Numerical stability

One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation.In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution.One of the common tasks of numerical analysis is to try to select algorithms which are robust – that is to say, do not produce a wildly different result for a very small change in the input data.The following definitions of forward, backward, and mixed stability are often used in numerical linear algebra.In other contexts, for instance when solving differential equations, a different definition of numerical stability is used.An algorithm for solving a linear evolutionary partial differential equation is stable if the total variation of the numerical solution at a fixed time remains bounded as the step size goes to zero.The Lax equivalence theorem states that an algorithm converges if it is consistent and stable (in this sense).These results do not hold for nonlinear PDEs, where a general, consistent definition of stability is complicated by many properties absent in linear equations.
Diagram showing the forward error Δ y and the backward error Δ x , and their relation to the exact solution map f and the numerical solution f* .
Mixed stability combines the concepts of forward error and backward error.
mathematicalnumerical analysisnumerical algorithmsnumerical linear algebraeigenvaluesoppositefunctionround-off errortruncation errorcondition numberrelative errororders of magnitudeunit round-offdifferential equationsnumerical ordinary differential equationsdynamical systemsLyapunov stabilitystiff equationnumerical partial differential equationspartial differential equationtotal variationLax equivalence theoremnumerical diffusionVon Neumann stability analysisfinite difference schemeswell-posed problemBabylonian methodloss of significancecatastrophic cancellationAlgorithms for calculating varianceStability theoryChaos theoryPropagation of uncertaintyfixed point iterationNicholas J. HighamBibcodeLinear algebraOutlineGlossaryScalarVectorVector spaceScalar multiplicationVector projectionLinear spanLinear mapLinear projectionLinear independenceLinear combinationMultilinear mapChange of basisRow and column vectorsRow and column spacesKernelEigenvalues and eigenvectorsTransposeLinear equationsMatricesDecompositionInvertibleMultiplicationTransformationCramer's ruleGaussian eliminationProductive matrixBilinearOrthogonalityDot productHadamard productInner product spaceOuter productKronecker productGram–Schmidt processMultilinear algebraDeterminantCross productTriple productSeven-dimensional cross productGeometric algebraExterior algebraBivectorMultivectorTensorOutermorphismQuotientSubspaceTensor productNumericalFloating-pointBasic Linear Algebra SubprogramsSparse matrixComparison of linear algebra libraries