Bisection, one interval at a time
Watch the bracket contract, inspect every midpoint, and scrub backward and forward through the iterations.
Open the bisection laboratory →Method atlas
The algorithms behind the answers.
This is not a general encyclopedia of numerical analysis. It is a map of the methods that belong to the book and its companion software, with room for demonstrations where seeing the method work is better than another paragraph.
8 domains75 methods80 R functions
CMNA Laboratory
Some algorithms should be watched, not merely read.
The laboratory pages turn selected methods into experiments. Change a tolerance, choose a function, walk through individual iterations, and compare the path different algorithms take toward the same answer.
Three roads to a root
Put bisection, Newton, and secant on the same problem and compare convergence by residual.
Compare root-finding methods →Method finder
Start with the computational problem and walk toward the CMNA methods that fit the information you actually have.
Open the method finder →Simpson's rule, panel by panel
Give R your own integrand and watch the quadratic panels accumulate into a numerical integral.
Open the Simpson laboratory →Watch a system converge
Paste your own matrix and compare Jacobi with Gauss-Seidel as their residuals shrink—or fail to.
Open the iterative solver lab →Four slopes make a step
Supply an arbitrary R ODE and compare Euler, midpoint, and RK4 while playing through RK4's internal stages.
Open the IVP laboratory →Follow the gradient downhill
Define your own two-dimensional objective and gradient in R, then watch every descent step cross the sampled landscape.
Open the gradient descent lab →Visit the Numerical Zoo
Wilkinson, Hilbert, Runge, Himmelblau, cancellation, unstable stepping, and other examples kept around because they expose numerical assumptions.
Open the Numerical Zoo →Error and elementary computation
Small algorithms that expose evaluation order, accumulated error, and the difference between a formula and a numerically sensible implementation.
Naive polynomial evaluation
naivepolyCached polynomial evaluation
betterpolyHorner's method
hornerrhornerNaive summation
naivesumKahan summation
kahansumNaive division
naivedivLong division
longdivPrimality test
isPrimeNth root
nthrootQuadratic formula
quadraticquadratic2Fibonacci sequence
fibonacciWilkinson's polynomial
wilkinsonHimmelblau function
himmelblauLinear algebra
Row operations, matrix reduction, decompositions, and iterative methods implemented close enough to the mathematics to inspect.
Row replacement
replacerowRow scaling
scalerowRow swapping
swaprowsVector norm
vecnormDeterminant
detmatrixMatrix inverse
invmatrixRow-echelon form
refmatrixReduced row-echelon form
rrefmatrixSolve by row reduction
solvematrixCholesky decomposition
choleskymatrixLU decomposition
lumatrixConjugate gradient
Gauss-Seidel iterationLABcgmmatrixgaussseidelJacobi iterationLABjacobiTridiagonal matrix solver
tridiagmatrixInterpolation and extrapolation
Polynomial interpolation, splines, Bézier curves, multidimensional interpolators, and an image-resizing application.
Linear interpolation
Polynomial interpolationLABlinterppolyinterpPiecewise linear interpolationLABpwiselinterpCubic splineLABcubicsplineQuadratic Bézier curve
qbezierCubic Bézier curve
cbezierBilinear interpolation
bilinearNearest neighbor
nnNearest-neighbor image resizing
resizeImageNNBilinear image resizing
resizeImageBLDifferentiation
Finite-difference formulas for first and second derivatives.
Finite differenceLAB
findiffSymmetric differenceLABsymdiffRight differenceLABrdiffSecond derivative
findiff2Numerical integration
Newton-Cotes rules, Gaussian quadrature, adaptive integration, Romberg integration, Monte Carlo methods, and applications.
Midpoint rule
midptTrapezoid rule
Simpson's ruleLABtrapsimpSimpson's 3/8 rule
Gaussian integration driverLABsimp38gaussintGauss-Hermite quadrature
gauss.hermiteGauss-Laguerre quadrature
Gauss-Legendre quadratureLABgauss.laguerregauss.legendreRecursive adaptive integration
adaptintRomberg integration
Monte Carlo integration, 1DLABrombergmcintMonte Carlo integration, 2D
mcint2Shell method for volume
shellmethodDisc method for volume
discmethodGini coefficient
giniquintileRoot finding
Three classic approaches that make assumptions, robustness, and convergence behavior especially easy to compare.
Optimization
Continuous and discrete searches, from golden-section and gradient methods through simulated annealing.
Golden-section maximumLAB
goldsectmaxGolden-section minimumLABgoldsectminGradient descentLABgdgdlsgraddscGradient ascent
gradascHill climbing
Simulated annealingLABhillclimbingsaTraveling salesperson by simulated annealing
tspsaDifferential equations
Initial-value methods, systems, elementary PDE examples, and boundary-value applications.
Euler method
eulerMidpoint method for IVPs
Fourth-order Runge-KuttaLABmidptivprungekutta4Adams-Bashforth
adamsbashforthEuler method for systems
Heat equation, 1DLABeulersysheatWave equation, 1DLABwaveBoundary-value examples
bvpexamplebvpexample10