Markus A. Hitz

Efficient algorithms for computing the nearest polynomial with a real root and related problems

We present three new algorithms in the general area of input-sensitivity analysis: a problem formulation, possibly with floating point coefficients, lacks an expected property because the inputs are slightly perturbed. A task is to efficiently compute the nearest problem that has the desired property. Nearness to the desired property can lead to problems for numerical algorithms: for example, an almost singular linear system cannot be solved by classical numerical techniques. In such case one can approach the problem of locating the nearest problem with the desired property by symbolic computation techniques, for instance, by exact arithmetic. Our three properties are:

Efficient algorithms for computing the nearest polynomial with constrained roots

The location of polynomial roots is sensitive to perturbations of the coefficients. Continuous changes of the coefficients of a polynomial move the roots continuously. We consider the problem of finding the minimal perturbations to the coefficients to move one or several roots to given loci. We measure minimality in the Euclidean distance to the coefficient vector, as well as the maximal coefficient-wise change in absolute value (infinity norm), and in the Manhattan norm (

Integer division in residue number systems

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Process scheduling in DSC and the large sparse linear systems challenge

-norm). In the Euclidean norm the computational task reduces to a least squares problem, in the infinity norm and the

On computing nearest singular hankel matrices

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Process Scheduling in DSC and the Large Sparse Linear Systems Challenge

-norm it can be formulated as a linear program. We can derive symbolic solutions in closed form for the Euclidean norm in the case of complex coefficients and a single complex root. Our new result is a formula for the minimum change in the infinity norm for the case of real coefficients and a single real root. Based on the principle of parametric minimization we develop hybrid symbolic-numeric algorithms to constrain one root of a complex or real polynomial to a curve in the complex plane. As an application to robust control, we give a polynomial-time algorithm to compute the radius of stability in the Euclidean norm for a variety of stability domains.

Implementation of residue number systems on GPUs

This contribution to the ongoing discussion of division algorithm for residue number systems (RNS) is based on Newton iteration for computing the reciprocal. An extended RNS with twice the number of moduli provides the range required for multiplication and scaling. Separation of the algorithm description from its RNS implementation achieves a high level of modularity, and makes the complexity analysis more transparent. The number of iterations needed is logarithmic in the size of the quotient for a fixed start value. With preconditioning it becomes the logarithm of the input bit size. An implementation of the conversion to mixed radix representation is outlined in the appendix. >

Cost-effective heterogeneous web clusters

Abstract New features of our DSC system for distributing a symbolic computation task over a network of processors are described. A new scheduler sends parallel subtasks to those compute nodes that are best suited in handling the added load of CPU usage and memory. Furthermore, a subtask can communicate back to the process that spawned it by a co-routine style calling mechanism. Two large experiments are described in this improved setting. In the first we have implemented an algorithm that can prove a number of more than 1,000 decimal digits prime in about 2 months elapsed time on some 20 computers. In the second a parallel version of a sparse linear system solver is used to compute the solution of sparse linear systems over finite fields. We are able to find the solution of a 100,000 by 100,000 linear system with about 10.3 million non-zero entries over the Galois field with 2 elements using 3 computers in about 54 hours CPU time.

The Kharitonov theorem and its applications in symbolic mathematical computation

We explore the problem of computing a nearest singular matrix to a given regular Hankel matrix while preserving the structure of the matrix. Nearness is measured in a matrix norm, or a componentwise norm. A recent result for structured condition numbers leads to an efficient algorithm in the spectral norm. We devise a parametrization of singular Hankel matrices, to discuss other norms.

Proceedings of the second international symposium on Parallel symbolic computation

New features of our DSC system for distributing a symbolic computation task over a network of processors are described. A new scheduler sends parallel subtasks to those compute nodes that are best suited in handling the added load of CPU usage and memory. Furthermore, a subtask can communicate back to the process that spawned it by a co-routine style calling mechanism. Two large experiments are described in this improved setting. We have implemented an algorithm that can prove a number of more than 1,000 decimal digits prime in about 2 months elapsed time on some 20 computers. A parallel version of a sparse linear system solver is used to compute the solution of sparse linear systems over finite fields. We are able to find the solution of a 100,000 by 100,000 linear system with about 10.3 million non-zero entries over the Galois field with 2 elements using 3 computers in about 54 hours CPU time.