Newton-Raphson Method
Find roots of f(x)=0 by Newton-Raphson with numerical derivative.
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Newton–Raphson method for finding roots
Newton–Raphson solves f(x) = 0 by iterating x_{n+1} = x_n − f(x_n)/f'(x_n). Each step replaces f by its tangent at x_n and jumps to where that tangent crosses zero. Convergence is quadratic: the number of correct digits roughly doubles per iteration — extremely fast when it works. Requirements: f must be differentiable and f' should not be near zero at the iterate; a bad initial guess can diverge or cycle. Example — compute √2 by solving x²−2=0, giving x_{n+1} = (x_n + 2/x_n)/2. From x₀=1: 1.5 → 1.4167 → 1.41422 → 1.41421356… (8 digits in 4 iterations). When f' is expensive, quasi-Newton methods like BFGS approximate it from past steps.
Applications and context
Newton–Raphson powers spreadsheet solvers (Excel Solver and Goal Seek), CAD/CAM (curve intersections, IK in robotics), IRR in finance, machine learning optimization (Newton steps and quasi-Newton variants), and computer graphics (ray marching, distance fields). It's the default root-finder when you can supply a derivative cheaply.
FAQ
What if Newton diverges? Bracket the root first with bisection, then switch to Newton once you're close — the basis of Brent's method.
How do I stop? Standard tests: |x_{n+1} − x_n| < tol, |f(x_n)| < tol, or a max-iterations cap (usually 50 is plenty thanks to quadratic convergence).
What if I don't have f'? Use the secant method (estimates f' from the last two iterates) — superlinear convergence (~1.618) but no derivative needed; or autodiff if you have source code for f.
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Settling Velocity (Newton)
Calculate the settling (terminal) velocity of a particle in turbulent regime by Newton's law, v_t = √(4·g·d·(s − 1) ÷ (3·C_d)), from the particle diameter d (m), the solids relative density s = ρ_s/ρ_w and the drag coefficient C_d (dimensionless, ≈ 0.44 for spheres in turbulent regime). While Stokes' Law holds for SMALL particles (laminar regime, particle Reynolds < 1), Newton's law holds for LARGE, dense particles — gravel, crushed stone, coarse sand — that sink fast, generating TURBULENT flow around them (particle Reynolds > ~1000). In this regime, drag is no longer proportional to velocity (Stokes) but to its SQUARE, and the terminal velocity grows with the SQUARE ROOT of diameter (not the square, as in Stokes) — large particles sink fast, but the size dependence is milder. The drag coefficient C_d ≈ 0.44 is roughly constant in this range (the 'Newton region' of the sphere drag curve). This calculation is fundamental in designing coarse-particle classifiers and separators, sizing settling basins for coarse solids, coarse-sediment transport and hydraulic transport of gravel and granular ore. For the intermediate range between Stokes and Newton, transition correlations are used. Enter the diameter, the relative density and the drag coefficient.
Particle Reynolds Number
Calculate the particle Reynolds number in settling, Re_p = (ρ_w·v_s·d) ÷ μ, from the fluid density ρ_w (kg/m³), the settling velocity v_s (m/s), the particle diameter d (m) and the dynamic viscosity μ (Pa·s). The particle Reynolds number characterizes the flow regime around a particle settling (or being transported) in a fluid, comparing inertial and viscous forces. Its value sets WHICH settling-velocity formula is valid: for Re_p < ~1, the flow around the particle is LAMINAR and Stokes' Law holds (drag proportional to velocity); for Re_p > ~1000, the flow is TURBULENT and Newton's law holds (drag proportional to velocity squared); in the intermediate range, transition correlations are used (such as Allen's or drag-coefficient expressions vs Re_p). So when computing a settling velocity by Stokes' Law, it is ESSENTIAL to verify afterwards that Re_p < 1 — if not, the Stokes result is wrong and the correct regime's formula must be used. The particle Reynolds number is thus the 'checker' that validates the settling calculation, and it is central in designing settling tanks, classifying particles and hydraulic solids transport. Enter the fluid density, the settling velocity, the particle diameter and the viscosity.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.