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Interactive 3D/Root-Finding Algorithms
Function
Initial Guess x₀
x₀ = 0.50
Tolerance
Playback
Convergence
Newton iters:0 / 21
Bisect iters:0 / 20
Newton |f(x)|:2.38e+0
Newton-Raphson: quadratic convergence - doubles correct digits each step. But requires f'(x) and fails if derivative is zero or starting point is bad.
Bisection: linear convergence - always works if f(a)·f(b) less than 0. Slower but guaranteed. Watch the green bracket shrink.

Root-Finding Algorithms - Interactive Visualization

Root finding solves f(x) = 0. Newton-Raphson converges quadratically (errors square each iteration) but can diverge with bad initialization. Bisection converges linearly but always works if you bracket the root. This visualization animates both simultaneously: Newton's tangent-line method vs bisection's bracket-halving, showing why practitioners often combine both strategies.

  • Watch Newton-Raphson draw tangent lines and jump to x-intercepts
  • Watch bisection halve the bracket each iteration
  • Choose from different functions to see convergence behavior
  • Adjust initial guess to see Newton-Raphson diverge on tricky functions
  • Foundation for optimization, inverse functions, and implicit solvers

Part of the EngineersOfAI Interactive 3D - free interactive visualizations covering every major concept in machine learning and AI engineering. Hover any element for a plain-English explanation. No code required.