Linear algebraic systems Gaussian elimination

For a tridiagonal linear system, Gaussian elimination simplifies to a simple algebraic factorization followed by back-substitution. The time taken is linearly proportional to the number of equations. 

Consequently, the approximative solution to the BVP is calculated on a computational mesh, and it results in a system of algebraic equations. In this example, die problem is linear and the Gaussian elimination can be used to solve the equation system (n — 1 algebraic equations). A Dirichlet boundary condition was specified in dus problem. Note that the boundary conditions are accounted for in two of diese equations, i.e. 

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. 

GAUSS Subroutine GAUSS solves a system of simultaneous linear algebraic equations by Gaussian elimination and back substitution. The number of equations (equal to the number of unknowns) is NROW. The coefficients are in array SLEQ(NR0W,NR0W+1), where the last column is the constants. 

The steady-state problem yields a system of simultaneous linear algebraic equations that can be solved by Gaussian elimination and back substitution. I shall turn now to calculating the time evolution of this system, starting from a phosphate distribution that is not in steady state. In this calculation, assume that the phosphate concentration is initially the same in all reservoirs and equal to the value in river water, 10 I 3 mole P/m3. How do the concentrations evolve from this starting value to the steady-state values just calculated ... 

GAUSSO Subroutine GAUSSO solves a system of simultaneous linear algebraic equations by Gaussian elimination and back substitution. 

HS, S, HCCU, CO3, RR NH, RR NCOO", H+, OH- and H2O. Hence there are twenty-three unknowns (m and Yj for all species except water plus x ). To solve for trie unknowns there are twenty-three independent equations Seven chemical equilibria, three mass balances, electroneutrality, the use of Equation (6) for the eleven activity coefficients and the phase equilibrium for xw. The problem is one of solving a system of nonlinear algebraic equations. Brown s method (21, 22) was used for this purpose. It is an efficient procedure, based on a partial pivoting technique, and is analogous to Gaussian elimination in linear systems of equations. 

Due to the special structure of MATLAB, readers should be familiar with the mathematical concepts pertaining to matrices, such as systems of linear equations, Gaussian elimination, size and rank of a matrix, matrix eigenvalues, basis change in n-dimensional space, matrix transpose, etc. For those who need a refresher on these topics there is a concise Appendix on linear algebra and matrices at the end of the book. 

This chapter gives a brief summary of properties of linear algebraic equation systems, in elementary and partitioned form, and of certain elimination methods for their solution. Gauss-Jordan elimination, Gaussian elimination, LU factorization, and their use on partitioned arrays are described. Some software for computational linear algebra is pointed out, and references for further reading are given. 

The fact that chemical reactions are expressed as linear homogeneous equations allows us to exploit the properties of such equations and to use the associated algebraic tools. Specifically, we use elementary row operations to reduce the stoichiometric matrix to a reduced form, using Gaussian elimination. A reduced matrix is defined as a matrix where all the elements below the diagonal (elements 1,1 2,2 3,3 etc.) are zero. The number of nonzero rows in the reduced matrix indicates the number of independent chemical reactions. (A zero row is defined as a row in which all elements are zero.) The nonzero rows in the reduced matrix represent one set of independent chemical reactions (i.e., stoichiometric relations) for the system. 

Equations 20-16a,b,c,d,e constitute five equations in five unknowns and easily yield to solution, using standard (but tedious) determinant or Gaussian elimination methods from elementary algebra. We could stop here, but we take the solution of Equation 20-16 one step further in order to develop efficient solution techniques. The simplicity seen here suggests that we can rewrite the system shown in Equations 20-16a,b,c,d,e in the matrix or linear algebra form... 

With Gaussian elimination and partial pivoting, we have a method for solving linear systems that either finds a solution or fails under conditions in which no unique solution exists. In this section, we consider at more depth the question of when a linear system possesses a real solution (existence) and if so, whether there is exactly one (uniqueness). These questions are vitally important, for linear algebra is the basis upon which we build algorithms for solving nonlinear equations, ordinary and partial differential equations, and many other tasks. 

The primary tool to solve systems of linear algebraic equations is Gaussian elimination. In MATLAB, this calculation is performed using the backslash operator . The following code demonstrates its use ... 

I.B.I. Compare the number of FLOPS necessary to solve a system of N linear algebraic equations by Gaussian and Gauss-Jordan elimination. Which one requires less work ... 

Set up the system of linear algebraic equations that is solved to obtain the least-squares parameter estimate. Then, solve this system by Gaussian elimination. Provide 95% confidence intervals for each of the model parameters. Do all calculations by hand and show complete results. 

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