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Numerical Methods

Root finding, approximation, numerical integration and error.

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Core Root finding: bisection and Newton's method Finding where f(x) = 0 to any accuracy, and how fast each method gets there. newton's method on x^2 - 2 from 1 Core Numerical integration: trapezoid and Simpson Approximating an integral from sampled values, with error bounds. integrate e^(-x^2) dx from 0 to 1 Core Interpolation and Taylor approximation Fitting polynomials through points, and approximating functions near a centre. taylor series of e^x Core Floating point and error Why 0.1 + 0.2 ≠ 0.3 on a computer, and how error propagates. 0.1 + 0.2

Most equations have no closed form; numerical methods find answers to as many decimals as you like and, crucially, tell you how wrong they might be. Every iteration here is shown as a step.

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Finding roots

Numerical analysisNewton's methodBisection methodSecant methodFixed-point iterationRate of convergenceRoot-finding algorithm

Integration and interpolation

Simpson's ruleTrapezoidal ruleGaussian quadratureInterpolationPolynomial interpolationSpline interpolationLeast squaresCurve fitting

Linear systems and ODEs

Gaussian eliminationLU decompositionQR decompositionCondition numberIterative methodEuler methodRunge–Kutta methodsFinite difference methodFinite element method

Error

Floating-point arithmeticRound-off errorTruncation errorNumerical stabilityMachine epsilonCatastrophic cancellation

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Why not just solve exactly?

Most equations have no closed-form solution at all, and many that do are unusable in practice. A numerical method delivers as many correct digits as you need, and a good one tells you how many that is.

Why can Newton's method fail?

If it starts where the tangent is nearly flat it shoots far away; near a repeated root it slows to a crawl; and with several roots it may land on the wrong one. A bracketing method like bisection is slower but cannot fail.

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