A Lookup Table (LUT) is a computationally inexpensive piecewise function used to approximate computationally expensive mathematical functions. Evaluating a LUT can be as quick as using Horner’s method to evaluate a polynomial after looking up its coefficients. A common choice of LUT is a piecewise constant or piecewise linear function; however, high-degree interpolating polynomials can also be valuable. Here, we describe the functionality of FunC 2.0, a C++ library designed to streamline the process of building, comparing, and implementing univariate LUTs in practical applications. In particular, FunC 2.0 can build relatively small LUTs satisfying user-provided absolute and relative tolerances for error. Furthermore, FunC 2.0 can build nonuniform LUTs, it provides utilities to quickly determine reasonable LUT bounds and tolerances for error, and it provides a way to quickly profile a set of LUTs. We demonstrate FunC ’s utility in application by reducing the total runtime of a simulation performed by the Canadian Hydrological Model (CHM). This simulation modeled the snow mass distribution across Western Canada over 1 month. Now, the CHM can evaluate the mathematical function of interest about 28 times faster, allowing the necessary algorithm to finish two times faster, and the overall simulation is about 9% faster.

Legendre elliptic integrals and Jacobi elliptic functions arise in multiple applications within the physical sciences, including oscillations, celestial mechanics, and geodynamics. In this study, we describe the Fortran library ellipFor capable of evaluating the following for generalized input values: (1) the complete Legendre elliptic integrals of the first and second kinds, (2) the incomplete Legendre elliptic integrals of the first and second kinds, and (3) the principal Jacobi elliptic functions. Our software builds upon previously developed Fortran routines, which were designed with restrictions on input parameters that may be limiting in applications. Our routines apply multiple transformations to allow for more general input values, such as elliptic moduli greater than unity for points 1–3, arbitrary real Jacobi amplitudes for points 1–2, and complex first arguments for point 3. In addition, our routines are thread-safe, allowing for parallel computations. Our routines were compared with values from the computer algebra system SageMath over a wide range of input parameters. Values from ellipFor and SageMath agreed to within tolerances commensurate with the limitations of floating-point arithmetic used for the elliptic integrals and Jacobi elliptic functions listed in points 1, 2, and 3 above for generalized input arguments.

Polyploidy is common in flowering plants, resulting in extra sets of chromosomes, known as subgenomes. These events are widespread in plant evolution, making the assignment of synteny blocks to subgenomes challenging due to gene fractionation and gene order rearrangements. Current methods for subgenome identification are labor-intensive and require expertise, lacking automation. To address this challenge, we introduce the SyntenyLink algorithm, which automates subgenome reconstruction from synteny blocks. This algorithm considers differences in substitution and fractionation patterns in synteny blocks and maintains the continuity of gene order. SyntenyLink starts by identifying synteny blocks using BLASTP and DAGchainer, then Automatically partitions them into subgenomes by traversing the maximum weighted path on the "super-synteny graph". We validated the SyntenyLink algorithm using verified subgenomes of Brassica rapa, Brassica oleracea, Brassica nigra, Brassica napus, and Sinapis alba. The results demonstrate its effectiveness, especially for subgenome 1, with accuracy ranging from 82% to 88%. Subgenomes 2 and 3 showed slightly lower accuracy (60%-85%) due to their similar fractionation patterns. Furthermore, we applied SyntenyLink to separate the six subgenomes in Brassica juncea and Brassica carinata, illustrating its versatility. In summary, the SyntenyLink algorithm offers a powerful and automated approach for reconstructing subgenomes in complex polyploid genomes. This advancement has significant implications for studying the evolutionary history of flowering plants and other polyploid organisms.

This paper introduces a new swarm intelligence algorithm called Mother Tree Optimization (MTO) for solving continuous optimization problems. MTO uses a set of cooperating agents that evolve based on the communication between Douglas fir trees mediated by the mycorrhizal fungi network that transfers nutrients between plants of the same or different species. In order to assess the performance of the MTO algorithm, we conducted extensive experiments on its variants, with and without climate change. In this regard, we run several statistical and graphical analyses on the resulting solutions when solving well-known test functions. In the statistical analysis, the average, standard deviation, and minimum number of function evaluations are calculated for various levels of solution quality. In the graphical analysis, qualified run-length distributions are used to show the probability of solving a suite of well-known test functions at different levels of solution quality. The results demonstrate that MTO with climate change is able to reach the global solution for all the problems considered. In addition, this MTO variant generally requires fewer function evaluations than Particle Swarm Optimization and Bacterial Foraging to reach a solution of a given quality.

BACOLI is a Fortran software package for solving one-dimensional parabolic partial differential equations (PDEs) with separated boundary conditions by B-spline adaptive collocation methods. A distinguishing feature of BACOLI is its ability to estimate and control error and correspondingly adapt meshes in both space and time. Many models of scientific interest, however, can be formulated as multiscale parabolic PDE systems, that is, models that couple a system of parabolic PDEs describing dynamics on a global scale with a system of ordinary differential equations describing dynamics on a local scale. This article describes the Fortran software eBACOLI, the extension of BACOLI to solve such multiscale models. The performance of the extended software is demonstrated to be statistically equivalent to the original for purely parabolic PDE systems. Results from eBACOLI are given for various multiscale models from the extended problem class considered.

