A representation of intervals is proposed that, instead of both end points, uses the low point and the width of the interval. This proposed representation is more efficient since both end points have many corresponding bits that are equal. Consequently, the width of the interval can be represented with a smaller number of bits than an end point, resulting in a better utilization of the number o...

In Group Synchronization, one attempts to find a collection of unknown group elements from noisy measurements of their pairwise differences. Several important problems in vision and data analysis reduce to group synchronization over various compact groups. Spectral Group Synchronization is a commonly used, robust algorithm for solving group synchronization problems, which relies on diagonalizat...

Finite diierence methods, like the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial diierential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in computer approximations, are strongly ampliied i...

In various applications, computers are required to compute approximations to univariate elementary and special functions such as exp and arctan to modest accuracy. This paper proposes a new heuristic for automating the design of such implementations. This heuristic takes a certain restricted specification of program structure and the desired error properties as input and takes explicit account ...

We suggest and analyze a technique by combining the variational iteration method and the homotopy perturbation method. This method is called the variational homotopy perturbation method VHPM . We use this method for solving higher dimensional initial boundary value problemswith variable coefficients. The developed algorithm is quite efficient and is practically well suited for use in these prob...

An iterative algorithm is constructed to solve the linear matrix equation pair AXB E, CXD F over generalized reflexive matrix X. When the matrix equation pair AXB E, CXD F is consistent over generalized reflexive matrix X, for any generalized reflexive initial iterative matrix X1, the generalized reflexive solution can be obtained by the iterative algorithm within finite iterative steps in the ...

We propose an algorithm for solving two polynomial equations in two variables. Our algorithm is based on the Macaulay resultant approach, combined with new techniques including randomization to make the algorithm accurate in the presence of roundoff error. The ultimate computation is the solution of a generalized eigenvalue problem via the QZ method. We analyze the error due to roundoff of the ...

A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalus 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 s...

Interval arithmetic was introduced by Ramon Moore [Moo66] in the 1960s as an approach to bound rounding errors in mathematical computation. The theory of interval analysis emerged considering the computation of both the exact solution and the error term as a single entity, i.e. the interval. Though a simple idea, it is a very powerful technique with numerous applications in mathematics, compute...

Scientific computation has unavoidable approximations built into its very fabric. One important source of error that is difficult to detect and control is round-off error propagation which originates from the use of finite precision arithmetic. We propose that there is a need to perform regular numerical ‘health checks’ on scientific codes in order to detect the cancerous effect of round-off er...