This paper contains the rounding error analysis for the Chebyshev method for the solution of large linear systems Ax+g = 0 where A = A is positive definite. We prove that the Chebyshev method in floating point arithmetic is numerically stable, which means that the computed sequence {x^} approximates the solution a such that lim|k is of order C||a||.||A~ \\.\\y\\ where £ is the relative computer...

We develop techniques for computing the (un)stable manifold at a hyperbolic equilibrium of an analytic vector field. Our approach is based on the so-called parametrization method for invariant manifolds. A feature of this approach is that it leads to a-posteriori analysis of truncation errors which, when combined with careful management of round off errors, yields a mathematically rigorous encl...

For the familiar Fibonacci sequence (defined by f1 = f2 = 1, and fn = fn−1 + fn−2 for n > 2), fn increases exponentially with n at a rate given by the golden ratio (1 + √ 5)/2 = 1.61803398 . . . . But for a simple modification with both additions and subtractions — the random Fibonacci sequences defined by t1 = t2 = 1, and for n > 2, tn = ±tn−1 ± tn−2, where each ± sign is independent and eithe...

This paper describes three numerical methods to collapse a formal product of p pairs of matrices P = Q p?1 k=0 E ?1 k A k down to the product of a single pair ^ E ?1 ^ A. In the setting of linear relations, the product formally extends to the case in which some of the E k 's are singular and it is impossible to explicitly form P as a single matrix. The methods diier in op count, work space, and...

A rounding error analysis is presented for a divide-and-conquer algorithm to solve linear systems with block Hessenberg matrices. Conditions are derived under which the algorithm computes a stable solution. The algorithm is shown to be stable for block diagonally dominant matrices and for M-matrices.

Linear prediction (LP) is the most prevalent method for spectral modelling of speech, and line spectrum pair (LSP) decomposition is the standard method to robustly represent the coefficients of LP models. Specifically, the angles of LSP polynomial roots, i.e. line spectrum frequencies (LSFs), encode exactly the same information as LP coefficients. The conversion of LP coefficients to LSFs and b...

The present investigation is concerned with estimating the rounding error in numerical solution of stochastic differential equations. A statistical rounding error analysis of Euler’s method for stochastic differential equations is performed. In particular, numerical evaluation of the quantities EjXðtnÞ2 Ŷnj and E1⁄2FðŶnÞ2 FðXðtnÞÞ is investigated, where X(tn) is the exact solution at the nth ti...

For the familiar Fibonacci sequence | deened by f 1 = f 2 = 1, and f n = f n?1 + f n?2 for n > 2 | f n increases exponentially with n at a rate given by the golden ratio (1 + p 5)=2 = 1:61803398 : : :. But for a simple modiication with both additions and subtractions | the random Fibonacci sequences deened by t 1 = t 2 = 1, and for n > 2, t n = t n?1 t n?2 , where each sign is independent and e...

This paper proposes a memristor SPICE model using the Tukey window (or tapered cosine window) function. Compared with the previously proposed models based on the boundary function, the proposed model is resistant to numerical errors. From the SPICE result of a relaxation oscillator, we observe that the proposed model can be effectively used to minimise the effect of round-off errors.

This paper reports on early-stage research into using dynamic scale-space representation of image point reprojection error data obtained during calibration of a single camera. In particular, we employ time-dependent simulation of the heat equation, diffusing the point reprojection errors over the entire image plane. Initial experiments show the expected effect of an originally large number of p...