## What is/are Linear Algebraic?

Linear Algebraic - The model couples a set of partial differential equations, describing the transport of chemical species, to nonlinear algebraic or differential equations, describing the chemical reactions.^{[1]}The proposed model is nonlinear algebraic and equations require iterative numerical solution.

^{[2]}For this purpose, this study traces the development of a novel numerical iterative method of an open method for solving nonlinear algebraic and transcendental application equations.

^{[3]}

## boundary value problem

We consider several versions of incomplete nested factorization methods for solving the large systems of linear algebraic equations (SLAEs) with sparse matrices which arise in grid approximations of the multi-dimensional boundary value problems.^{[1]}We consider parallel iterative processes in Krylov subspaces for solving symmetric positive definite systems of linear algebraic equations (SLAEs) with sparse ill-conditioned matrices arising under grid approximations of multidimensional initial-boundary value problems.

^{[2]}When using the finite element method, the main computational load, as a rule, falls on the solution of systems of linear algebraic equations obtained as a result of the finite element approximation of the boundary value problem.

^{[3]}An accurate fulfillment of the imperfect interface conditions reduces the model boundary value problem to the linear algebraic system for multipole strengths.

^{[4]}The solvability of the boundary value problem for the loaded differential equation with a parameter is reduced to the solvability of a system of linear algebraic equations with respect to arbitrary vectors of the introduced general solution.

^{[5]}The general solution of the Neumann boundary value problem for Helmholtz equation is obtained in a form of two coupled infinite systems of linear algebraic equations of the second kind.

^{[6]}The irregular arrangement of collocation nodes in the problem solving domain can sharply increase the accuracy of the numerical solution by improving the quality of the linear algebraic equations system, to which the solved boundary value problem leads.

^{[7]}The main idea behind the proposed technique is established on converting the governed boundary-value problem into a system of linear algebraic equations via the application of the spectral Galerkin method.

^{[8]}By accurate fulfillment of the contact conditions, the model boundary value problem is reduced to the linear algebraic system.

^{[9]}Fulfillment of the matrix-to-inhomogeneity elastic contact conditions reduces the boundary value problem to the linear algebraic system for multipole strengths and provides a highly efficient algorithm for numerical study.

^{[10]}

## Solving Linear Algebraic

This is the first part of our paper that attempts to give an answer with focus on solving linear algebraic equations (LAEs) from the perspective of systems and control, where it mainly introduces the controllability-based design results.^{[1]}This article considers the problem of solving linear algebraic equations of the form

^{[2]}If SARE is solvable the higher degree terms can be fouuding by solving linear algebraic equations.

^{[3]}

## Dimensional Linear Algebraic

Fitting such an LMM in a large scale cohort study, however, is tremendously challenging due to its high dimensional linear algebraic operations.^{[1]}In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety.

^{[2]}The geometric multigrid method has been implemented using the V-cycle for solving the high-dimensional linear algebraic equation systems that appear when applying the finite element and finite difference methods for the problems of mathematical physics.

^{[3]}

## Connected Linear Algebraic

Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups.^{[1]}Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.

^{[2]}Let G be a simple, simply connected linear algebraic group of exceptional type defined over F q with Frobenius endomorphism F : G → G.

^{[3]}

## Simple Linear Algebraic

We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field.^{[1]}By transforming the nonlinear input–output relationship of the device into a matrix of learned weights, a set of simple linear algebraic and nonlinear activation calculations can be made to predict the device outputs 1830 times faster than numerical simulations, within 93.

^{[2]}

## Standard Linear Algebraic

After that, a set of standard linear algebraic equations was obtained.^{[1]}This has caused computational science communities to now face the computational limitations of standard linear algebraic methods that have been relied upon for decades to run quickly and efficiently on modern computing hardware.

^{[2]}

## N Linear Algebraic

Systems of n linear algebraic equations with n unknowns are addressed.^{[1]}The method is based on (1) linear interpolation of velocity values in drag terms, (2) implicit approximation of drag terms that conserves momentum with machine precision, and (3) solution of system of N linear algebraic equations with O ( N 2 ) arithmetic operation instead of O ( N 3 ).

