Gauss Elimination Method

The most widely used method for solution of simultaneous linear algebraic equations is the Gauss elimination method. This is based on the principle of converting the set of n equations in n unknowns  

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The system thus obtained involves N — n + 1 variables, including t//, related by the same number of equations. Since N — n of these are nonlinear equations because of the aq term, an iteration procedure is needed. One starts from a set of q values obtained for a = 0. The equations then become linear and the Gauss elimination method may thus be used to obtain these starting q values. In a second round, these values are used in the aq term and a new set of q values are obtained by... 

Depending on the space discretisation techniques used, the set of equations to be solved may be different, but for FD- and FV- based methods, the discretisation results in a set of linear or non-linear algebraic equations. These depend on the nature of these partial differential equations and how they are derived. For linear equations, it is well known that a Gauss elimination method can be used as a basic method to solve them. Further details of the Gauss method can be found in [60],... 

GdUSS Elniliation Netliod We will b in our discussion by demonstrating the Gauss elimination method using an example. Consider the following three linear equations with three unknowns Xi, X2, and X3. 

Here, the matrices H and V are symmetric and positive definite matrices, which are each, after suitable permutation of indices, tridiagonal matrices. The matrix S is a non-negative diagonal matrix. Recalling that tridiagonal matrix equations are efficiently solved by the Gauss elimination method, we consider now the Peaceman-Rachford iterative method [27], a particular variant of the lAD methods, which is defined by... 

The matrix inversion J is performed by the Gauss elimination method applied to linear equations. The derivatives d//dy, are calculated from analytical expressions or numerical approximations. The approximation of the derivatives d//dy, with forward differences is... 

Gauss Elimination Method with Partial Pivoting... 

In order to demonstrate the application of the Gauss elimination method, apply the triangularization procedure to obtain the solution of the following set of three equations ... 

The above formulas complete the solution of the equations by the Gauss elimination method by calculating all the unknowns from to x,. The Gauss elimination algorithm requires n /3 multiplications to evaluate the vector jc. 

The second type of matrices used by the Gauss elimination method are unit lower triangular matrices of the form... 

Therefore, the entire Gauss elimination method, which reduces a nonsingular matrix A to an upper triangular matrix U, can be represented by the following series of matrix multiplications ... 

The Gauss elimination method is also very useful in the calculation of determinants of matrices. The elementary operations used in the Gauss method are consistent with the... 

Example 2.1 demonstrates the Gauss elimination method with complete pivoting strategy in solving a set of simultaneous linear algebraic equations and in calculating the determinant of the matrix of coefficients. 

Solving the set of equations by Gauss elimination method T = Gauss (A, c) ... 

GAUSS Solves a set of linear algebraic equations by the Gauss % elimination method. 

Discussion of Results The Gauss elimination method finds the interface temperatures as T, = 129.79°C, Tj = 129.77°C, and = 48.12°C. These values are quite predictable, because the heat transfer coefficient of steam and the heat conductivity of steel are very high. Therefore, the temperatures at steam-pipe interface and pipe-insulation interface are very close to the steam temperature. The main resistance to heat transfer is due to insulation. 

The Gauss-Jordan reduction method is an extension of the Gauss elimination method. It reduces a set of n equations from its canonical form of... 

The Gauss-Jordan reduction method applies the same series of elementary operations that are used by the Gauss elimination method. It applies these operations both below and above the diagonal in order to reduce all the off-diagonal elements of the matrix to zero. In addition, it converts the elements on the diagonal to unity. 

In Sec. 2.5.2, we showed that the Gauss elimination method can be represented in matrix form as... 

We, therefore, conclude that if the Gauss elimination method is extended so that matrix A is postmultiplied by L , at each step of the operation, in addition to being premulliplied by L, the resulting matrix B is similar to A. Tliis operation is called the elementary similarity... 

Apply the Gauss elimination method with complete pivoting to the matrix (A - XT) to evaluate the eigenvectors corresponding to each eigenvalue. Several different possibilities exist when the eigenvalues are real ... 

Solves a set of simultaneous linear algebraic equations that model the heat transfer in a steel pipe using the Gauss Elimination method (Gauss.m). 

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