Full linear multistep methods as root-finders

Root-finders based on full linear multistep methods (LMMs) use previous function values, derivatives and root estimates to iteratively find a root of a nonlinear function. As ODE solvers, full LMMs are typically not zero-stable. However, used as root-finders, the interpolation points are convergent so that such stability issues are circumvented. A general analysis is provided based on inverse polynomial interpolation, which is used to prove a fundamental barrier on the convergence rate of any LMM-based method. We show, using numerical examples, that full LMM-based methods perform excellently. Finally, we also provide a robust implementation based on Brent's method that is guaranteed to converge.Comment: 20 pages, 1 figur

Computing Simple Roots by an Optimal Sixteenth-Order Class

The problem considered in this paper is to approximate the simple zeros of the function by iterative processes. An optimal 16th order class is constructed. The class is built by considering any of the optimal three-step derivative-involved methods in the first three steps of a four-step cycle in which the first derivative of the function at the fourth step is estimated by a combination of already known values. Per iteration, each method of the class reaches the efficiency index , by carrying out four evaluations of the function and one evaluation of the first derivative. The error equation for one technique of the class is furnished analytically. Some methods of the class are tested by challenging the existing high-order methods. The interval Newton's method is given as a tool for extracting enough accurate initial approximations to start such high-order methods. The obtained numerical results show that the derived methods are accurate and efficient

The Eight Epochs of Math as Regards Past and Future Matrix Computations

This survey paper gives a personal assessment of epoch-making advances in matrix computations, from antiquity and with an eye toward tomorrow. It traces the development of number systems and elementary algebra and the uses of Gaussian elimination methods from around 2000 BC on to current real-time neural network computations to solve time-varying matrix equations. The paper includes relevant advances from China from the third century AD on and from India and Persia in the ninth and later centuries. Then it discusses the conceptual genesis of vectors and matrices in Central Europe and in Japan in the fourteenth through seventeenth centuries AD, followed by the 150 year cul-de-sac of polynomial root finder research for matrix eigenvalues, as well as the superbly useful matrix iterative methods and Francis’ matrix eigenvalue algorithm from the last century. Finally, we explain the recent use of initial value problem solvers and high-order 1-step ahead discretization formulas to master time-varying linear and nonlinear matrix equations via Zhang neural networks. This paper ends with a short outlook upon new hardware schemes with multilevel processors that go beyond the 0–1 base 2 framework which all of our past and current electronic computers have been using

Optimized Steffensen-Type Methods with Eighth-Order Convergence and High Efficiency Index

Steffensen-type methods are practical in solving nonlinear equations. Since, such schemes do not need derivative evaluation per iteration. Hence, this work contributes two new multistep classes of Steffensen-type methods for finding the solution of the nonlinear equation ()=0. New techniques can be taken into account as the generalizations of the one-step method of Steffensen. Theoretical proofs of the main theorems are furnished to reveal the eighth-order convergence. Per computing step, the derived methods require only four function evaluations. Experimental results are also given to add more supports on the underlying theory of this paper as well as lead us to draw a conclusion on the efficiency of the developed classes

Iron homeostasis is hierarchically regulated by multiple inputs : evidence for the role of reactive oxygen species and iron-zinc cross talk

Iron (Fe) is a heavy metal micronutrient vital for all forms of life. In plants, Fe deficiency results in chlorosis and reduced growth, while Fe excess results in lipid peroxidation through the generation of reactive oxygen species. Hence, Fe homeostasis must be tightly regulated. Plants have been shown to use multiple sensing mechanisms to regulate whole plant Fe demand (systemically) and at through protein level changes at the root epidermis (locally). The companion cell of the phloem has recently been strongly implicated as the site of systemic Fe sensing. In this work I demonstrate that leaves and roots are subject to multiple regulatory inputs which modulate Fe dependent gene expression in a hierarchical fashion, and was able to separate these responses into reactive oxygen species (ROS) dependent and independent groups. Excess heavy metal has been shown to generate ROS, hence plants must also balance relative abundances of each heavy metal to prevent deficiency/toxicity. We identified bZIP23, which was previously described as an inducer of Zn uptake, as a likely candidate to mediate the mediate Fe-Zn crosstalk through the characterization of the double mutant bzip23-1/opt3-2 which suppresses opt3 dependent induction of Fe deficiency responses, likely by directly regulating the Fe uptake machinery. To facilitate the identification of time dependent changes in root growth phenotypes, such as under heavy metal stress, I designed and constructed a Small Plant Imaging Platform (SPIP) which able to capture high quality images for automated time course analysis which we aim to distribute throughout the plant science community. Finally, I have performed the two complementary experiments to first identify Fe dependent changes in gene translation in companion cells which is paired with the identification of transcription factor which directly regulate OPT3. Initial results indicate a novel mechanism of Fe release from the cell wall in the leaf vasculature during Fe deficiency, and implicate transcription factors known to mediate Fe deficiency responses as being responsible for the rapid induction of OPT3 upon Fe deficiency.Includes bibliographical reference

GEAR-RT: Towards Exa-Scale Moment Based Radiative Transfer For Cosmological Simulations Using Task-Based Parallelism And Dynamic Sub-Cycling with SWIFT

The development and implementation of GEAR-RT, a radiative transfer solver using the M1 closure in the open source code SWIFT, is presented, and validated using standard tests for radiative transfer. GEAR-RT is modeled after RAMSES-RT (Rosdahl et al. 2013) with some key differences. Firstly, while RAMSES-RT uses Finite Volume methods and an Adaptive Mesh Refinement (AMR) strategy, GEAR-RT employs particles as discretization elements and solves the equations using a Finite Volume Particle Method (FVPM). Secondly, GEAR-RT makes use of the task-based parallelization strategy of SWIFT, which allows for optimized load balancing, increased cache efficiency, asynchronous communications, and a domain decomposition based on work rather than on data. GEAR-RT is able to perform sub-cycles of radiative transfer steps w.r.t. a single hydrodynamics step. Radiation requires much smaller time step sizes than hydrodynamics, and sub-cycling permits calculations which are not strictly necessary to be skipped. Indeed, in a test case with gravity, hydrodynamics, and radiative transfer, the sub-cycling is able to reduce the runtime of a simulation by over 90%. Allowing only a part of the involved physics to be sub-cycled is a contrived matter when task-based parallelism is involved, and is an entirely novel feature in SWIFT. Since GEAR-RT uses a FVPM, a detailed introduction into Finite Volume methods and Finite Volume Particle Methods is presented. In astrophysical literature, two FVPM methods are written about: Hopkins (2015) have implemented one in their GIZMO code, while the one mentioned in Ivanova et al. (2013) isn't used to date. In this work, I test an implementation of the Ivanova et al. (2013) version, and conclude that in its current form, it is not suitable for use with particles which are co-moving with the fluid, which in turn is an essential feature for cosmological simulations.Comment: PhD Thesi

Can Automated Vehicles "See" in Minnesota? Ambient Particle Effects on LiDAR

(c)1035427This project will use a combination of laboratory experimentation and road demonstrations to better understand the reduction of LiDAR signal and object detection capability under adverse weather conditions found in Minnesota. It will also lead to concepts to improve LiDAR systems to adapt to such conditions through better signal processing image recognition software