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Stability is sometimes achieved by including numerical diffusion. Numerical diffusion is a mathematical term which ensures that roundoff and other errors in the calculation get spread out and do not add up to cause the calculation to "blow up". Von Neumann stability analysis is a commonly used procedure for the stability analysis of finite ...
In certain cases, von Neumann stability is necessary and sufficient for stability in the sense of Lax–Richtmyer (as used in the Lax equivalence theorem): The PDE and the finite difference scheme models are linear; the PDE is constant-coefficient with periodic boundary conditions and has only two independent variables; and the scheme uses no ...
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 numerical linear algebra, the biconjugate gradient stabilized method, often abbreviated as BiCGSTAB, is an iterative method developed by H. A. van der Vorst for the numerical solution of nonsymmetric linear systems.
[1] [2] Variants of the latter have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems). [3] The algorithms can also be used to find the determinant of a Toeplitz matrix in O ( n 2 ) {\displaystyle O(n^{2})} time.
The reduction in the number of arithmetic operations however comes at the price of a somewhat reduced numerical stability, [9] and the algorithm also requires significantly more memory compared to the naive algorithm. Both initial matrices must have their dimensions expanded to the next power of 2, which results in storing up to four times as ...
Zero-stability, also known as D-stability in honor of Germund Dahlquist, [1] refers to the stability of a numerical scheme applied to the simple initial value ...
In numerical linear algebra, the Bartels–Stewart algorithm is used to numerically solve the Sylvester matrix equation =.Developed by R.H. Bartels and G.W. Stewart in 1971, [1] it was the first numerically stable method that could be systematically applied to solve such equations.