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In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation.
The condition number is a property of the problem. Paired with the problem are any number of algorithms that can be used to solve the problem, that is, to calculate the solution. Some algorithms have a property called backward stability; in general, a backward stable algorithm can be expected to accurately solve well-conditioned problems ...
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, [5] as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.
The Crank–Nicolson stencil for a 1D problem. The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time.For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method [citation needed] —the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator.
Numerical stability is the central criterion for judging the usefulness of implementing an algorithm on a computer with roundoff. For the Lanczos algorithm, it can be proved that with exact arithmetic , the set of vectors v 1 , v 2 , ⋯ , v m + 1 {\displaystyle v_{1},v_{2},\cdots ,v_{m+1}} constructs an orthonormal basis, and the eigenvalues ...
In computer science, a stable sorting algorithm preserves the order of records with equal keys. In numerical analysis, a numerically stable algorithm avoids magnifying small errors. An algorithm is stable if the result produced is relatively insensitive to perturbations during computation.
The numerical stability is reduced compared to the naive algorithm, [5] but it is faster in cases where n > 100 or so [6] and appears in several libraries, such as BLAS. [7] Fast matrix multiplication algorithms cannot achieve component-wise stability, but some can be shown to exhibit norm-wise stability. [8]
Cannon's algorithm — a distributed algorithm, especially suitable for processors laid out in a 2d grid; Freivalds' algorithm — a randomized algorithm for checking the result of a multiplication; Matrix decompositions: LU decomposition — lower triangular times upper triangular; QR decomposition — orthogonal matrix times triangular matrix