^{[2]}

## Establish Linear Algebraic

In this method, a fast time discrete scheme is first derived by applying the L1 and the fast H2N2 approximations to discretize the time Caputo fractional derivative, and then the Nitsche’s technique and the stabilized moving least squares approximation are adopted to establish linear algebraic systems.^{[1]}Then, a stabilized EFG method is proposed to establish linear algebraic systems.

^{[2]}

## Fractional Linear Algebraic

This paper is devoted to the study of numerical methods for solving large, sparse or full fractional linear algebraic systems (FLAS).^{[1]}This paper is devoted to the computation of the solution to fractional linear algebraic systems using a differential-based strategy to evaluate matrix-vector products Aαx, with α ∈ R+.

^{[2]}

## Infinite Linear Algebraic

The solution of the problem is reduced to the two systems of infinite linear algebraic equations (ISLAE) of the second kind using the mode–matching technique and the analytical regularization procedure.^{[1]}Then, based on the conditions of free boundary stress, continuous magnetic induction intensity and continuous magnetic potential around the circular cavity, the infinite linear algebraic equations are established.

^{[2]}

## linear algebraic equation

It is shown that to solve this problem it is necessary to solve an interval system of nonlinear algebraic equations, the solutions area of which is nonconvex and incoherent.^{[1]}The synchronization problem of two HR coupled neurons is ultimately converted into the stability problem of roots to a nonlinear algebraic equation.

^{[2]}Hence, this article has introduced a new method suitable for the direct solution of first order differential equation models without the need to simplify to a system of linear algebraic equations.

^{[3]}This is the first part of our paper that attempts to give an answer with focus on solving linear algebraic equations (LAEs) from the perspective of systems and control, where it mainly introduces the controllability-based design results.

^{[4]}Infinite systems of linear algebraic equations obtained by satisfying the boundary conditions and conjugation conditions of a layer with inclusions have been solved by the reduction method.

^{[5]}Rational Chebyshev collocation (RCC) method is used to transform the problem to a system of nonlinear algebraic equations.

^{[6]}The algorithm for numerical solving an overdetermined system of linear algebraic equations arising in the analysis of a magnetic circuit is constructed.

^{[7]}The proposed scheme converts the underlying fractional initial value problems into a matrix equation, which corresponds to a set of linear algebraic equations consist of polynomial coefficients.

^{[8]}The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations.

^{[9]}Using the fundamental matrix of the differentialpart of the integro-differential equation and assuming the solvability of the special Cauchy problem, the originalboundary value problem is reduced to a system of linear algebraic equations with respect to the introduced additionalparameters.

^{[10]}To develop an automated system for static calculation of flexible round plates, we use central finite-difference schemes that approximate derivatives with second-order accuracy, we obtain a system of quasilinear algebraic equations.

^{[11]}By substituting this approximation into the equation, and collocating the resulting equation at a set of collocation points, a system of linear algebraic equations is obtained.

^{[12]}A simple introduction to systems of two linear algebraic equations with two unknowns is included.

^{[13]}The solution method is based on finding integer solutions of simultaneous linear algebraic equations (SLAE).

^{[14]}In each adaptive step, the matrix decomposition methods are adopted to solve linear algebraic equations, which is obviously time-consuming and computationally expensive for large-scale problems.

^{[15]}A novel approach to the solution of the inverse kinematic problem is formulated by adding a set of m target equations to obtain 4 m + 3 nonlinear algebraic equations solved in terms of an equal number of unknowns.

^{[16]}After discretizing in time with an implicit Euler scheme, the resulting systems of nonlinear algebraic equations are solved with Newton’s method and the systems of linear equations are solved efficiently and in parallel with an algebraic multigrid method.

^{[17]}These equations are reduced to a system of two nonlinear algebraic equations using the plane-sections hypothesis.

