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Advanced - Arrays, Sorting and Searching Ā· 4:15
Root Finding: Bisection vs Newton's Method
Bisection finds a root of a continuous function from a sign change and gains about a third of a digit per step; Newton's method follows the tangent and doubles the correct digits near a simple root. We race them on the square root of 2, take Newton to integers with the right stopping rule and exact checks, watch it cycle on a cubic, and combine both into safeguarded Newton, the tool behind bond-yield solvers.
You will learn
- Use bisection on a continuous function with a sign change and estimate the number of steps for a given accuracy
- Derive the Newton update x - f(x)/f'(x), its square-root form (x + n/x)/2, and explain quadratic convergence
- Compute integer roots exactly: the integer Newton stopping rule, a good starting point and exact correction of floating-point estimates
- Recognise Newton failures (cycles, flat slopes, leaving the domain) and fix them with safeguarded Newton