Shoug Fawaz Alomran
Implementation, documentation and the digital report.
Octave · MkDocs · publishingMost nonlinear equations have no closed-form answer, so we iterate towards one. This project implements the Bisection, Newton–Raphson and Secant methods in GNU Octave and compares how fast, how accurately and how reliably each one converges on f(x) = x² − 4.
SA SH MA Three-person team · Math 221 Numerical AnalysisRoot-finding sits underneath engineering models, optimisation, finance and simulation. When an equation can't be solved exactly, a numerical method starts from a guess and improves it step by step until the change falls below a tolerance.
We studied the three classical approaches, a bracketing method and two open methods, and ran each on the same function with the same tolerance of 10⁻⁶.
All three reached the exact root x = 2. Newton–Raphson showed the quadratic convergence the theory predicts, Secant came close without needing a derivative, and Bisection remained the most dependable.
Each method trades reliability against speed differently, and that trade-off shows up in the iteration formula itself.
Halve the interval and keep the half where the sign changes. The Intermediate Value Theorem guarantees a root stays inside.
Follow the tangent line at the current guess down to the x-axis. Very fast near the root, but it needs the derivative.
Replace the derivative with the slope through the last two points. Almost Newton's speed, with no f′ required.
Absolute error |xn − 2| at every iteration, on a log scale, using the same starting points as the Octave code. On this scale, a steeper downward curve means faster convergence.
Bisection's single step is a special case: the midpoint of [1, 3] is exactly the root. In general it gains only one binary digit per step. Newton–Raphson roughly doubles the number of correct digits with every iteration.
See the full resultsMath 221 · Numerical Analysis · Dr. Nahid Fatima · submitted 16 November 2025.
Implementation, documentation and the digital report.
Octave · MkDocs · publishingHandwritten derivations and the theoretical explanation.
theory · derivationsHandwritten calculations and verification of numerical results.
calculation · verificationRead the complete report online or download the PDF, with the method derivations, Octave listings, figures and references.