^{[18]}This article considers the problem of solving linear algebraic equations of the form

^{[19]}With the method of dimensional analysis, the solution of the nonlinear algebraic equations for the Lagrange multiplier is superfluous.

^{[20]}5, and the system of linear algebraic equations is symmetric and positive definite.

^{[21]}Currently, modeling of processes by numerical solution of differential equations is widely used in every field of Science, the most common methods bring the differential problem to a system of linear algebraic equations, methods that solve such systems include various startup options.

^{[22]}It is shown that this task at each iteration is a task the forming and solving interval systems of nonlinear algebraic equations.

^{[23]}If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

^{[24]}Combining the performance index and problem’s constraints in a scalar function, necessary optimality conditions are achieved in a form of nonlinear algebraic equations.

^{[25]}The originality of the research lies in obtaining implicit relations connecting the coefficients of the spline expansion at the initial level with the spline coefficients and wavelet coefficients at the embedded level by a band system of linear algebraic equations with a nondegenerate matrix.

^{[26]}It is reduced to the system of linear algebraic equations and solved in a numerical manner by the mechanical quadrature method.

^{[27]}Based on this expressions, a system of linear algebraic equations of the 7-th order with respect to the unknown coefficients ai and bi was compiled.

^{[28]}Previously, a method was proposed for synthesizing radar images in the coordinates “time delay-Doppler frequency shift” for passive location with extraneous illumination, based on solving the problem of minimizing the discrepancy of an overdetermined system of linear algebraic equations using non-square regularization.

^{[29]}The problem is reduced to a system of linear algebraic equations with fuzzy variables.

^{[30]}While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic elastic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications.

^{[31]}This is necessary for the convenience of solving a system of linear algebraic equations in a matrix polynomial calculation.

^{[32]}We consider several versions of incomplete nested factorization methods for solving the large systems of linear algebraic equations (SLAEs) with sparse matrices which arise in grid approximations of the multi-dimensional boundary value problems.

^{[33]}In this paper, a new adaptive Monte Carlo algorithm is proposed to solve systems of linear algebraic equations (SLAEs).

^{[34]}However, sparse estimation is often studied in the framework of linear algebraic equations, whereas model-based fault diagnosis is usually investigated for dynamic systems modeled with state equations involving internal states.

^{[35]}For the case of nonlinear differential equations with polynomial nonlinearities, SEsM can reduce the solved equations to a system of nonlinear algebraic equations.

^{[36]}For steady states responses, the Newton-Raphson shooting technique is used to solve the three systems of six parametric nonlinear algebraic equations obtained.

^{[37]}Thus, an infinite system of linear algebraic equations is derived for the unknown coefficients of the series representing the solution for each case.

^{[38]}In practical computations, the (preconditioned) conjugate gradient (P)CG method is the iterative method of choice for solving systems of linear algebraic equations Ax = b with a real symmetric positive definite matrix A.

^{[39]}A semi-implicit discretization is employed to convert the enthalpy based partial differential equations into a system of nonlinear algebraic equations.

^{[40]}Full characterisation of the exact optimal solution requires numerically solving a set of four nonlinear algebraic equations with respect to four unknowns.

^{[41]}The latter relies on non-linear algebraic equations and, thus, requires a very short computational time.

^{[42]}The current computational scheme coverts a mathematical model to a system of linear algebraic equations that are easier to solve.

^{[43]}In the proposed method, the \begin{document}$ L2 $\end{document} scheme and the Grunwald-Letnikov formula is used for the approximation of the Caputo fractional derivative and the right Riemann-Liouville fractional derivative, respectively, which converts the optimality system into a system of linear algebraic equations.

^{[44]}Application of Orthogonal Polynomial Method and Analytical Regularization Method allowed the problem to be reduced to a well-conditioned infinite system of linear algebraic equations of the second kind.

^{[45]}When explicit time integration is employed, it is beneficial to use a lumped mass matrix because it provides a larger critical time step and a straightforward solution of the linear algebraic equation.

