A new algorithm for calculating two-dimensional differential transform of nonlinear functions

A new algorithm for calculating the two-dimensional differential transform of nonlinear functions is developed in this paper. This new technique is illustrated by studying suitable forms of nonlinearity. Three strongly nonlinear partial differential equations are then solved by differential transform method to demonstrate the validity and applicability of the proposed algorithm. The present framework offers a computationally easier approach to compute the transformed function for all forms of nonlinearity. This gives the technique much wider applicability.     

An algorithm for calculating continued fractions

A new algorithm is described in the paper for calculating continued fractions. The conditions are given under which this algorithm is faster than the hitherto fastest algorithm for handling this problem. Also the interrelation between the suggested algorithm and that for calculating a tridiagonal system of linear equations is investigated.     

A new derivation of eigenvalue inequalities for the multinomial distribution

We consider the covariance matrix of the multinomial distribution. We suggest a new derivation of inequalities for the eigenvalues of this matrix using a classical result on the product of two positive semi-definite matrices.     

A new derivation of Painlevé hierarchies

We give a new derivation of two Painlevé hierarchies. This is done by extending the accelerating-wave reductions of the Korteweg-de Vries and dispersive water wave equations to their respective hierarchies. We also consider the extension of this reduction of Burgers equation to the Burgers hierarchy.     

A new derivation of the counting formula for Young tableaux

A new very short proof of the counting formula for Young tableaux is given. Its equivalence with the hook formula is easy to establish.     

A new algorithm for optimization

A procedure is presented for solving problems of a specified kind.     

On an algorithm for calculating diffraction integrals

An algorithm for calculating integrals of rapidly oscillating functions given on a smooth two-dimensional surface is proposed. The surface is approximated by a collection of flat triangles with the values of the integrand known at their vertices. These values are used as reference ones to extend the function to other points of a triangle. The integral of the extended function over the surface of a triangle is calculated exactly. The desired value of the full diffraction integral is determined as the sum of the integrals calculated over the surfaces of all triangles. The resulting formulas for integral calculation involve singularities (indeterminate forms). Much attention is given to representations of these formulas in such a way that the indeterminate forms are automatically evaluated. Numerical results are presented.     

An algorithm for calculating logarithms through displacement

A procedure is proposed for the factorial expansion of special real numbers, each of which in turn allows a similar expansion. Problems of executing the program for calculating a natural logarithm are discussed. Approaches to generalizing the method are considered.     

An algorithm for calculating generalized continued fractions

In this paper the generalization of a continued fraction in the sense of the Jacobi-Perron algorithm (called an n-fraction) is considered.Apart from the known algorithms to calculate an n-fraction a new one is derived and the algorithms are compared with respect to the number of operations required and the time to execute these operations.     

A harmonic restarted Arnoldi algorithm for calculating eigenvalues and determining multiplicity

A restarted Arnoldi algorithm is given that computes eigenvalues and eigenvectors. It is related to implicitly restarted Arnoldi, but has a simpler restarting approach. Harmonic and regular Rayleigh-Ritz versions are possible.For multiple eigenvalues, an approach is proposed that first computes eigenvalues with the new harmonic restarted Arnoldi algorithm, then uses random restarts to determine multiplicity. This avoids the need for a block method or for relying on roundoff error to produce the multiple copies.     

A new approach for calculating the real stability radius

We present a new fast algorithm to compute the real stability radius with respect to the open left half plane which is an important problem in many engineering applications. The method is based on a well-known formula for the real stability radius and the correspondence of singular values of a transfer function to pure imaginary eigenvalues of a three-parameter Hamiltonian matrix eigenvalue problem. We then apply the implicit determinant method, used previously by the authors to compute the complex stability radius, to find the critical point corresponding to the desired singular value. This corresponds to a two-dimensional Jordan block for a pure imaginary eigenvalue in the parameter dependent Hamiltonian matrix. Numerical results showing quadratic convergence of the algorithm are given.     

A quasi-linear algorithm for calculating the infimal convolution of convex quadratic functions

In this paper we present an algorithm of quasi-linear complexity to exactly calculate the infimal convolution of convex quadratic functions. The algorithm exactly and simultaneously solves a separable uniparametric family of quadratic programming problems resulting from varying the equality constraint.     

An optimal parallel connectivity algorithm

A synchronized parallel algorithm of depth O(n²/p) for p (≤n²/log²n) processors is given for the problem of computing connected components of an undirected graph. The speed-up of this algorithm is optimal in the sense that the depth of the algorithm is of the order of the running time of the fastest known sequential algorithm over the number of processors used.     

A new algorithm for generating derangements

A new algorithm for generating derangements based on a well known permutation generation method is presented and analysed. The algorithm is shown to be superior in storage and time requirements to the existing method.This work was supported by the Natural Sciences and Engineering Research Council of Canada under Grant NSERC-A3336.     

A new algorithm for digraph isomorphism

A new and efficient procedure for testing a pair of digraphs for isomorphism is developed. It is based on conducting a depth-first search on one of the digraphs followed by a systematic matching of edges using backtracking with very effective pruning. It is proved that for digraphs (ofn vertices) the expected time complexity of this procedure isO(n ^logn ). This theoretical result is verified empirically on more than 300 large random digraphs. This procedure is shown to be more efficient than any of the existing general isomorphism procedures.