Merrell L. Patrick

A family of root finding methods

SummaryA one parameter family of iteration functions for finding roots is derived. The family includes the Laguerre, Halley, Ostrowski and Euler methods and, as a limiting case, Newtons method. All the methods of the family are cubically convergent for a simple root (except Newtons which is quadratically convergent). The superior behavior of Laguerres method, when starting from a pointz for which |z| is large, is explained. It is shown that other methods of the family are superior if |z| is not large. It is also shown that a continuum of methods for the family exhibit global and monotonic convergence to roots of polynomials (and certain other functions) if all the roots are real.

Parallel, iterative solution of sparse linear systems - Models and architectures

Abstract Solving large, sparse, linear systems of equations is a fundamental problems in large scale scientific and engineering computation. A model of a general class of asynchronous, iterative solution methods for linear systems is developed. In the model, the system is solved by creating several cooperating tasks that each compute a portion of the solution vector. A data transfer model predicting both the probability that data must be transferred between two tasks and the amount of data to be transferred is presented. This model is used to derive an execution time model for predicting parallel execution time and an optimal number of tasks given the dimension and sparsity of the coefficient matrix and the costs of computation, synchronization, and communication. The suitability of different parallel architectures for solving randomly sparse linear systems is discussed. Based on the complexity of task scheduling, one parallel architecture, based on a broadcast bus, is presented and analyzed.

Estimating the multiplicity of a root

SummaryOne point iteration functions for approximating roots of a function of a single variable and sequences generated by them are considered. Methods of estimating the multiplicity of the roots to which such sequences are converging are given. In particular, multiplicity estimators are given for Newtons method, the modified Newtons method, and Laguerres method. It is shown that the estimators for Newtons method and Laguerres method offer improvement over those of Rall and Dekker, respectively.

A highly parallel algorithm for approximating all zeros of a polynomial with only real zeros

An algorithm is described based on Newtons method which simultaneously approximates all zeros of a polynomial with only real zeros. The algorithm, which is conceptually suitable for parallel computation, determines its own starting values so that convergence to the zeros is guaranteed. Multiple zeros and their multiplicity are readily determined. At no point in the method is polynomial deflation used.

A globally convergent algorithm for determining approximate real zeros of a class of functions

It is shown that Newtons method can be used to define a globally convergent algorithm for approximating real zeros of a certain class of functions. Included in this class are the polynomials with only real zeros.

Polynomial evaluation with scaling

To overcome difficulties in polynomial evaluation caused by overflow or unnecessary underflow, we introduce a simple scaling procedure into Horners method.

The impact of domain partitioning on the performance of a shared memory multiprocessor

Abstract The impact of the choice of domain partition/stencil pairs in solving elliptic partial differential equations on a multiprocessor system with both global shared memory and locl memory is considered. A parallel execution time model is given for square and hexagonal partitions using both five-point and nine-point star discretizations. Results from the model for a particular multiprocessor system show that keeping copies of partition boundaries in local memory significantly improves performance.