^{[46]}We consider parallel iterative processes in Krylov subspaces for solving symmetric positive definite systems of linear algebraic equations (SLAEs) with sparse ill-conditioned matrices arising under grid approximations of multidimensional initial-boundary value problems.

^{[47]}The advantages of working with linear algebraic equations of the second kind instead of the first kind are underlined in terms of reduction in the number of iterations for the iterative solver for a specific residual error.

^{[48]}Systems of n linear algebraic equations with n unknowns are addressed.

^{[49]}At last, the state-space equation combined with the boundary conditions of the flexible manipulator is transformed to a system of linear algebraic equations, from which the response of the flexible manipulator can be easily solved.

^{[50]}

## linear algebraic system

The paper discusses a reuse of matrix factorization as a building block in the Augmented Lagrangian (AL) and modified AL preconditioners for a non-symmetric saddle point linear algebraic systems.^{[1]}The Collocation method converts the given fractional differential equation into a matrix equation, which yields a linear algebraic system.

^{[2]}A cubic nonlinear algebraic system is obtained and solved numerically using an approximation method (the so-called second formulation) is applied to resolve various nonlinear vibration problems.

^{[3]}Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented.

^{[4]}A nonlinear algebraic system has to be solved at every time step for the LagM based scheme, hence the SAV scheme is usually more efficient than the LagM one.

^{[5]}First, the nonequivalence between direct and indirect method is analytically studied by using the degenerate kernel and examined by using the linear algebraic system.

^{[6]}With this variation, the problem transformed into a nonlinear algebraic system that its coefficients must be determined.

^{[7]}The compact determinant form of the N-fold Darboux transformation is obtained by iterating the onefold Darboux transformation and solving a complex linear algebraic system.

^{[8]}The implementation of the algorithm requires only the solution of a linear algebraic system with multiple right-hand sides in each time step.

^{[9]}Specifically, resorting to computational algebraic geometry concepts and tools, such as Grobner basis, it is shown that the complete solution set of a nonlinear algebraic system of coupled multivariate polynomial equations is determined exactly, even for a relatively large number of unknowns.

^{[10]}The postprocessing technique is independently implemented by solving a 3-by-3 and 4-by-4 local linear algebraic system on each triangular and quadrilateral element, respectively.

^{[11]}In this paper, a class of multilevel preconditioning schemes is presented for solving the linear algebraic systems resulting from the application of Morley nonconforming element approximations to the biharmonic Dirichlet problem.

^{[12]}The Frechet derivative of the objective function vanishes at the optimal solution and it gives a resulting nonlinear algebraic system for the optimal solution.

^{[13]}When constructing stable synthesis algorithms, a recurrent method for determining pseudo-solutions of linear algebraic systems of equations is used.

^{[14]}An accurate fulfillment of the imperfect interface conditions reduces the model boundary value problem to the linear algebraic system for multipole strengths.

^{[15]}Combination of the proposed method with a semi-implicit time integration method such as the Leapfrog–Crank–Nicolson scheme leads to reducing the complexity of computations and obtaining a linear algebraic system of equations.

^{[16]}With the increasing complexity of integrated circuits, it is becoming cumulatively challenging to solve the entire large-scale nonlinear algebraic system in DC analysis within reasonable simulation time and without accuracy lost.

^{[17]}In this method, a fast time discrete scheme is first derived by applying the L1 and the fast H2N2 approximations to discretize the time Caputo fractional derivative, and then the Nitsche’s technique and the stabilized moving least squares approximation are adopted to establish linear algebraic systems.

^{[18]}This paper is devoted to the study of numerical methods for solving large, sparse or full fractional linear algebraic systems (FLAS).

^{[19]}Consequently, computational grid distributions and linear algebraic systems arising from finite difference schemes for them are also asymmetric.

^{[20]}To satisfy the boundary conditions at every boundary node and the governing equation at every node, a sparse linear algebraic system can be obtained.

