MATH221 · Chapter 2 · Annotated SlidesNewton's Method and Its Extensions
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Newton's Method and Its Extensions

Newton–Raphson, the secant method and the method of false position (regula falsi): formulas, worked examples and how they compare.

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Every original slide is shown unchanged. Yellow marks on a slide point to a numbered note: hover or tap a mark to jump to its note, and hover a note to see its marks. Questions tagged Past exam appeared on a real paper; Prediction questions are likely angles, not confirmed. Pick an answer to each question: a wrong pick tells you why it is wrong, so you can try again.

All 19 slides
  1. 1 Slide 1
  2. 2 Newton Raphson Method
  3. 3 Formula Used in Newton Raphson Method
  4. 4 Worked Example – Solving a Problem
  5. 5 Slide 5
  6. 6 Solution
  7. 7 Solution cont’d
  8. 8 Slide 8
  9. 9 A Secant Method refers to an iterative technique used in numerical analysis
  10. 10 Formula Used in Secant Method
  11. 11 False Position
  12. 12 The False Position Method, also called Regula Falsi Method, is a numerical method to solve equations
  13. 13 Formula Used in False Position Method
  14. 14 Slide 14
  15. 15 Slide 15
  16. 16 Slide 16
  17. 17 Slide 17
  18. 18 Slide 18
  19. 19 Slide 19
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Newton Raphson Method

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Formula Used in Newton Raphson Method

Slide 3: Formula Used in Newton Raphson Method

Notes

  1. Newton: pₙ = pₙ₋₁ − f(pₙ₋₁)/f′(pₙ₋₁). Quadratic convergence near a simple root, but needs f′ and a good starting point.

Exam angle

PredictionQ1

One Newton step for f(x) = x² − 2 from p₀ = 1.

Compare with the model answer

p₁ = 1 − (1 − 2)/(2·1) = 1.5.

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Worked Example – Solving a Problem

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Solution

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Solution cont’d

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A Secant Method refers to an iterative technique used in numerical analysis

Slide 9: A Secant Method refers to an iterative technique used in numerical analysis
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Formula Used in Secant Method

Slide 10: Formula Used in Secant Method

Notes

  1. Secant: replaces f′ with the slope through the last two points: pₙ = pₙ₋₁ − f(pₙ₋₁)(pₙ₋₁ − pₙ₋₂)/(f(pₙ₋₁) − f(pₙ₋₂)). No derivative needed.

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False Position

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The False Position Method, also called Regula Falsi Method, is a numerical method to solve equations

Slide 12: The False Position Method, also called Regula Falsi Method, is a numerical method to solve equations

Notes

  1. False position: secant formula but keeps the two points that bracket the root (like bisection), so it always converges.

  2. Memory trick Newton needs f′; secant drops f′; false position = secant + bracketing.

Exam angle

PredictionQ1

Which method keeps the root bracketed at every step?

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Formula Used in False Position Method

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