2. Numerical Solutions of Algebraic and Transcendental Equations
The chapter discusses several numerical methods for finding the roots of algebraic and transcendental equations, emphasizing the Bisection Method, Newton-Raphson Method, Secant Method, and Fixed-Point Iteration. Each method is described in terms of its workings, advantages, disadvantages, and practical examples. A comparison of the methods aids in understanding their respective performance in various scenarios.
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Sections
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What we have learnt
- The Bisection Method guarantees convergence when bracketed correctly but has slow convergence.
- The Newton-Raphson Method offers faster convergence but requires the derivative and may not work well with poor initial guesses.
- The Secant Method avoids the derivative but requires two initial guesses and may also fail to converge.
Key Concepts
- -- Bisection Method
- A reliable root-finding technique requiring two initial points where the function changes sign, guaranteeing convergence.
- -- NewtonRaphson Method
- An iterative method that uses derivatives to find roots more quickly, converging quadratically if close to the root.
- -- Secant Method
- An alternative to Newton-Raphson that approximates derivatives using secant lines rather than requiring actual derivatives.
- -- FixedPoint Iteration
- An iterative method that rewrites an equation into a form suitable for successive approximations, although convergence is not guaranteed.
Additional Learning Materials
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