This paper describes modifications to many of the standard algorithms used in computing eigenvalues and eigenvectors of matrices. These modifications can dramatically increase the performance of the underlying software on high-performance computers without resorting to assembler language, without significantly influencing the floating-point operation count, and without affecting the roundoff-er...

Rounded Arithmetic is the traditional way of presenting previous year accounts. The method of showing rounded sums of vectors in which the arithmetic still “adds up” and the errors in the display of the components are bounded by twice the maximum rounding error have been well known for at least 30 years in spite of there being little or nothing in the literature and no attempt to implement this...

We devise an analytically simple as well as invertible approximate expression, which describes the relation between the minimum distance of a binary code and the corresponding maximum attainable code-rate. For example, for a rate-(1/4), length-256 binary code the best known bounds limit the attainable minimum distance to 65 ≤ d̃(n = 256, k = 64) ≤ 90, while our solution yields d(n = 256, k = 64)...

In this paper we approach the problems of rounding errors in geometric algorithms. Particularly, we are interested in geometric algorithms used by TerraLib – a spatial component library used to construct small GIS. We describe some techniques that were used to (re)implement and adapt some TerraLib functions to produce robust (roundoff error free) functions.

When we estimate the Mahalanobis distance of one point from a cluster of points, there are a number of potential sources of error. This is particularly so if the cluster is strongly correlated, which implies that its covariance matrix is ill-conditioned. Here we analyse the magnitude of the errors introduced by round-off, inversion of nearsingular matrices, and empirical estimation of mean and ...

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations AXB − CYD,EXF − GYH M,N , which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices X and Y . When the matrix equations are consistent, for any initial generalized reflexive matrix pair X1, Y1 , the generalized reflexive solutions can be obtained b...

An important task in solving second order linear ordinary differential equations by the finite difference is to choose a suitable stepsize h. In this paper, by using the stochastic arithmetic, the CESTAC method and the CADNA library we present a procedure to estimate the optimal stepsize hopt, the stepsize which minimizes the global error consisting of truncation and round-off error. Keywords—o...

To study the heat or diffusion equation it is often used the Crank-Nicolson method which is unconditionally stable and has order of convergence O(k + h ), where k and h are mesh con2 2 stants. Unfortunately, using this method in conventional floatingpoint arithmetic we get solutions including not only the method error, but also representation and rounding error, Therefore, we propose an interva...

Buckling and certain contact situations cause scattering results of numerical crash simulation: For a BMW model differences between the position of a node in two simulation runs of up to 10 cm were observed, just as a result of round-off differences in the case of parallel computing. An engineer has to measure this scatter, to check whether important parts of the car show such indeterministic b...

A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalues and both right and left eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Restarting also makes it possible to deal with roundoff error in new ways. We give a scheme for avoiding near-breakdown and discuss maintaining biorthogonality. A system of linear equations can be solved ...