Bisection, Newton-Raphson and secant methods solve f(x)=0 iteratively.
Bisection halves an interval containing a sign change (slow but sure). Newton-Raphson x_{n+1}=xₙ-f/f' converges quadratically near a simple root. The secant method avoids derivatives. Choose by robustness versus speed.
√2 by Newton on x²-2: 1.5, 1.4167, 1.41422 in three steps.
Root-finding methods solve equations with no closed form, which is common in design, circuit and fluid problems.
Bisection, Newton-Raphson and secant methods solve f(x)=0 iteratively.
Root-finding methods solve equations with no closed form, which is common in design, circuit and fluid problems.
It can diverge or jump to a wrong root if the starting guess is poor or the derivative is near zero. Bracketing methods are safer.
If the function changes sign over an interval, bisection always converges to a root. It is slower but very reliable.
Typical applications include finding the operating point where a pump curve meets a system curve, solving the Colebrook equation for friction factor and finding the current in a diode circuit with a nonlinear model.
Study "Numerical error and conditioning" first (topic 90 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Bracketing methods always converge. Newton needs a good start. Stop on tolerance or iteration limit.
√2 by Newton on x²-2: 1.5, 1.4167, 1.41422 in three steps.
The next topic is "Interpolation", which builds directly on this one.
It is classed as Advanced in Engineering Mathematics (topic 91 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.
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