Problem-solving environments (PSEs) offer a powerful yet flexible and convenient means for general experimentation with computational methods, algorithm prototyping, and visualization and manipulation of data. Consequently, PSEs have become the modus operandi of many computational scientists and engineers. However, despite these positive aspects, PSEs typically do not offer the level of granularity required by the specialist or algorithm designer to conveniently modify the details. In other words, the level at which PSEs are black boxes is often still too high for someone interested in modifying an algorithm as opposed to trying an alternative. In this article, we describe odeToJava, a Java-based PSE for initial-value problems in ordinary differential equations. odeToJava implements explicit and linearly implicit implicit-explicit Runge--Kutta methods with error and stepsize control and intra-step interpolation (dense output), giving the user control and flexibility over the implementational aspects of these methods. We illustrate the usage and functionality of odeToJava by means of computational case studies of initial-value problems (IVPs).

Boundary value ordinary differential equations (BVODEs) are systems of ODEs with boundary conditions imposed at two or more distinct points. The global error (GE) of a numerical solution to a BVODE is the amount by which the numerical solution differs from the exact solution. The defect is the amount by which the numerical solution fails to satisfy the ODEs and boundary conditions. Although GE control is often familiar to users, the defect controlled numerical solution can be interpreted as the exact solution to a perturbation of the original BVODE. Software packages based on GE control and on defect control are in wide use. The defect control solver, BVP_SOLVER, can provide an a posteriori estimate of the GE using Richardson extrapolation. In this article, we consider three more strategies for GE estimation based on (i) the direct use of a higher-order discretization formula (HO), (ii) the use of a higher-order discretization formula within a deferred correction (DC) framework, and (iii) the product of an estimate of the maximum defect and an estimate of the BVODE conditioning constant, and demonstrate that the HO and DC approaches have superior performance. We also modify BVP_SOLVER to introduce GE control .

Recent developments of Particle Swarm Optimization (PSO) have successfully trended towards Adaptive PSO (APSO). APSO changes its behavior during the optimization process based on information gathered at each iteration. It has been shown that APSO is able to solve a wide range of difficult optimization problems efficiently and effectively. In classical PSO, all parameters remain constant for the entire swarm during the iterations. In particular, all particles share the same settings for their velocity weights. We propose a Step-Optimized PSO (SOPSO) algorithm in which every particle has its own velocity weights and an inner PSO iteration is used to take a step towards optimizing the settings of the velocity weights of every particle at every iteration. We compare SOPSO to four known PSO variants (global best PSO, decreasing weight PSO, time-varying acceleration coefficients PSO, and guaranteed convergence PSO). Experiments are conducted to compare the performance of SOPSO to the known PSO variants on 22 benchmark problems. The results show that SOPSO outperforms the known PSO variants on difficult optimization problems that require large numbers of function evaluations for their solution. This suggests that the SOPSO strategy of optimizing the settings of the velocity weights of every particle improves the robustness and performance of the algorithm.

The method of programmed constraints has recently been proposed as an executable specification language for robot programming. The mathematical structures behind such problems are viability problems for control systems described by ordinary differential equations (ODE) subject to user-defined inequality constraints. This paper describes a method for the numerical solution of such problems, improving and extending some of our previous results. The algorithm presented is composed of three parts: delay-free discretization, local control, and local planning. Delay-free discretizations are consistent discretizations of control systems described by ODEs with discontinuous inputs. The local control is based on the minimization of an artificial, logarithmic barrier potential function. Local planning is a computationally inexpensive way to increase the robustness of the solution procedure, making it a refinement to a strategy based on viability alone. Simulations of a mobile robot are used to demonstrate the proposed strategy. Some complementarity is shown between the programmed-constraints approach to robot programming and optimal control. Moreover, we demonstrate the relative efficiency of our algorithm compared to optimal control: Typically, our method is able to find a solution on the order of 100 times faster than an optimal-control solver.

Recently, new robot programming approaches have proposed the use of programmed constraints as an executable specification language for the desired behavior of a robot. The constraint-based approaches are intermediate level languages, promising a higher, more declarative level of programming than trajectory-based approaches, while being more tractable computationally than motion planning. This paper considers a numerical algorithm for solution of differential systems subject to algebraic inequality constraints. These are the mathematical structures behind the constraint-based approach. Our approach is based on a principle of 'least constraint', consisting of a dynamic integration of the equations of motion coupled with invocation of a control mechanism to ensure that the robot trajectory avoids all constraint boundaries. This is achieved by minimization of a barrier function defined using buffer zones near the constraint boundaries. Determination of the buffer zones is done dynamically, corresponding to a local planning strategy.