^{[21]}Indeed, the linear algebraic system formed by the steady-state mass balance equations around the intracellular metabolites and the equality constraints related to the measurements of extracellular fluxes do not define a unique solution for the distribution of intracellular fluxes, but instead a set of solutions belonging to a convex polytope.

^{[22]}An analytical formulation of the governing nonlinear algebraic system of equations is developed for the case of piecewise polynomial systems.

^{[23]}The periodic response is described in terms of a set of temporal nodes of all spatial degrees of freedom of the system yielding a block-diagonal nonlinear algebraic system to be solved iteratively.

^{[24]}A fixed-point iterative method is proposed for solving the nonlinear algebraic system.

^{[25]}Finally, for discretization, a stable P2P1 finite element pair is employed to approximate the displacement, velocity and pressure spaces independently and the resulting nonlinear algebraic system is linearized by implementing the Newtons procedure.

^{[26]}The numerical algorithm implemented in the software is based on the approximation of the variable-order derivative by finite differences and the subsequent solution of the corresponding nonlinear algebraic system.

^{[27]}Because an increase of $m$ results in an increase of $N_c$, an appropriate range of $\left[ {{N_b}/{N_c},{\rm{ }}2{N_b}/{N_c}}\right]$ for $m$ is suggested to prevent too much additional computations for solving the linear algebraic system and computing the displacements of volume nodes induced by the increase of $N_c $.

^{[28]}Therefore, the Nystrom method is applied to the linear Volterra integral equation with the discontinuous kernel to convert it to a linear algebraic system.

^{[29]}Hamilton’s principle and spectral analysis are used to reduce the problem to a non-linear algebraic system solved using a previously developed approximate method.

^{[30]}Then, a stabilized EFG method is proposed to establish linear algebraic systems.

^{[31]}Taking into account the harmonic response, the transverse displacement function of the non-linear beam determined by applying Hamilton’s principle, the problem is reduced to a non-linear algebraic system solved by an approximate method.

^{[32]}By accurate fulfillment of the contact conditions, the model boundary value problem is reduced to the linear algebraic system.

^{[33]}The integral equation is discretized by Gaussian approximating functions and reduced to a linear algebraic system for the coefficients of the approximation (the discretized problem).

^{[34]}Discrete expressions of the strain energy and kinetic energies are derived, and Hamilton’s principle is applied to reduce the problem to a solution of a nonlinear algebraic system and then solved by an approximate method.

^{[35]}By using a collocation method, the integral equations are reduced to a linear algebraic system of equations for the unknown coefficients of the series.

^{[36]}Fulfillment of the matrix-to-inhomogeneity elastic contact conditions reduces the boundary value problem to the linear algebraic system for multipole strengths and provides a highly efficient algorithm for numerical study.

^{[37]}This paper is devoted to the computation of the solution to fractional linear algebraic systems using a differential-based strategy to evaluate matrix-vector products Aαx, with α ∈ R+.

^{[38]}The fully discrete scheme leads to a linear algebraic system to solve at each time step.

^{[39]}The fast solver significantly reduces the computational work of solving the discrete linear algebraic systems from O ( M N 3 + M 2 N ) by a direct solver to O ( M N log ( M N ) ) per Krylov subspace iteration and a memory requirement from O ( M N 2 ) to O ( N log M ).

^{[40]}The solution of the CLEBVP is approximated by a linear combination of piecewise linear and continuous basis functions, upon substituting this approximated solution into the CLEBVP the problem then reduces to solve a “Galerkin” linear algebraic system (GALAS).

^{[41]}

## linear algebraic group

These formulas are then applied to the case of linear algebraic groups.^{[1]}This is a survey article on some recent developments in the arithmetic theory of linear algebraic groups over higher-dimensional fields, written for the Notices of the AMS.

^{[2]}We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups.

^{[3]}Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups.

^{[4]}Let G be a connected, simply connected, simple, complex, linear algebraic group.

^{[5]}We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields.

^{[6]}Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.

^{[7]}We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field.

