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Numerical method comparisons

These tables do not replace the guides: they are a quick view for choosing a method by assumptions, cost and expected behaviour.

Newton vs secant vs bisection

Three scalar root-finding strategies: global safety, local speed and per-iteration cost.

Method	Assumptions	Order	Cost	When to use it
Bisection	Continuity and sign change on $[a,b]$	Linear; the interval is halved	One new evaluation of $f$ per iteration	To bracket the root and get a reliable initial estimate
Newton-Raphson	Simple root, available $f'$ and nearby starting point	Local quadratic	$f$ and $f'$ per iteration	When the seed is good and the derivative is cheap
Secant	Two initial estimates and sufficiently regular $f$	Superlinear, $p\approx1.618$	One new evaluation of $f$ per iteration	When $f'$ is unavailable or undesirable

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Euler vs Heun vs RK4

One-step IVP methods: accuracy, evaluations of $f$ and practical use.

Method	Idea	Global error	Evaluations	When to use it
Explicit Euler	Advances with the slope at the start of the interval	$\mathcal{O}(h)$	1 per step	For quick estimates or as a predictor
Heun	Averages Euler's slope with a corrected slope	$\mathcal{O}(h^2)$	2 per step	When a cheap improvement over Euler is enough
RK4	Combines four slopes with weights $1,2,2,1$	$\mathcal{O}(h^4)$	4 per step	For high accuracy without solving implicit equations

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Jacobi vs Gauss-Seidel vs SOR

Linear iterative methods viewed as splittings $A=M-N$ and through their iteration matrix.

Method	Splitting	Update	Advantage	Risk
Jacobi	$M=D$	Uses only values from the previous iteration	Simple and parallelizable	May converge slowly
Gauss-Seidel	$M=D-L$	Reuses new values as soon as they are computed	Often improves Jacobi without an extra parameter	Depends on equation ordering
SOR	$M=\frac{1}{\omega}D-L$	Relaxes Gauss-Seidel with a weight $\omega$	Can accelerate strongly with a good $\omega$	A poor $\omega$ can slow or break convergence

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Lagrange vs Newton vs Hermite vs splines

Four ways to interpolate the same data: what information they use, what they cost to build and when each one pays off.

Interpolant	Data it uses	Construction	Strength	Limitation
Lagrange	Values $f(x_i)$ at $n+1$ nodes	Explicit basis $\ell_i(x)$ , no systems to solve	Closed form, ideal for theory and quadrature derivations	Adding a node forces rebuilding the whole basis
Newton	Values $f(x_i)$ at $n+1$ nodes	Divided-difference table, nested form	Adding a node only costs one more diagonal	Same polynomial as Lagrange: inherits its high-degree limits
Hermite	Values $f(x_i)$ and derivatives $f'(x_i)$	Divided differences with repeated nodes	Doubles the information per node: degree $2n+1$ and higher accuracy	Requires derivatives, which the data do not always provide
Cubic spline	Values $f(x_i)$ at all nodes	Tridiagonal system for the moments, one cubic per piece	Smooth ( $C^2$ ) and free of the Runge phenomenon with many nodes	No single global polynomial, and a system must be solved

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Trapezoid vs Simpson vs midpoint vs Gauss

The everyday quadratures compared by nodes, composite error order and the situation where each wins.

Formula	Nodes	Composite error	Degree of precision	When to use it
Composite trapezoid	Equispaced, closed (uses the endpoints)	$\mathcal{O}(h^2)$	1	Tabulated data; simple and robust
Composite Simpson	Equispaced, closed, even number of subintervals	$\mathcal{O}(h^4)$	3	Smooth tabulated integrand: two extra orders at the same cost per node
Composite midpoint	Equispaced, open (avoids the endpoints)	$\mathcal{O}(h^2)$	1	When the integrand cannot be evaluated at the endpoints
Gauss-Legendre	Roots of Legendre polynomials, not equispaced	Degree $2n-1$ with $n$ nodes	$2n-1$	When you can evaluate $f$ anywhere and evaluations are expensive

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AB2 vs AM2 vs predictor-corrector

The order-2 multistep methods head to head: cost per step, error constant and the price of implicit stability.

Method	Type	Local error	Cost per step	Notes
AB2	Explicit, two-step	$\frac{5}{12}h^3y'''(\xi)$	1 new evaluation of $f$	Needs a startup value (Heun or RK4) and a uniform mesh
AM2 (implicit trapezoid)	Implicit, one new step	$-\frac{1}{12}h^3y'''(\xi)$	Solve one equation in $y_{k+1}$	Error constant 5 times smaller than AB2 and far better stability
Predictor-corrector AB2+AM2	Explicit in practice	Of the corrector's order	2 evaluations of $f$	AM2's accuracy without solving the implicit equation; the predictor-corrector gap estimates the error

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