A multiplication algorithm is an algorithm to multiply two numbers. Depending on the size of the numbers, different algorithms are in use. Efficient multiplication algorithms have existed since the advent of the decimal system.
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. Once stationary points have been identified from the first-order necessary conditions, the definiteness of the bordered Hessian matrix determines whether those points are maxima, minima, or saddle points.
In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion. A divide-and-conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computing, especially digital signal processing, the multiply–accumulate operation is a common step that computes the product of two numbers and adds that product to an accumulator. The hardware unit that performs the operation is known as a multiplier–accumulator ; the operation itself is also often called a MAC or a MAC operation. The MAC operation modifies an accumulator a:
CORDIC, also known as Volder's algorithm, is a simple and efficient algorithm to calculate hyperbolic and trigonometric functions, typically converging with one digit per iteration. CORDIC is therefore also an example of digit-by-digit algorithms. CORDIC and closely related methods known as pseudo-multiplication and pseudo-division or factor combining are commonly used when no hardware multiplier is available, as the only operations it requires are addition, subtraction, bitshift and table lookup. As such, they belong to the class of shift-and-add algorithms.
In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968.
Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization. SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable.
In computational biology, a Cellular Potts model (CPM) is a computational model of the collective behavior of cellular structures. It allows modelling of many phenomena, such as cell migration, clustering, and growth taking adhesive forces, environment sensing as well as volume and surface-area constraints into account. The first CPM was proposed for the simulation of cell sorting by Fraçois Graner and James Glazier as a modification of a large-Q Potts model. Although the model was developed to model biological cells, it can also be used to model individual parts of a biological cell, or even regions of fluid.
In physical simulations, sweep and prune is a broad phase algorithm used during collision detection to limit the number of pairs of solids that need to be checked for collision, i.e. intersection. This is achieved by sorting the starts and ends of the bounding volume of each solid along a number of arbitrary axes. As the solids move, their starts and ends may overlap. When the bounding volumes of two solids overlap in all axes they are flagged to be tested by more precise and time-consuming algorithms.
Multibody system is the study of the dynamic behavior of interconnected rigid or flexible bodies, each of which may undergo large translational and rotational displacements.
In computational chemistry, a constraint algorithm is a method for satisfying the Newtonian motion of a rigid body which consists of mass points. A restraint algorithm is used to ensure that the distance between mass points is maintained. The general steps involved are; (i) choose novel unconstrained coordinates, (ii) introduce explicit constraint forces, (iii) minimize constraint forces implicitly by the technique of Lagrange multipliers or projection methods.
The Gauss pseudospectral method (GPM), one of many topics named after Carl Friedrich Gauss, is a direct transcription method for discretizing a continuous optimal control problem into a nonlinear program (NLP). The Gauss pseudospectral method differs from several other pseudospectral methods in that the dynamics are not collocated at either endpoint of the time interval. This collocation, in conjunction with the proper approximation to the costate, leads to a set of KKT conditions that are identical to the discretized form of the first-order optimality conditions. This equivalence between the KKT conditions and the discretized first-order optimality conditions leads to an accurate costate estimate using the KKT multipliers of the NLP.
In numerical analysis, mortar methods are discretization methods for partial differential equations, which use separate finite element discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution. Mortar discretizations lend themselves naturally to the solution by iterative domain decomposition methods such as FETI and balancing domain decomposition In the engineering practice in the finite element method, continuity of solutions between non-matching subdomains is implemented by multiple-point constraints.
Sequential minimal optimization (SMO) is an algorithm for solving the quadratic programming (QP) problem that arises during the training of support vector machines (SVM). It was invented by John Platt in 1998 at Microsoft Research. SMO is widely used for training support vector machines and is implemented by the popular LIBSVM tool. The publication of the SMO algorithm in 1998 has generated a lot of excitement in the SVM community, as previously available methods for SVM training were much more complex and required expensive third-party QP solvers.
Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems and add a penalty term to the objective; the difference is that the augmented Lagrangian method adds yet another term, designed to mimic a Lagrange multiplier. The augmented Lagrangian is not the same as the method of Lagrange multipliers.
Surprisal analysis is an information-theoretical analysis technique that integrates and applies principles of thermodynamics and maximal entropy. Surprisal analysis is capable of relating the underlying microscopic properties to the macroscopic bulk properties of a system. It has already been applied to a spectrum of disciplines including engineering, physics, chemistry and biomedical engineering. Recently, it has been extended to characterize the state of living cells, specifically monitoring and characterizing biological processes in real time using transcriptional data.
In computational fluid dynamics, the Stochastic Eulerian Lagrangian Method (SELM) is an approach to capture essential features of fluid-structure interactions subject to thermal fluctuations while introducing approximations which facilitate analysis and the development of tractable numerical methods. SELM is a hybrid approach utilizing an Eulerian description for the continuum hydrodynamic fields and a Lagrangian description for elastic structures. Thermal fluctuations are introduced through stochastic driving fields.
PLUMED is an open-source library implementing enhanced-sampling algorithms, various free-energy methods, and analysis tools for molecular dynamics simulations. It is designed to be used together with ACEMD, AMBER, DL_POLY, GROMACS, LAMMPS, NAMD, OpenMM, ABIN, CP2K, i-PI, PINY-MD, and Quantum ESPRESSO, but it can also be used to together with analysis and visualization tools VMD, HTMD, and OpenPathSampling.
