MATH221 · Annotated SlidesChapter 2 — Newton's Method and Its Extensions
Chapter 2 · Annotated slides

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.

Slide text Study notes Highlight: key words Tips, tricks & traps Seen again elsewhere How it can come up
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    Slide 2: Newton Raphson Method
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      Slide 3: Formula Used in Newton Raphson Method
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      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.
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      Written

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

      Slide 4: Worked Example – Solving a Problem
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          Slide 6: Solution
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            Slide 7: Solution cont’d
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                Slide 9: A Secant Method refers to an iterative technique used in numerical analysis
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                  Slide 10: Formula Used in Secant Method
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                  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.
                  Slide 11: False Position
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                    Slide 12: The False Position Method, also called Regula Falsi Method, is a numerical method to solve equations
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                    1. False position: secant formula but keeps the two points that bracket the root (like bisection), so it always converges.
                    Memory trickNewton needs f′; secant drops f′; false position = secant + bracketing.
                    🎯 How this can come up · 1 question
                    MCQ

                    Which method keeps the root bracketed at every step?

                    Slide 13: Formula Used in False Position Method
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