Gaussian elimination method

The most important direct solution algorithms used in finite element computations are based on the Gaussian elimination method. 

To describe the basic concept of the Gaussian elimination method we consider the following system of simultaneous algebraic equations... 

The Gaussian elimination method provides a systematic approach for implementation of the described forward reduction and back substitution processes for large systems of algebraic equations. 

SOLUTION ALGORITHMS BASED ON THE GAUSSIAN ELIMINATION METHOD... 

The most frequently used modifications of the basic Gaussian elimination method in finite element analysis are the LU decomposition and frontal solution techniques. 

SOLUTION ALGORITHMS GAUSSIAN ELIMINATION METHOD 205 6.4.2 Frontal solution technique... 

Procedure. Write a program for solving simultaneous equations by the Gaussian elimination method and enter the absorptivity matiix above to solve Eqs. (2-51). Set up and solve the problem resulting from a new set of experimental observations on a new unknown solution leading to the nonhomogeneous veetor b = 0.327,0.810,0.673. ... 

We solve the system (284) by the Gaussian elimination method. To this end, we multiply the last line Anxn-1 + Bnxn = Dn in Eq. (284) by C i and write ... 

Gaussian elimination method is based upon the principle of converting a set of N equations of N unknowns represented by... 

Program PROG 14 uses the Gaussian elimination method to determine the quantities of each of a, [3, y, and 5 and required to meet the blending specification. Answers ... 

THIS PROGRAM SOLVES A SET OF LINEAR EQUATIONS USING THE GAUSSIAN ELIMINATION METHOD. A SEARCH IS MADE IN EACH COLUMN FOR THE LARGEST ELEMENT. 

In the final step DHPCG calculates the NSIM derivatives for the reactants that are being simulated. Since the derivative forms of T37pe (1), (2) and (3) equations are all linear with respect to the CP (i) s, they are solved simultaneously for the CP(iys by the Gaussian elimination method. The partial derivatives used in the evaluation of the derivative form of the Type (1) equation are calculated with the same numerical differentiation formula that is used in the NONLIN module. 

Now some of the manipulations that have been employed to this point in arranging the finite-difference form of this mass conservation equation may seem rather arbitrary, however, there was a good reason for the procedure. If we write out the set of equations (xvii) in matrix form we find that the matrix of coefficients on the left-hand side is tridiagonal, that is, all elements off the three main diagonals are zero. The overall set of equations can then be solved directly by a Gaussian elimination method for the set of /i /v (recalling that /q is fixed). Thus, writing the set out... 

The approximate solution to Equation 9.41 is readily available, since the multipliers for the Gaussian elimination method have been calculated (and retained). Now, then y, the approximate solution of Ay = r, satisfies... 

The direct solvers listed in Table 11.1 solve the assembled set of equations by the Gaussian elimination method [1, 8] and deliver solutions often faster than the iterative solvers for ID and 2D problems. Direct solvers can be used for 3D models if the degrees of freedom is less than 10 . Iterative solvers, on the other hand, are used in models with degrees of freedom above 10 and in the solution of 3D problems, for which the memory requirements of the direcrt solvers are excessive. The readers are directed to Ref. [15] for further details of the solvers listed in Table 11.1. 

The Gaussian elimination method corresponds to the scaling of the matrix coefficients, obtaining an upper diagonal matrix, which is solved by retrosubsti-tution. When the system is very ill conditioned, it is convenient to use Gaussian elimination. 

The set of linear equations (A2.8) can be solved by e.g. the Gaussian elimination method. The procedure for finding a new approximation x will be, for the generalized case,... 

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... 

Use the Gaussian elimination method to solve the system of equations ... 

Solve the linear system using the Gaussian elimination method. 

Figure 8.1 gauss elim M-file for calculating the unknown vector x using the Gaussian elimination method with partial pivoting. 

---