^{[8]}For $X$ a smooth scheme acted on by a linear algebraic group $G$ and $p$ a prime, the equivariant Chow ring $CH^*_G(X)/p$ is an unstable algebra over the Steenrod algebra.

^{[9]}Let G be a simple, simply connected linear algebraic group of exceptional type defined over F q with Frobenius endomorphism F : G → G.

^{[10]}

^{[11]}These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus.

^{[12]}

## linear algebraic method

Hence, the above computational problems of nonlinear DSFF are solvable by linear algebraic methods.^{[1]}The methods avoid any (1) fitting of parameters to outcome or data, (2) use of linear algebraic methods, (3) determinations of scale factor values, and (4) use of some typically inaccurate types of experimentally estimated probability values.

^{[2]}We compare the capabilities of two approaches to approximating graph isomorphism using linear algebraic methods: the invertible map tests (introduced by Dawar and Holm) and proof systems with algebraic rules, namely polynomial calculus, monomial calculus and Nullstellensatz calculus.

^{[3]}This has caused computational science communities to now face the computational limitations of standard linear algebraic methods that have been relied upon for decades to run quickly and efficiently on modern computing hardware.

^{[4]}This leads us to introduce the Symmetric Subspace Decomposition strategy to identify linear evolutions using efficient linear algebraic methods.

^{[5]}

## linear algebraic model

A nonlinear formulation is adopted to capture the nonlinear thermal models and the nonlinear algebraic model of the heat supply network.^{[1]}Originality/value The paper presents a modified Chun-Hui He’s algorithm for solving the nonlinear algebraic models exist in various area.

^{[2]}In this methodology, the data fidelity term is developed using a statistical linear algebraic model of quasi-static equilibrium equation revealing the relationship of the observed displacement fields to the unobserved elastic modulus.

^{[3]}

## linear algebraic operation

Fitting such an LMM in a large scale cohort study, however, is tremendously challenging due to its high dimensional linear algebraic operations.^{[1]}We also present a variant that applies to the case of zonotopic uncertainties, uses only linear algebraic operations, and yields zonotopic over-approximations.

^{[2]}

## linear algebraic solver

To solve the set of equations subject to the boundary and the inlet conditions of the pipeline, the nonlinear algebraic solver library DNSQE in Fortran is used to study the behaviour of 90 vol % CO2, and 10 vol % single impure component (N2, H2, H2S, O2 and Ar).^{[1]}We design and study its adaptive numerical approximation interconnecting a finite element discretization, the Banach-Picard linearization, and a contractive linear algebraic solver.

^{[2]}

## linear algebraic approach

This serves as a basis to introduce a multilinear algebraic approach, inherent in tensors, to the modeling of the global term structure underlying multiple interest rate curves.^{[1]}Inspired by the geometric approach of Maschke and van der Schaft and the linear algebraic approach of Mehl, Mehrmann and Wojtylak, we present another view by using the theory of linear relations.

^{[2]}

## linear algebraic structure

We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations.^{[1]}Leveraging the fiber sampling scheme and the multilinear algebraic structure of low-rank tensors, the proposed lightweight algorithm also ensures global convergence to a stationary point under mild conditions.

^{[2]}

## linear algebraic tool

Finally, we show that the negotiation function is piecewise linear and can be analyzed using the linear algebraic tool box.^{[1]}Then, the linearized infective part of the model is discussed through a positivity/stability viewpoint from linear algebraic tools.

^{[2]}

## linear algebraic computation

DarKnight relies on cooperative execution between trusted execution environments (TEE) and accelerators, where the TEE provides privacy and integrity verification, while accelerators perform the bulk of the linear algebraic computation to optimize the performance.^{[1]}We thus provide a framework for compiling instructions for linear algebraic computations into gate sequences on actual quantum computers.

^{[2]}

## linear algebraic analysi

When both^{[1]}In the proposed method a simple decomposition of the traditional iterative linear finite element analysis and the nonlinear algebraic analysis of the plate cross-section is used.

^{[2]}