EC-4.7 Numerical Methods: Roots, Interpolation, and Stepping an Equation Forward
You can find a root numerically and say when the method will fail, integrate and interpolate from sampled values, and step a differential equation forward while controlling the error.
Before:EC-3. Differential EquationsUnlocks:EC-1. Circuit AnalysisEC-7. Signals and Systems
Most equations that describe real hardware have no closed-form solution, so a numerical answer is the normal case rather than a fallback. What separates a useful numerical answer from a confident wrong one is knowing where the error comes from: truncation from the method, rounding from the arithmetic, and amplification from a badly conditioned problem. This topic is deliberately about that judgement rather than about implementing the algorithms, which any library already does better.
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Why a numerical answer at all, and what error means here
Truncation error comes from the method stopping early, rounding error from finite arithmetic, and the two behave oppositely as the step size shrinks. Knowing which one dominates is the difference between improving an answer and ruining it.
NPTEL: Numerical Analysis · CourseRoot finding: bisection and Newton, and when each one fails
Bisection always converges if the root is bracketed but converges slowly, while Newton converges fast and can diverge entirely from a poor start or a flat derivative. Choosing between them is a judgement about what is known in advance.
NPTEL: Numerical Analysis · CourseSolving a linear system numerically, and what conditioning means
Elimination with partial pivoting solves the system, but a poorly conditioned matrix will amplify small input errors into large output ones no matter how the solving is done. The condition number is the warning, and it comes from the linear algebra module.
Interpolation, and why a high-order polynomial misbehaves
Fitting a single polynomial through many points produces wild oscillation near the ends, which is why piecewise fits are used instead in practice. Meeting that failure once prevents reaching for the obvious method later.
Numerical integration: the trapezium and Simpson rules, and their error orders
Both approximate the area under sampled values, with Simpson gaining accuracy by fitting a curve rather than a straight line between points. The error orders say how much halving the spacing actually buys.
Stepping a differential equation: Euler, Runge-Kutta, and choosing the step
Euler takes the derivative at the current point and steps blindly, while Runge-Kutta samples several points within the step and is far more accurate for the same size. Too large a step gives a wrong answer and too small a step accumulates rounding error.
NPTEL: Numerical Analysis · Course
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