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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.
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite. The conjugate gradient method is often implemented as an iterative algorithm , applicable to sparse systems that are too large to be handled by a direct ...
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 ...
Some algorithms have a property called backward stability; in general, a backward stable algorithm can be expected to accurately solve well-conditioned problems. Numerical analysis textbooks give formulas for the condition numbers of problems and identify known backward stable algorithms.
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.
(Figure 2) Illustration of numerical integration for the equation ′ =, = Blue is the Euler method; green, the midpoint method; red, the exact solution, =. The step size is =
Numerical analysis is not only the design of numerical methods, but also their analysis. Three central concepts in this analysis are: convergence: whether the method approximates the solution, order: how well it approximates the solution, and; stability: whether errors are damped out. [22]
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.