Calculus I · 2B · exploration
Newton's Method: Convergence and Failure Modes
Explore newton's method: convergence and failure modes in an optional advanced note connecting Calculus I to later analysis and mathematical theory.
Section overview
Optional advanced explorationsWhat this section is building
Explore newton's method: convergence and failure modes in an optional advanced note connecting Calculus I to later analysis and mathematical theory.
Advanced results connect derivative evidence to error amplification, convergence speed, or global optimality under explicit hypotheses.
State the hypotheses before the conclusion and test the result on a concrete numerical or graphical example.
Quoting elasticity, quadratic convergence, or convexity without checking units, root simplicity, or the relevant domain.
A Tangent-Line Iteration as a Dynamical System
Newton's iteration is
A root is a fixed point because . Near a simple root, Taylor expansion shows
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.
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 , 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.
Apply two Newton steps to from and compare the errors with .
Exercise 1 answer
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Explain why starting Newton's method for near zero is problematic.
Exercise 2 answer
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Investigate numerically what happens to from several starting values.
Exercise 3 answer
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advanced-newton-cycle-01For and , describe the first two Newton iterates.
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Compute and .
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