accuracy and stability of numerical algorithms

By Mei An. 1991

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Citations in the corpus

Bayesian Solution Uncertainty Quantification for Differential Equations (2013, methodology): “tem of diﬀerential equations yields a deterministic approximation, whose properties depend on the numerical method employed and the speciﬁed error tolerance and implied discretisation (Butcher, 2008; Higham, 1996). Local and global errorsare related to the asymptotic behaviour of the deterministic a”
A Theoretical Foundation for the Nelson and Siegel Class of Yield Curve Models (2012, unknown): “From Higam (1996) p. 223 a non-symmetric matrix may be written as the sum of a symmetric matrix A S and an antisymmetric matrix A K , and A will be positive de…nite if A S is positive de…nite.”
Efficient and Modular Implicit Differentiation (2021, background): “B is bounded and Lipschitz: Under these conditions, when (cid:107) ˆ x − x (cid:63) ( θ ) (cid:107) ≤ ε , we have This result is inspired by [52, Theorem 7.2], that is concerned with the stability of solutions to inverse problems.”
On the Approximation Order and Numerical Stability of Local Lagrange Interpolation by Polyharmonic Splines (2003, background): “For a more general discussion on the relevant principles and concepts from error analysis, especially the condition number of a given problem versus the stability of a numerical algorithm, we recommend the textbook [13].”
MINRES-QLP: A Krylov Subspace Method for Indefinite or Singular Symmetric Systems (2010, methodology): “[25] Matlab function NORMEST(A), which is based on the power method 5.”
From density-matrix renormalization group to matrix product states (2007, methodology): “Arithmetic manipulations on such quantities are a recipe for catastrophic cancellation [27] resulting in loss of precision.”
SuperLU Users'' Guide (1997, methodology): “The algorithm used to compute FERR is an approximation; see [2, 18] for a discussion.”
On significance of the least significant bits for differential privacy (2012, methodology): “This is a manifestation of the accumulated error phenomenon, which can be counteracted by application of the Kahan summation algo-rithm [22, 20].”
Convergence Speed in Distributed Consensus and Averaging (2009, unknown): “There are two possible drawbacks of the scheme we have just described: (a) If some of the nπ i are very small, then some of the initial x i (0) will be very large, which can lead to numerical diﬃculties [16].”
Numerical Reliability for Mixed Methods Applied to Flow Problems in Porous Media (2002, methodology): “The matrix-inversion function of Maple is based on Cramer’s rule which is well known to be numerically unstable [12].”
On Multivariate Interpolation (2006, background): “In addition, intriguing explicit formulae that connect the classical finite difference interpolation coefficients for univariate curves with multivariate interpolation coefficients for higher dimensional submanifolds are established.”
LSMR: An Iterative Algorithm for Sparse Least-Squares Problems (2010, unknown): “…x measures the perturbation to A that would make x an exact LS solution: It is known to be the smallest singular value of a certain m × ( n + m ) matrix C ; see Wald´en et al. [26] and Higham [10, pp. 392–393]: Since it is generally too expensive to evaluate µ ( x ), we need to ﬁnd approximations.”
Precimonious: Tuning assistant for floating-point precision (2013, background): “Compensated summation [17] fends oﬀ that corruption at an added cost negligible compared with the cost of computing term(k) .”