Approximating cube roots of integers, after Heron's Metrica III.20

Heron, in Metrica III.20-22, is concerned with the the division of solid figures - pyramids, cones and frustra of cones - to which end there is a need to extract cube roots. We report here on some of our findings on the conjecture by Taisbak in C.M.Taisbak, Cube roots of integers. A conjecture about Heron's method in Metrika III.20. Historia Mathematica, 41 (2014), 103-104

Computational science in the eighteenth century. Test cases for the methods of Newton, Raphson, and Halley: 1685 to 1745

This is an overview of examples and problems posed in the late 1600s up to the mid 1700s for the purpose of testing or explaining the two different implementations of the Newton-Raphson method, Newton’s method as described by Wallis in 1685, Raphson’s method from 1690, and Halley’s method from 1694 for solving nonlinear equations. It is demonstrated that already in 1745, it was shown that the methods of Newton and Raphson were the same but implemented in different ways.acceptedVersio

Fibonacci and digit-by-digit computation; An example of reverse engineering in computational mathematics

The Fibonacci numbers are familiar to all of us. They appear unexpectedly often in mathematics, so much there is an entire journal and a sequence of conferences dedicated to their study. However, there is also another sequence of numbers associated with Fibonacci. In The On-Line Encyclopedia of Integer Sequences, a sequence of numbers which is an approximation to the real root of the cubic polynomial. Fibonacci gave the first few numbers in the sequence in the manuscript Flos from around 1215. Fibonacci stated an error in the last number and based on this error we try, in this paper to reconstruct the method used by Fibonacci. Fibonacci gave no indication on how he determined the numbers and the problem of identifying possible methods was raised already the year after the first transcribed version of the manuscript was published in 1854. There are three possible methods available to Fibonacci to solve the cubic equation; two of the methods have been shown to give Fibonacci's result. In this paper we show that also the third method gives the same result, and we argue that this is the most likely method.Comment: Presented at NIK2022, 28 November - 1 December Kristiansand, Norwa

An interior-point trust-region-based method for large-scale non-negative regularization

Abstract We present a new method for solving large-scale quadratic problems with quadratic and nonnegativity constraints. Such problems arise for example in the regularization of ill-posed problems in image restoration where, in addition, some of the matrices involved are very ill-conditioned. The new method uses recently developed techniques for the large-scale trust-region subproblem

The CPR Method and Beyond : Prologue

In $1974$ A.R. Curtis, M.J.D. Powell, and J.K.Reid published a seminal paper on the estimation of Jacobian matrices which was later coined as the CPR method. Central to the CPR method is the effective utilization of a priori known sparsity information. It is only recently that the optimal CPR method in its general form is characterized and the theoretical underpinning for the optimality is shown. In this short note we provide an overview of the development of computational techniques and software tools for the estimation of Jacobian matrices

A New Generating Set Search Algorithm for Partially Separable Functions

A new derivative-free optimization method for unconstrained optimization of partially separable functions is presented. Using average curvature information computed from sampled function values the method generates an average Hessian-like matrix and uses its eigenvectors as new search directions. For partially separable functions, many of the entries of this matrix will be identically zero. The method is able to exploit this property and as a consequence update its search directions more often than if sparsity is not taken into account. Numerical results show that this is a more effective method for functions with a topography which requires frequent updating of search directions for rapid convergence. The method is an important extension of a method for nonseparable functions previously published by the authors. This new method allows for problems of larger dimension to be solved, and will in most cases be more efficient.publishedVersio

A minimum requiring angle trisection

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Approximating cube roots of integers, after Heron’s Metrica III.20

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A polynomial-time algorithm for LO based on generalized logarithmic barrier functions

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On a New Method for Derivative Free Optimization

A new derivative-free optimization method for unconstrained optimization of partially separable functions is presented. Using average curvature information computed from sampled function values the method generates an average Hessian-like matrix and uses its eigenvectors as new search directions. Numerical experiments demonstrate that this new derivative free optimization method has the very desirable property of avoiding saddle points. This is illustrated on two test functions and compared to other well known derivative free methods. Further, we compare the efficiency of the new method with two classical derivative methods using a class of testproblems