Consistency
A method is consistent if its local error tends to zero as the step size shrinks.
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Numerical methods glossary
Terms that appear across several topics. Each entry links to the methods where the concept is used concretely.
Stability
Control over how small errors are amplified during the computation.
Convergence
The property that approximations approach the target value.
Order
The asymptotic rate at which the error decreases.
Conditioning
Sensitivity of the exact problem to perturbations in the data.
Residual
The defect left after substituting an approximation into the original equation.
Truncation error
Error introduced by cutting off an expansion or replacing a continuous object with a discrete one.
Absolute and relative error
The two basic measures of the distance between an exact value and its approximation.
Rounding error
Error caused by representing and operating on numbers with finite precision.
Machine epsilon
The rounding unit of the floating-point system: the maximum relative error of representing a number.
Significant figures
The digits of an approximation that carry reliable information about the exact value.
Taylor polynomial
The local polynomial approximation of a function built from its derivatives at a point.
Nodes
The points of the domain where the function is known or evaluated.
Divided differences
The recursive coefficients that build the interpolating polynomial in Newton form.
Spline
A piecewise-polynomial interpolant with smoothness conditions at the joints.
Runge phenomenon
Growing oscillations near the endpoints when interpolating with high degree at equispaced nodes.
Quadrature
Approximating a definite integral by a weighted sum of function values.
Degree of precision
The highest polynomial degree that a quadrature formula integrates exactly.
Richardson extrapolation
Combining two approximations with different steps to cancel the leading error term.
Step size ($h$)
The distance between consecutive points of the discretization.
Explicit and implicit methods
The distinction between computing the new step directly or by solving an equation that contains it.
Stiffness
A property of an ODE that forces explicit methods to take tiny steps for stability.
Region of absolute stability
The set of values for which a method damps the test equation.
Fixed point
A value the iteration function leaves unchanged: .
Stopping criterion
The rule that decides when an iteration has reached sufficient accuracy.
Diagonally dominant matrix
A matrix in which each diagonal element dominates the sum of the rest of its row.
Spectral radius
The largest absolute value of a matrix's eigenvalues; it decides the convergence of linear iterations.
Jacobian matrix
The matrix of partial derivatives that generalizes to systems of several variables.
Vector and matrix norms
The size measures used to quantify errors and convergence in several dimensions.