near a suitable simple root but not universally magical.

Visual study

For a well-chosen starting point, the number of correct digits can roughly double from one iteration to the next. The table visualizes this rapid error collapse for x 2-2=0.

Failure can occur when $f'(x_n)=0$ , when tangent intercepts jump into a different region, when the starting point lies near a cycle, or when the root has multiplicity greater than one. For a multiple root, the ordinary method often converges only linearly. A modified iteration can restore faster convergence when the multiplicity is known.

Exercise

Apply two Newton steps to $f(x)=x^2-2$ from $x_0=1.5$ and compare the errors with $\sqrt2$ .

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Exercise 1 answer

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Exercise

Explain why starting Newton's method for $f(x)=x^{1/3}$ near zero is problematic.

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Exercise 2 answer

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Exercise

Investigate numerically what happens to $f(x)=x^3-2x+2$ from several starting values.

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Exercise 3 answer

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After the explanation

Use the section idea

Reading lens

Use local sensitivity, convexity, and convergence results to explain when familiar application methods become reliable global tools.

Mental model

Advanced results connect derivative evidence to error amplification, convergence speed, or global optimality under explicit hypotheses.

Decision

State the hypotheses before the conclusion and test the result on a concrete numerical or graphical example.

Common trap

Quoting elasticity, quadratic convergence, or convexity without checking units, root simplicity, or the relevant domain.

Check yourself

Can you describe both what the theorem guarantees and the failure mode its hypotheses exclude?

Interactive checkadvanced-newton-cycle-01

For $f(x)=x^3-2x+2$ and $x_0=0$ , describe the first two Newton iterates.

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Compute $x_1$ and $x_2$ .

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