A famous lower bound for the bilinear complexity of the multiplication in associative algebras is the Alder–Strassen bound. Algebras for which this bound is tight are called algebras of minimal rank. After 25 years of research, these algebras are now well understood. We here start the investigation of the algebras for which the Alder–Strassen bound is off by one. As a first result, we completel...

Shiu derives two formulas for calculating the probability of eventual ruin in a collective risk model. This paper implements one of the formulas by incorporating an algorithm to reduce round-off error due to convolution. It gives the results of the programs for some claim distributions. The usefulness of these two formulas is also discussed.

The composite step biconjugate gradient method (CSBCG) is a simple modification of the standard biconjugate gradient algorithm (BCG) which smooths the sometimes erratic convergence of BCG by computing only a subset of the iterates. We show that 2 × 2 composite steps can cure breakdowns in the biconjugate gradient method caused by (near) singularity of principal submatrices of the tridiagonal ma...

In this paper, we propose a combined form of the sumudu transform method with the homotopy perturbation method to solve nonlinear equations. This method is called the homotopy perturbation sumudu transform method (HPSTM). The nonlinear terms can be easily handled by the use of He’s polynomials. The proposed scheme finds the solution without any discretization or restrictive assumptions and avoi...

Compensated summation algorithms are designed to improve the accuracy of ill-conditioned sums. They are based on algorithms, such as FastTwoSum, which are proved to provide, with rounding to nearest, the sum of two floating-point numbers and the associated rounding error. Discrete stochastic arithmetic enables one to estimate rounding error propagation in numerical codes. It requires a random r...

A watermark hidden in an image is retrieved differently from the original watermark due to the frequently used rounding approach. The simple rounding will cause numerous errors in the embedded watermark especially when it is large. A novel technique based on genetic algorithms (GAs) is presented in this paper to correct the rounding errors. The fundamental is to adopt a fitness function for cho...

Abstract The Formal Linear Algebra Methods Environment (FLAME) project at UT-Austin pursues the mechanical derivation of algorithms for linear algebra operations. Rather than proving loop based algorithms correct a posteriori, a systematic methodology is employed that determines loop invariants from a mathematical specification of a given linear algebra operation. Algorithms and their correctne...

Exponential windowing is a widely used technique for suppressing the effects of old data as new data is added to a matrix. Specifically, given an n x p matrix Xn and a "forgetting factor" ß € (0, 1 ), one works with the matrix diag(/i'1_1, ß"~2, ... , \)Xn ■ In this paper we examine an updating algorithm for computing the QR factorization of diag(/j"-1, ß"~2 , ... , \)X„ and show that it is unc...

Division-free arithmetic computations can be boiled down to summation due to Dekker/Veltkamp’s algorithm of 1971. The known double-precision numerical algorithms for summation are highly effective but limited by rounding errors. Our new summation algorithms relaxe this limitation, although they still almost entirely amount to double-precision additions. The efficiency of the algorithms is confi...

imperialist competitive algorithm (ica) is a new socio-politically motivated global search strategy. the ica is applied to hybrid composite laminates to obtain minimum weight and cost. the approach which is chosen for conducting the multi-objective optimization was the weighted sum method (wsm). the hybrid composite laminates are made of glass/epoxy and carbon/epoxy to combine the lightness and...