Gaussian elimination technique

If a 33 = 0, we have a i3 = 0, and the function 03 is then a linear combination of the functions 0X and 0 2 and should be omitted in the orthogonalization process, which is here simply accomplished by means of the Gaussian elimination technique developed for solving equation systems. The connection between the matrices a and a may be written in the form ... 

Gaussian elimination technique, 291 Gaussian wave function, 276 Gegen ions, 160... 

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... 

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. 

There are two basic families of solution techniques for linear algebraic equations Direct- and iterative methods. A well known example of direct methods is Gaussian elimination. The simultaneous storage of all coefficients of the set of equations in core memory is required. Iterative methods are based on the repeated application of a relatively simple algorithm leading to eventual convergence after a number of repetitions (iterations). Well known examples are the Jacobi and Gauss-Seidel point-by-point iteration methods. 

Use of the LU factorization technique (demonstrated above) for the calculation of the partial derivatives of the /s with respect to the 0/s and 7 s materially reduces the time required to compute the partial derivatives of the F/s and G/s which appear in the jacobian matrix. Additional speed is also achieved by performing numerical operations on only those elements lying on the principal and two adjacent diagonals of the tridiagonal matrices. The remaining elements are zero at the outset of the gaussian elimination process and are not altered by this process. A summary of the steps of the proposed calcula-tional procedure follows. 

Linear Combination Add multiples of one equation to the others in order to get rid of one variable. This is the basis for Gaussian elimination which is one of the faster techniques to use with a computer. 

Linear equations result naturally when we conduct material and energy balances, but most applications occur when we implement other numerical methods. One of the most basic solutirm techniques for systems such as Equation 9.15 is Gaussian elimination [3,5,9,13,14], which is illustrated using the System of Equations 9.15. 

Although a large number of techniques exist for solving a well-defined set of linear equations (one having a unique solution), the most efficient methods are those based on the method of Gaussian elimination. 

The system of equations Is solved using Gaussian elimination with partial pivoting but since the Newton-Raphson technique requires relatively few iterations to achieve convergence (typically 15 Iterations) this does not require large quantities of CPU time. 

Iterative methods. Since an objective of this book is the development of portable tools, we will not discuss direct solvers. Suffice it to say that such solvers, the most notorious being Gaussian elimination, are well documented in the literature (e g., see Carnahan, Luther, and Wilkes, 1969). We will, by contrast, emphasize iterative techniques, since these require minimal computer resources and allow the greatest flexibility. As we will show, they are also very useful in designing smart and robust algorithms. For reasons that will become obvious, let us rewrite Equation 7-15 in the form... 

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... 

All measurements are accompanied by a certain amount of error, and an estimate of its magnitude is necessary to validate results. The error cannot be eliminated completely, although its magnitude and nature can be characterized. It can also be reduced with improved techniques. In general, errors can be classified as random and systematic. If the same experiment is repeated several times, the individual measurements cluster around the mean value. The differences are due to unknown factors that are stochastic in nature and are termed random errors. They have a Gaussian distribution and equal probability of being above or below the mean. On the other hand, systematic errors tend to bias the measurements in one direction. Systematic error is measured as the deviation from the true value. 

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