Systems of linear algebraic equations

We will discuss here Cramer s systems of n linear equations with n unknowns which may be written 

Systems of linear equations are to be solved during the stoichiometric analysis of a reaction, but also on the occasion of the use of other numerical methods (see below). 

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Preliminary comments. By applying approximate methods the problem of solving differential equations leads to the systems of linear algebraic equations ... 

The starting point in more a detailed exploration is the simplest systems of linear algebraic equations, namely, difference equations with special matrices in simplified form, for example, with tridiagonal matrices. 

Difference schemes as operator equations. After replacing differential equations by difference equations on a certain grid we obtain a system of linear algebraic equations that can be written in matrix form. The outcome of this is... 

Direct and iterative methods. Recall that the final results of the difference approximation of boundary-value problems associated with elliptic equations from Chapter 4 were various systems of linear algebraic equations (difference or grid equations). The sizes of the appropriate matrices are extra large and equal the total number N of the grid nodes. For... 

Seidel method. As we have mentioned above, implicit schemes are rather stable in comparison with explicit ones. Seidel method, being the simplest implicit iterative one, is considered first. The object of investigation here is the system of linear algebraic equations... 

This system of linear algebraic equations is easy to solve to find the estimates of model parameters b,. It can be rewritten in more general matrix notation ... 

The more interesting problems tend to be neither steady state nor linear, and the reverse Euler method can be applied to coupled systems of ordinary differential equations. As it happens, the application requires solving a system of linear algebraic equations, and so subroutine GAUSS can be put to work at once to solve a linear system that evolves in time. The solution of nonlinear systems will be taken up in the next chapter. 

For implicit schemes, we will obtain a system of linear algebraic equations that must be solved. As mentioned in Example 8.1, one-dimensional diffusion problems generate tri-diagonal matrices, that can be solved for using the Thomas algorithm or other fast matrix routines. Equation (8.83) can be written as... 

Consequently, there are 3(N + NIP) velocity unknowns and 3N traction unknowns. This makes eqn. (10.95) a system of 3(N+NIP) equations with 3(N+NIP) + 3Nunknowns. Each boundary nodal point has either traction or velocity specified for each direction as a boundary condition, thus the system in eqn. (10.95) can ultimately be arranged into a solvable system of linear algebraic equations as... 

This allows us to represent partial differential equations as found in the balance equations using the collocation method. Equation (11.47) is a solution to a partial differential equation represented by a system of linear algebraic equations, formed by the interpolation coefficients, oij, and the operated radial functions. The interpolation coefficients are solved for using matrix inversion techniques to approximately satisfy the partial differential equation... 

As a rule, more than two dimensionless numbers are necessary to describe a phys-ico-technological problem they cannot be produced as shown in the first three examples. The classical method to approach this problem involved a solution of a system of linear algebraic equations. They were formed separately for each of the base dimensions by exponents with which they appeared in the physical quantities. J. Pawlowski [5] replaced this relatively awkward and involved method by a simple and transparent matrix transformation ( equivalence transformation ) which will be presented in detail in the next example. 

As a result, we have the aggregate of N connected infinite systems of linear algebraic equations. The systems described by Eq. (11) contain only the coefficients of external potentials, and the problem of the interaction of N... 

The resulting system of linear algebraic equations and the accompanying boundary conditions can be expressed in matrix notation as... 

In this paper, an inverse problem for galvanic corrosion in two-dimensional Laplace s equation was studied. The considered problem deals with experimental measurements on electric potential, where due to lack of data, numerical integration is impossible. The problem is reduced to the determination of unknown complex coefficients of approximating functions, which are related to the known potential and unknown current density. By employing continuity of those functions along subdomain interfaces and using condition equations for known data leads to over-determined system of linear algebraic equations which are subjected to experimental errors. Reconstruction of current density is unique. The reconstruction contains one free additive parameter which does not affect current density. The method is useful in situations where limited data on electric potential are provided. 

The system of linear algebraic equations (SLAE) is formed by equations (13)-... 

Steady state linear elliptic PDEs in finite domains are solved by applying finite difference technique in both x and y coordinates in this section. When finite differences are applied, a linear elliptic PDE is converted to a system of linear algebraic equations. This resulting system of linear equations can be directly solved using Maple s solve or fsolve command. This is best illustrated with the following examples. 

Hindmarsh, A. C. Solution of Block-Tridiagonal Systems of Linear Algebraic Equations Lawrence Livermore National Laboratory Report UCID-30150, 1977. 

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