Why does Newton's method fail when the function is flat?
Conditions
- The function is locally flat at the current point.
- The slope is too small or zero to provide a useful Newton step.
Reasoning, step by step
- Observe that the function is flat at the current guess.
- Recognize that the derivative (slope) is zero or near-zero.
- Understand that the tangent line is horizontal.
- Realize that a horizontal tangent does not produce a new x-intercept closer to the root.
- Conclude that the iteration cannot progress.
Example
The video shows a yellow dot on a nearly horizontal portion of the curve with a horizontal green dotted tangent line, illustrating that following the slope won't get us anywhere.
Common misconceptions
- Thinking that a flat function means the root is at infinity; it means the method cannot find the root from that starting point.
- Assuming the method will automatically adjust the step size to overcome flatness.
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Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.