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Joseph F. Grcar

Publications and source records attributed to Joseph F. Grcar.

13 recordsLinked to original sources
arXiv2012

Operator Coefficient Methods for Linear Equations

New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high order recurrence formulas with scalars for coefficients, as in truncated orthomin, or have 1st order recurrence formulas with matrix polynomials for coefficients, as in restarted gcr/gmres. The new methods include both: high order recurrence formulas and matrix polynomials for coefficients. These methods provide a trade-off between recurrence order and polynomial degree that can be exploited to achieve greater efficiency. Convergence results are obtained for both constant coefficient and varying coefficient methods.

math.NA↗
arXiv2012

Matrix Stretching for Linear Equations

Stretching is a new sparse matrix method that makes matrices sparser by making them larger. Stretching has implications for computational complexity theory and applications in scientific and parallel computing. It changes matrix sparsity patterns to render linear equations more easily solved by parallel and sparse techniques. Some stretchings increase matrix condition numbers only moderately, and thus solve linear equations stably. For example, these stretchings solve arrow equations with accuracy and expense preferable to other solution methods.

math.NA↗
arXiv2011

Education for Computational Science and Engineering

Computational science and engineering (CSE) has been misunderstood to advance with the construction of enormous computers. To the contrary, the historical record demonstrates that innovations in CSE come from improvements to the mathematics embodied by computer programs. Whether scientists and engineers become inventors who make these breakthroughs depends on circumstances and the interdisciplinary extent of their educations. The USA currently has the largest CSE professorate, but the data suggest this prominence is ephemeral.

math.HO↗
arXiv2011

Unattainability of A Perturbation Bound for Indefinite Linear Least Squares Problems

Contrary to an assumption made by Bojanczyk, Higham, and Patel [SIAM J. Matrix Anal. Appl., 24(4):914-931, 2003], a perturbation bound for indefinite least square problems is capable of arbitrarily large overestimates for all perturbations of some problems. For these problems, the hyperbolic QR factorization algorithm is not proved to be forward stable because the error bound systematically overestimates the solution error of backward stable methods.

math.NA↗
arXiv2011

SIAM 2020

Some observations are made about how the Society for Industrial and Applied Mathematics (SIAM) might be better oriented to serve the industrial and interdisciplinary mathematics community in the future.

math.HO↗
arXiv2011

A Brief Review of SIAM Review

SIAM Review is examined for referee delay, citations, and paper length after the reorganization of the journal in 1999. A single, very-highly cited article was responsible for all the increase to the impact factor during the past decade; the reorganization did not improve the journal overall. Some suggestions are made for additional changes.

math.HO↗
arXiv2011

Mathematics Turned Inside Out: The Intensive Faculty Versus the Extensive Faculty

Research universities in the United States have larger mathematics faculties outside their mathematics departments than inside. Members of this "extensive" faculty conduct most mathematics research, their interests are the most heavily published areas of mathematics, and they teach this mathematics in upper division courses independent of mathematics departments. The existence of this de facto faculty challenges the pertinence of institutional and national policies for higher education in mathematics, and of philosophical and sociological studies of mathematics that are limited to mathematics departments alone.

math.HO↗
arXiv2010

Spectral Condition Numbers of Orthogonal Projections and Full Rank Linear Least Squares Residuals

A simple formula is proved to be a tight estimate for the condition number of the full rank linear least squares residual with respect to the matrix of least squares coefficients and scaled 2-norms. The tight estimate reveals that the condition number depends on three quantities, two of which can cause ill-conditioning. The numerical linear algebra literature presents several estimates of various instances of these condition numbers. All the prior values exceed the formula introduced here, sometimes by large factors.

math.NA↗
arXiv2010

Topical Bias in Generalist Mathematics Journals

Generalist mathematics journals exhibit bias toward the branches of mathematics by publishing articles about some subjects in quantities far disproportionate to the production of papers in those areas within all of mathematics.

math.HO↗
arXiv2010

Nuclear Norms of Rank 2 Matrices for Spectral Condition Numbers of Full Rank Linear Least Squares Solutions

The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares coefficient matrix and 2-norms. It depends on three quantities each of which can contribute ill-conditioning. The literature presents several estimates for this condition number with varying results; even standard reference texts contain serious overestimates. The use of the nuclear norm affords a single derivation of the best known lower and upper bounds on the condition number and shows why there is unlikely to be a closed formula.

math.NA↗
arXiv2010

How Ordinary Elimination Became Gaussian Elimination

Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method that Euler did not recommend, that Legendre called "ordinary," and that Gauss called "common" - is now named after Gauss: "Gaussian" elimination. Gauss's name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.

math.HO↗